Systems of Linear Equations
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- Systems of Linear Equations - SLE
- Solution of SLEs
- Number of Solutions of an SLE
- Row Echelon Form of an SLE
- Solution of the Row Echelon Form
- Transforming an SLE to Row Echelon Form
- Solution of a General SLE
- Using Matrices to Solve an SLE
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17 part a
- Exercise 17 part b

SLE with Parameter
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- Row-echelon Form with Parameter
- Number of Solutions of SLE with Parameters I
- Number of Solutions of SLE with Parameters II
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13 Parts a-c
- Exercise 13 Parts d-f
- Exercise 13 Parts g-h

SLE over Zp
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Systems of Linear Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Systems of Linear Equations - SLE","Duration":"8m ","ChapterTopicVideoID":9459,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9459.jpeg","UploadDate":"2017-07-26T08:19:30.9730000","DurationForVideoObject":"PT8M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.440 ","End":"00:06.390","Text":"We\u0027re starting a new topic called solving a system of linear equations."},{"Start":"00:06.390 ","End":"00:11.985","Text":"There is an abbreviation of system of linear equations, SLE."},{"Start":"00:11.985 ","End":"00:15.330","Text":"There\u0027s going to be 8 sections."},{"Start":"00:15.330 ","End":"00:17.100","Text":"This clip is Part 1,"},{"Start":"00:17.100 ","End":"00:18.855","Text":"which is an introduction,"},{"Start":"00:18.855 ","End":"00:20.700","Text":"and then later on,"},{"Start":"00:20.700 ","End":"00:22.740","Text":"there\u0027ll be 7 others."},{"Start":"00:22.740 ","End":"00:25.290","Text":"Here\u0027s a sneak preview of the titles, doesn\u0027t matter."},{"Start":"00:25.290 ","End":"00:30.670","Text":"Let\u0027s just move on to Part 1, that is."},{"Start":"00:30.680 ","End":"00:34.250","Text":"What is a system of linear equations?"},{"Start":"00:34.250 ","End":"00:35.870","Text":"You fully seen them before,"},{"Start":"00:35.870 ","End":"00:37.340","Text":"maybe not under this name."},{"Start":"00:37.340 ","End":"00:39.320","Text":"Let\u0027s start straight away with an example."},{"Start":"00:39.320 ","End":"00:44.060","Text":"A system has a certain number of equations and a certain number of unknowns,"},{"Start":"00:44.060 ","End":"00:46.775","Text":"and we\u0027ll start with 2 equations and 2 unknowns."},{"Start":"00:46.775 ","End":"00:50.450","Text":"Often we put curly braces at the side."},{"Start":"00:50.450 ","End":"00:52.340","Text":"This is what the thing looks like."},{"Start":"00:52.340 ","End":"00:57.650","Text":"Now here, x and y are the variables and they stay as x and y."},{"Start":"00:57.650 ","End":"01:00.140","Text":"But instead of the other letters which are parameters,"},{"Start":"01:00.140 ","End":"01:01.955","Text":"we put actual numbers."},{"Start":"01:01.955 ","End":"01:06.240","Text":"For example, here\u0027s this with a is 4,"},{"Start":"01:06.240 ","End":"01:07.800","Text":"b is 10, and so on,"},{"Start":"01:07.800 ","End":"01:10.640","Text":"and this is an actual system of linear equations with"},{"Start":"01:10.640 ","End":"01:14.780","Text":"2 unknowns and 2 variables, x and y."},{"Start":"01:14.780 ","End":"01:16.520","Text":"It doesn\u0027t have to be x and y,"},{"Start":"01:16.520 ","End":"01:17.690","Text":"but those are the most common."},{"Start":"01:17.690 ","End":"01:21.110","Text":"As I said, we often put curly braces to show that this is"},{"Start":"01:21.110 ","End":"01:25.390","Text":"a system as a whole and not each equation separately."},{"Start":"01:25.390 ","End":"01:29.255","Text":"Here\u0027s another example which doesn\u0027t look quite like this,"},{"Start":"01:29.255 ","End":"01:31.355","Text":"because of course, if 1 of the coefficients,"},{"Start":"01:31.355 ","End":"01:33.455","Text":"say d is 0,"},{"Start":"01:33.455 ","End":"01:38.255","Text":"we just don\u0027t bother writing 0y and we would just write it this way."},{"Start":"01:38.255 ","End":"01:43.250","Text":"Or for example, if x was missing in the first equation,"},{"Start":"01:43.250 ","End":"01:45.860","Text":"then you might get something that looks like this."},{"Start":"01:45.860 ","End":"01:48.800","Text":"In general, I want to make a remark on"},{"Start":"01:48.800 ","End":"01:55.999","Text":"notation that instead of writing the system this way with or without the curly braces,"},{"Start":"01:55.999 ","End":"01:59.045","Text":"we often write it like this,"},{"Start":"01:59.045 ","End":"02:03.860","Text":"usually the letter x because x, y, z, and then what,"},{"Start":"02:03.860 ","End":"02:06.590","Text":"I mean you run out of letters even if you wrap around."},{"Start":"02:06.590 ","End":"02:08.915","Text":"To make it more general,"},{"Start":"02:08.915 ","End":"02:11.940","Text":"outside the 2-by-2 situation,"},{"Start":"02:11.940 ","End":"02:16.730","Text":"we just have subscripts on the same letter x,"},{"Start":"02:16.730 ","End":"02:18.620","Text":"x_1, x_2 instead of z,"},{"Start":"02:18.620 ","End":"02:20.390","Text":"you\u0027d have x_3 and so on."},{"Start":"02:20.390 ","End":"02:23.210","Text":"But to make it even more general,"},{"Start":"02:23.210 ","End":"02:26.149","Text":"so we don\u0027t run out of letters for the parameters,"},{"Start":"02:26.149 ","End":"02:30.035","Text":"the coefficients, we also put subscripts on them,"},{"Start":"02:30.035 ","End":"02:32.300","Text":"only here we need double subscript."},{"Start":"02:32.300 ","End":"02:39.425","Text":"For example, this here means that this is the coefficient of the second row,"},{"Start":"02:39.425 ","End":"02:43.325","Text":"the first variable or x_1 and so on."},{"Start":"02:43.325 ","End":"02:45.050","Text":"First row, second variable,"},{"Start":"02:45.050 ","End":"02:47.710","Text":"first row, first variable."},{"Start":"02:47.710 ","End":"02:52.280","Text":"Then it\u0027s completely general and we can go beyond the 2 by 2."},{"Start":"02:52.280 ","End":"02:55.070","Text":"When I say 2-by-2, I mean 2 equations and 2 unknowns."},{"Start":"02:55.070 ","End":"03:01.010","Text":"Let\u0027s move on. Here we have an example of 3 equations and 3 unknowns."},{"Start":"03:01.010 ","End":"03:03.860","Text":"Again, the curly braces are optional,"},{"Start":"03:03.860 ","End":"03:06.120","Text":"but I actually like them."},{"Start":"03:06.280 ","End":"03:12.439","Text":"It\u0027s fairly common situation to have the same number of equations as unknowns,"},{"Start":"03:12.439 ","End":"03:16.400","Text":"but it\u0027s certainly not the rule or anything,"},{"Start":"03:16.400 ","End":"03:19.100","Text":"it just is a common occurrence."},{"Start":"03:19.100 ","End":"03:24.410","Text":"Here\u0027s an example with actual numbers instead of the coefficients."},{"Start":"03:24.410 ","End":"03:26.810","Text":"Like with the case of x, y, x, y,"},{"Start":"03:26.810 ","End":"03:30.755","Text":"and z stay, they don\u0027t get substituted."},{"Start":"03:30.755 ","End":"03:33.665","Text":"The equation has variables in it."},{"Start":"03:33.665 ","End":"03:35.330","Text":"But the a, b, c,"},{"Start":"03:35.330 ","End":"03:40.035","Text":"and so on up through l are actual numbers as here."},{"Start":"03:40.035 ","End":"03:43.780","Text":"As before, if some of these coefficients are 0,"},{"Start":"03:43.780 ","End":"03:46.660","Text":"we just don\u0027t bother writing in the term like here,"},{"Start":"03:46.660 ","End":"03:50.065","Text":"there\u0027s 0y, so we don\u0027t bother."},{"Start":"03:50.065 ","End":"03:53.755","Text":"Here there\u0027s a missing y also,"},{"Start":"03:53.755 ","End":"03:56.140","Text":"and here there\u0027s a missing z term."},{"Start":"03:56.140 ","End":"04:00.070","Text":"Some of these might be absent, which means that they\u0027re 0."},{"Start":"04:00.070 ","End":"04:04.940","Text":"Now as before, we might want to generalize the notation."},{"Start":"04:06.150 ","End":"04:09.205","Text":"Instead of writing it this way,"},{"Start":"04:09.205 ","End":"04:11.310","Text":"we might go completely general,"},{"Start":"04:11.310 ","End":"04:13.455","Text":"and instead of x, y, z to have x_1,"},{"Start":"04:13.455 ","End":"04:16.330","Text":"x_2, x_3, like first variable, second variable,"},{"Start":"04:16.330 ","End":"04:20.860","Text":"third variable and the coefficients as before,"},{"Start":"04:20.860 ","End":"04:22.320","Text":"say this 1, for example,"},{"Start":"04:22.320 ","End":"04:27.260","Text":"means the coefficient of the third equation,"},{"Start":"04:27.260 ","End":"04:29.870","Text":"second variable, and so on."},{"Start":"04:29.870 ","End":"04:35.105","Text":"On the right-hand side are these constants like here, here and here."},{"Start":"04:35.105 ","End":"04:38.460","Text":"These just called b_1,"},{"Start":"04:38.460 ","End":"04:40.025","Text":"b_2, b_3 and so on,"},{"Start":"04:40.025 ","End":"04:43.050","Text":"depending on how many equations you have."},{"Start":"04:43.430 ","End":"04:48.680","Text":"Another example this time not the same number of equations as unknowns,"},{"Start":"04:48.680 ","End":"04:51.560","Text":"2 equations, 4 unknowns."},{"Start":"04:51.560 ","End":"04:59.930","Text":"I did it in the form where the coefficients have indexes but the variables x, y, z."},{"Start":"04:59.930 ","End":"05:02.045","Text":"Afterwards usually we take t,"},{"Start":"05:02.045 ","End":"05:04.320","Text":"t, u, v, and so on."},{"Start":"05:04.390 ","End":"05:09.495","Text":"But really it\u0027s better in such cases to just use x_1,"},{"Start":"05:09.495 ","End":"05:11.180","Text":"x_2, x_3, x_4,"},{"Start":"05:11.180 ","End":"05:13.670","Text":"and not worry about what letters."},{"Start":"05:13.670 ","End":"05:17.330","Text":"Then an actual example with numbers as you see,"},{"Start":"05:17.330 ","End":"05:21.080","Text":"x, y, z, and t stay from this 1."},{"Start":"05:21.080 ","End":"05:25.730","Text":"But the other constants get actual numbers."},{"Start":"05:25.730 ","End":"05:28.070","Text":"Or if we use the other form,"},{"Start":"05:28.070 ","End":"05:30.470","Text":"this 1, this format here,"},{"Start":"05:30.470 ","End":"05:34.950","Text":"then we have instead of"},{"Start":"05:34.950 ","End":"05:36.465","Text":"x, y, z, t, we have x_1,"},{"Start":"05:36.465 ","End":"05:38.415","Text":"x_2, x_3, x_4."},{"Start":"05:38.415 ","End":"05:42.195","Text":"These are the same as these in principle."},{"Start":"05:42.195 ","End":"05:49.010","Text":"Here\u0027s another example where some of the variables seem to be absent."},{"Start":"05:49.010 ","End":"05:51.215","Text":"It\u0027s because their coefficient is 0."},{"Start":"05:51.215 ","End":"05:58.350","Text":"Here there\u0027s no z and no t. Here there\u0027s no x or y."},{"Start":"05:59.840 ","End":"06:02.460","Text":"Another example."},{"Start":"06:02.460 ","End":"06:05.185","Text":"Sorry, not another example."},{"Start":"06:05.185 ","End":"06:08.750","Text":"I\u0027m just going to generalize what we said above."},{"Start":"06:08.750 ","End":"06:10.700","Text":"We had 2 equations, 2 unknowns, 3 equations,"},{"Start":"06:10.700 ","End":"06:12.860","Text":"3 unknowns, 2 equations, 4 unknowns."},{"Start":"06:12.860 ","End":"06:16.295","Text":"Let\u0027s in general see what in m equations,"},{"Start":"06:16.295 ","End":"06:19.540","Text":"in n unknowns might look like."},{"Start":"06:19.540 ","End":"06:24.275","Text":"Here we are. Lot of dot-dot-dot."},{"Start":"06:24.275 ","End":"06:27.600","Text":"Dot-dot-dot is also called an ellipsis."},{"Start":"06:27.620 ","End":"06:32.480","Text":"Then we really have to use the index form,"},{"Start":"06:32.480 ","End":"06:38.720","Text":"double index on the coefficients a and x from 1 to n,"},{"Start":"06:38.720 ","End":"06:43.150","Text":"n is the number of unknowns and b_1 through b_m,"},{"Start":"06:43.150 ","End":"06:44.600","Text":"the constants on the right-hand side,"},{"Start":"06:44.600 ","End":"06:47.125","Text":"where m is the number of equations,"},{"Start":"06:47.125 ","End":"06:50.590","Text":"we have a double index,"},{"Start":"06:50.590 ","End":"06:53.835","Text":"where the first index say here,"},{"Start":"06:53.835 ","End":"06:59.705","Text":"m means the equation number and the second index is the variable number."},{"Start":"06:59.705 ","End":"07:02.195","Text":"Just like here. Second equation,"},{"Start":"07:02.195 ","End":"07:05.000","Text":"third unknown, and so on."},{"Start":"07:05.000 ","End":"07:09.340","Text":"Now an important word, homogeneous."},{"Start":"07:09.340 ","End":"07:12.660","Text":"If all these bs from 1 through m,"},{"Start":"07:12.660 ","End":"07:16.265","Text":"all the right-hand side constants are all 0,"},{"Start":"07:16.265 ","End":"07:19.300","Text":"the system is called homogeneous."},{"Start":"07:19.300 ","End":"07:22.670","Text":"It turns out to be a very important concept."},{"Start":"07:22.670 ","End":"07:25.855","Text":"I\u0027ll just give an example."},{"Start":"07:25.855 ","End":"07:29.220","Text":"My first example is this,"},{"Start":"07:29.220 ","End":"07:31.800","Text":"notice 0, 0, 0."},{"Start":"07:31.800 ","End":"07:34.345","Text":"This is a homogeneous system."},{"Start":"07:34.345 ","End":"07:39.730","Text":"The curly braces emphasize it\u0027s a system and not each equation separately."},{"Start":"07:39.730 ","End":"07:41.885","Text":"Here\u0027s another example."},{"Start":"07:41.885 ","End":"07:43.250","Text":"This is a 3-by-3,"},{"Start":"07:43.250 ","End":"07:45.005","Text":"this is a 2-by-4."},{"Start":"07:45.005 ","End":"07:46.640","Text":"I say 2-by-4. again,"},{"Start":"07:46.640 ","End":"07:49.980","Text":"I mean 2 equations, 4 unknowns."},{"Start":"07:52.040 ","End":"07:59.010","Text":"I guess that concludes this first clip and the continuation in the following ones."}],"ID":9803},{"Watched":false,"Name":"Solution of SLEs","Duration":"4m 52s","ChapterTopicVideoID":9460,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9460.jpeg","UploadDate":"2017-07-26T08:19:43.8070000","DurationForVideoObject":"PT4M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"This clip comes after the clip about what is a system of linear equations."},{"Start":"00:04.740 ","End":"00:10.890","Text":"In this clip, we want to know what is the solution to a system of linear equations."},{"Start":"00:10.890 ","End":"00:16.260","Text":"It\u0027s intuitive, but here\u0027s a semi-formal definition."},{"Start":"00:16.260 ","End":"00:19.440","Text":"A solution to a linear system or a system of"},{"Start":"00:19.440 ","End":"00:22.410","Text":"linear equations is an assignment of values to"},{"Start":"00:22.410 ","End":"00:24.585","Text":"the variables such that"},{"Start":"00:24.585 ","End":"00:29.100","Text":"all the equations are simultaneously satisfied, all the equations."},{"Start":"00:29.100 ","End":"00:31.470","Text":"Now, this seems a bit abstract,"},{"Start":"00:31.470 ","End":"00:33.450","Text":"let\u0027s just demonstrate it."},{"Start":"00:33.450 ","End":"00:35.190","Text":"Suppose, for example,"},{"Start":"00:35.190 ","End":"00:36.525","Text":"we take the system,"},{"Start":"00:36.525 ","End":"00:42.445","Text":"I say system instead of SLE for short then what is the solution to this?"},{"Start":"00:42.445 ","End":"00:45.710","Text":"Well, it\u0027s an assignment of values to the variables and that"},{"Start":"00:45.710 ","End":"00:49.370","Text":"means that each variable x and y gets assigned a value."},{"Start":"00:49.370 ","End":"00:53.720","Text":"In this case, I claim that x equals 1 and y equals 2."},{"Start":"00:53.720 ","End":"00:58.110","Text":"This assignment satisfies all the equations,"},{"Start":"00:58.110 ","End":"01:00.720","Text":"so we have to check them 1 by 1."},{"Start":"01:00.720 ","End":"01:03.355","Text":"For example, the first one,"},{"Start":"01:03.355 ","End":"01:05.870","Text":"we put x equals 1, y equals 2,"},{"Start":"01:05.870 ","End":"01:10.945","Text":"we get 4 times 1 plus 10 times 2 is indeed 24,"},{"Start":"01:10.945 ","End":"01:16.070","Text":"and 20 times 1 minus 2 is 18."},{"Start":"01:16.130 ","End":"01:18.750","Text":"This is a solution."},{"Start":"01:18.750 ","End":"01:22.260","Text":"For short, we write this as 1, 2,"},{"Start":"01:22.260 ","End":"01:25.530","Text":"where it\u0027s understood that the variables are taken in order,"},{"Start":"01:25.530 ","End":"01:28.605","Text":"so x is 1, y is 2."},{"Start":"01:28.605 ","End":"01:35.600","Text":"Another example, let\u0027s suppose the system is this 3 by 3,"},{"Start":"01:35.600 ","End":"01:38.200","Text":"I\u0027ll put curly braces here,"},{"Start":"01:38.200 ","End":"01:43.040","Text":"then it turns out that this is a solution."},{"Start":"01:43.040 ","End":"01:44.945","Text":"This means of course,"},{"Start":"01:44.945 ","End":"01:47.180","Text":"that x equals 1."},{"Start":"01:47.180 ","End":"01:48.500","Text":"That\u0027s the assignment."},{"Start":"01:48.500 ","End":"01:50.690","Text":"Y equals 0,"},{"Start":"01:50.690 ","End":"01:53.585","Text":"z equals minus 1."},{"Start":"01:53.585 ","End":"01:55.490","Text":"I\u0027ll leave it to you to verify,"},{"Start":"01:55.490 ","End":"01:56.795","Text":"maybe I\u0027ll do just one of them."},{"Start":"01:56.795 ","End":"01:58.805","Text":"X minus y plus z,"},{"Start":"01:58.805 ","End":"02:04.940","Text":"1 minus 0 plus negative 1 is indeed 0, so that\u0027s okay."},{"Start":"02:04.940 ","End":"02:08.135","Text":"All 3 of them have to be satisfied."},{"Start":"02:08.135 ","End":"02:11.870","Text":"Also note that I said a solution and not the solution,"},{"Start":"02:11.870 ","End":"02:14.795","Text":"and we\u0027ll talk more about that later."},{"Start":"02:14.795 ","End":"02:16.610","Text":"Here\u0027s another example."},{"Start":"02:16.610 ","End":"02:18.380","Text":"It\u0027s also a 3 by 3."},{"Start":"02:18.380 ","End":"02:21.720","Text":"I mean 3 equations in 3 unknowns."},{"Start":"02:22.220 ","End":"02:24.240","Text":"I claim that 1,"},{"Start":"02:24.240 ","End":"02:26.084","Text":"2, 4 is a solution."},{"Start":"02:26.084 ","End":"02:29.615","Text":"Don\u0027t worry about the fact that I\u0027m just pulling these solutions out of a hat."},{"Start":"02:29.615 ","End":"02:31.865","Text":"Later we\u0027ll learn how to find them."},{"Start":"02:31.865 ","End":"02:35.855","Text":"Meanwhile, I\u0027m just presenting them to you for verification."},{"Start":"02:35.855 ","End":"02:38.450","Text":"Well, let\u0027s see. I won\u0027t do all 3 of them."},{"Start":"02:38.450 ","End":"02:40.040","Text":"Let\u0027s try say the middle one."},{"Start":"02:40.040 ","End":"02:43.430","Text":"4 times 1, because of course,"},{"Start":"02:43.430 ","End":"02:45.260","Text":"this means that x equals 1,"},{"Start":"02:45.260 ","End":"02:49.110","Text":"y equals 2, z equals 4."},{"Start":"02:49.110 ","End":"02:51.855","Text":"4 times 1 is 4,"},{"Start":"02:51.855 ","End":"02:54.240","Text":"minus 2, we\u0027re down to 2,"},{"Start":"02:54.240 ","End":"02:56.370","Text":"plus twice 4,"},{"Start":"02:56.370 ","End":"02:57.660","Text":"2 plus 8 is 10,"},{"Start":"02:57.660 ","End":"03:01.165","Text":"that\u0027s okay, and similarly for the other two."},{"Start":"03:01.165 ","End":"03:05.855","Text":"Now I did say a solution and not the solution."},{"Start":"03:05.855 ","End":"03:11.720","Text":"Here\u0027s an example of why I do this because there might be more than 1."},{"Start":"03:11.720 ","End":"03:13.880","Text":"In fact, if you check,"},{"Start":"03:13.880 ","End":"03:15.395","Text":"then 3, 2,"},{"Start":"03:15.395 ","End":"03:17.540","Text":"0, meaning x equals 3, y equals 2,"},{"Start":"03:17.540 ","End":"03:21.350","Text":"z equals 0 will also satisfy all 3 equations,"},{"Start":"03:21.350 ","End":"03:23.495","Text":"and so will 5, 2,"},{"Start":"03:23.495 ","End":"03:27.535","Text":"negative 4, and perhaps many others."},{"Start":"03:27.535 ","End":"03:33.640","Text":"In a future clip, we\u0027ll discuss the question of how many solutions can a system have?"},{"Start":"03:33.640 ","End":"03:38.490","Text":"Here\u0027s another example of a system also 3 by 3."},{"Start":"03:38.490 ","End":"03:41.375","Text":"This one is homogeneous."},{"Start":"03:41.375 ","End":"03:42.860","Text":"Note the 0,"},{"Start":"03:42.860 ","End":"03:44.900","Text":"0, 0,"},{"Start":"03:44.900 ","End":"03:47.155","Text":"so it\u0027s homogeneous,"},{"Start":"03:47.155 ","End":"03:50.030","Text":"and for a homogeneous system,"},{"Start":"03:50.030 ","End":"03:55.320","Text":"you can always count on at least 1 solution because 0, 0, 0,"},{"Start":"03:55.320 ","End":"03:57.080","Text":"or however many variables there are,"},{"Start":"03:57.080 ","End":"04:00.665","Text":"all 0, is always a solution to the homogeneous."},{"Start":"04:00.665 ","End":"04:04.025","Text":"I mean, obviously, if I put x and y and z all 0,"},{"Start":"04:04.025 ","End":"04:07.770","Text":"then 2x minus 7y plus 16z,"},{"Start":"04:07.770 ","End":"04:11.915","Text":"you can see right away that if I put the 0, it\u0027ll be 0."},{"Start":"04:11.915 ","End":"04:15.290","Text":"0, 0, 0 is always a solution to the homogeneous."},{"Start":"04:15.290 ","End":"04:17.120","Text":"I\u0027m not saying it\u0027s the only one,"},{"Start":"04:17.120 ","End":"04:18.755","Text":"but it\u0027s certainly a solution."},{"Start":"04:18.755 ","End":"04:23.900","Text":"In fact, there\u0027s a rule that this is the if and only if,"},{"Start":"04:23.900 ","End":"04:25.985","Text":"if the solution is homogeneous,"},{"Start":"04:25.985 ","End":"04:30.920","Text":"then all 0s will be a solution and vice versa."},{"Start":"04:30.920 ","End":"04:34.660","Text":"If all 0 is a solution then the system is going to be homogeneous."},{"Start":"04:34.660 ","End":"04:37.985","Text":"I mean obviously, if I know that all 0 is a solution,"},{"Start":"04:37.985 ","End":"04:39.390","Text":"then if I put 0,"},{"Start":"04:39.390 ","End":"04:41.345","Text":"0, 0 on the left-hand side,"},{"Start":"04:41.345 ","End":"04:45.004","Text":"I have to get 0 on the right-hand side for each of the equations."},{"Start":"04:45.004 ","End":"04:48.845","Text":"That\u0027s a rule that you should know."},{"Start":"04:48.845 ","End":"04:52.320","Text":"I think that\u0027ll be enough for this clip."}],"ID":9804},{"Watched":false,"Name":"Number of Solutions of an SLE","Duration":"3m 54s","ChapterTopicVideoID":9461,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9461.jpeg","UploadDate":"2017-07-26T08:19:54.8370000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this clip, we address the question,"},{"Start":"00:02.400 ","End":"00:05.355","Text":"how many solutions can an SLE have?"},{"Start":"00:05.355 ","End":"00:07.170","Text":"What is the number of solutions?"},{"Start":"00:07.170 ","End":"00:12.240","Text":"Just for reference think of a quadratic equation from high school."},{"Start":"00:12.240 ","End":"00:18.780","Text":"We knew that it could either have 0 solutions 2 solutions or just 1 solution."},{"Start":"00:18.780 ","End":"00:20.550","Text":"What could an SLE have?"},{"Start":"00:20.550 ","End":"00:25.040","Text":"Could it have exactly 2 solutions, 47 solutions."},{"Start":"00:25.040 ","End":"00:29.665","Text":"What? Well, there\u0027s a definite answer to that."},{"Start":"00:29.665 ","End":"00:35.195","Text":"That is that there are only 3 possibilities for the number of solutions."},{"Start":"00:35.195 ","End":"00:37.355","Text":"Here are the 3 possibilities."},{"Start":"00:37.355 ","End":"00:43.580","Text":"It could have just 1 solution and single solution or unique solution,"},{"Start":"00:43.580 ","End":"00:45.005","Text":"or exactly 1 solution."},{"Start":"00:45.005 ","End":"00:47.870","Text":"It also could have an infinite number of solutions"},{"Start":"00:47.870 ","End":"00:50.215","Text":"or infinitely many solutions."},{"Start":"00:50.215 ","End":"00:53.945","Text":"It could be that it has no solutions at all."},{"Start":"00:53.945 ","End":"00:56.555","Text":"But these are the only 3 possibilities."},{"Start":"00:56.555 ","End":"01:00.170","Text":"It can\u0027t have just 10 solutions exactly,"},{"Start":"01:00.170 ","End":"01:01.415","Text":"or something like that."},{"Start":"01:01.415 ","End":"01:03.605","Text":"0, 1 or infinity."},{"Start":"01:03.605 ","End":"01:06.580","Text":"That\u0027s all the possibilities that we have."},{"Start":"01:06.580 ","End":"01:09.080","Text":"I\u0027m going to give an example of each starting"},{"Start":"01:09.080 ","End":"01:13.070","Text":"with a single solution or just 1 solution."},{"Start":"01:13.070 ","End":"01:16.285","Text":"Here is the system."},{"Start":"01:16.285 ","End":"01:22.580","Text":"I claim that it has just 1 solution, that x is 2 and y is 1."},{"Start":"01:22.580 ","End":"01:25.160","Text":"Or if you like that 2, 1 is the solution."},{"Start":"01:25.160 ","End":"01:26.600","Text":"How do I know?"},{"Start":"01:26.600 ","End":"01:29.660","Text":"Well, here it says that y is 1,"},{"Start":"01:29.660 ","End":"01:31.295","Text":"so that settles that."},{"Start":"01:31.295 ","End":"01:32.975","Text":"No other possibility."},{"Start":"01:32.975 ","End":"01:35.935","Text":"If y is 1 and you plug it into here,"},{"Start":"01:35.935 ","End":"01:37.800","Text":"10x plus 1 is 21,"},{"Start":"01:37.800 ","End":"01:39.344","Text":"so 10x is 20,"},{"Start":"01:39.344 ","End":"01:41.040","Text":"so x has to be 2."},{"Start":"01:41.040 ","End":"01:42.020","Text":"This is forced."},{"Start":"01:42.020 ","End":"01:44.575","Text":"There\u0027s no other possibility."},{"Start":"01:44.575 ","End":"01:48.875","Text":"Now give you an example with an infinite number of solutions."},{"Start":"01:48.875 ","End":"01:51.200","Text":"Here\u0027s the example."},{"Start":"01:51.200 ","End":"01:53.240","Text":"If you look at it for a moment,"},{"Start":"01:53.240 ","End":"01:56.330","Text":"you\u0027ll see that the second equation"},{"Start":"01:56.330 ","End":"02:00.395","Text":"is just the first equation multiplied by 10."},{"Start":"02:00.395 ","End":"02:07.000","Text":"Obviously, anything that satisfies the first will also satisfy the second."},{"Start":"02:07.000 ","End":"02:10.489","Text":"In a sense, it\u0027s as if we only had 1 equation."},{"Start":"02:10.489 ","End":"02:13.190","Text":"This is sort of redundant because of course,"},{"Start":"02:13.190 ","End":"02:16.270","Text":"if x plus y equals 4 then 10x plus 10y is 40,"},{"Start":"02:16.270 ","End":"02:19.275","Text":"just multiply both sides by 10."},{"Start":"02:19.275 ","End":"02:22.577","Text":"Now it\u0027s easy to see that it has infinitely many solutions,"},{"Start":"02:22.577 ","End":"02:27.725","Text":"1 example would be to take x equals 2 and y equals 2,"},{"Start":"02:27.725 ","End":"02:29.030","Text":"which I can write like this."},{"Start":"02:29.030 ","End":"02:31.265","Text":"But there are many other possibilities,"},{"Start":"02:31.265 ","End":"02:34.280","Text":"such as x equals 1 and y equals 3."},{"Start":"02:34.280 ","End":"02:37.385","Text":"That will also give me a sum of 4."},{"Start":"02:37.385 ","End":"02:40.560","Text":"I could take x is a 1/2, y is 3 and 1/2."},{"Start":"02:40.560 ","End":"02:46.645","Text":"That will give me that their sum is 4, or even 0, 4 will work."},{"Start":"02:46.645 ","End":"02:50.870","Text":"More generally, I could write it parametrically"},{"Start":"02:50.870 ","End":"02:56.300","Text":"as x is some t and y is 4 minus t."},{"Start":"02:56.300 ","End":"02:58.370","Text":"Whenever I put t as,"},{"Start":"02:58.370 ","End":"03:00.710","Text":"for example, if I put t equals 1,"},{"Start":"03:00.710 ","End":"03:04.105","Text":"then I get this 1 and 4 minus 1 is 3."},{"Start":"03:04.105 ","End":"03:07.115","Text":"But don\u0027t worry about how I got to this parametric form."},{"Start":"03:07.115 ","End":"03:12.230","Text":"Just see if you can understand that any value of t will give me a solution."},{"Start":"03:12.230 ","End":"03:16.465","Text":"T could be 1 of infinite number of possibilities."},{"Start":"03:16.465 ","End":"03:19.190","Text":"There are infinitely many solutions."},{"Start":"03:19.190 ","End":"03:21.350","Text":"Now let\u0027s look at the third case"},{"Start":"03:21.350 ","End":"03:25.540","Text":"where the system has no solutions at all."},{"Start":"03:25.540 ","End":"03:32.240","Text":"Here\u0027s an example, x plus y equals 10 and x plus y equals 20."},{"Start":"03:32.240 ","End":"03:34.850","Text":"I think you can see right away that this is impossible."},{"Start":"03:34.850 ","End":"03:38.270","Text":"I mean, if it were possible then 10 and 20"},{"Start":"03:38.270 ","End":"03:39.650","Text":"are both equal to the same thing,"},{"Start":"03:39.650 ","End":"03:40.970","Text":"there\u0027d be equal to each other,"},{"Start":"03:40.970 ","End":"03:43.130","Text":"but 10 is not equal to 20."},{"Start":"03:43.130 ","End":"03:46.040","Text":"There is no combination of x and y."},{"Start":"03:46.040 ","End":"03:49.099","Text":"The sum is 10, the same time 20."},{"Start":"03:49.099 ","End":"03:52.475","Text":"Here\u0027s an example of no solution."},{"Start":"03:52.475 ","End":"03:55.140","Text":"That concludes this clip."}],"ID":9805},{"Watched":false,"Name":"Row Echelon Form of an SLE","Duration":"3m 2s","ChapterTopicVideoID":9462,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9462.jpeg","UploadDate":"2017-07-26T08:20:04.2100000","DurationForVideoObject":"PT3M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this clip, I\u0027m going to talk about a very important concept,"},{"Start":"00:03.870 ","End":"00:09.015","Text":"the row-echelon form of a system of linear equations."},{"Start":"00:09.015 ","End":"00:15.450","Text":"This will be the first and giant step towards solving the system of linear equations,"},{"Start":"00:15.450 ","End":"00:17.310","Text":"which is actually our goal."},{"Start":"00:17.310 ","End":"00:19.380","Text":"Let me give a definition."},{"Start":"00:19.380 ","End":"00:21.930","Text":"This definition will be clearer as soon as I\u0027ve given you"},{"Start":"00:21.930 ","End":"00:25.770","Text":"an example but the system is in row-echelon form,"},{"Start":"00:25.770 ","End":"00:28.635","Text":"means that the leading coefficient in each row is"},{"Start":"00:28.635 ","End":"00:33.975","Text":"further to the right of the leading coefficient in the row above it."},{"Start":"00:33.975 ","End":"00:37.310","Text":"For example, this system,"},{"Start":"00:37.310 ","End":"00:40.805","Text":"I should explain what the leading coefficient means."},{"Start":"00:40.805 ","End":"00:43.415","Text":"It\u0027s the 1 leftmost."},{"Start":"00:43.415 ","End":"00:45.095","Text":"This is the leading coefficient."},{"Start":"00:45.095 ","End":"00:46.895","Text":"This is the leading coefficient,"},{"Start":"00:46.895 ","End":"00:49.860","Text":"and this is the leading coefficient."},{"Start":"00:50.210 ","End":"00:55.730","Text":"Notice that this 2 is further to the right than the 1 in the row above it,"},{"Start":"00:55.730 ","End":"00:58.235","Text":"and the 25 is further to the right than the 2."},{"Start":"00:58.235 ","End":"01:01.780","Text":"I guess this doesn\u0027t apply to the top row of course."},{"Start":"01:01.780 ","End":"01:05.300","Text":"The top row, each 1 leading coefficient has to get"},{"Start":"01:05.300 ","End":"01:08.935","Text":"further and further to the right as we go further down, basically."},{"Start":"01:08.935 ","End":"01:10.520","Text":"It could jump more than 1,"},{"Start":"01:10.520 ","End":"01:12.815","Text":"but it has to jump properly."},{"Start":"01:12.815 ","End":"01:14.855","Text":"Here\u0027s another example."},{"Start":"01:14.855 ","End":"01:18.260","Text":"Although the leading coefficient are not quite clear."},{"Start":"01:18.260 ","End":"01:21.350","Text":"They are clear but they\u0027re not written explicitly."},{"Start":"01:21.350 ","End":"01:23.540","Text":"This will be minus 1 here,"},{"Start":"01:23.540 ","End":"01:26.615","Text":"would be 1 here and 1 here."},{"Start":"01:26.615 ","End":"01:28.805","Text":"But in both cases,"},{"Start":"01:28.805 ","End":"01:33.620","Text":"you notice this staircase form."},{"Start":"01:33.620 ","End":"01:36.710","Text":"It keeps getting further to the right."},{"Start":"01:36.710 ","End":"01:42.275","Text":"I think the word echelon is some variation of the word for steps of stairs."},{"Start":"01:42.275 ","End":"01:45.820","Text":"I believe so. Anyway, this is what it is."},{"Start":"01:45.820 ","End":"01:49.565","Text":"Whoops, I forgot the last row. Silly me."},{"Start":"01:49.565 ","End":"01:51.890","Text":"But there it is. You see here the coefficient,"},{"Start":"01:51.890 ","End":"01:53.845","Text":"of course is visible, it\u0027s 4."},{"Start":"01:53.845 ","End":"01:59.960","Text":"This staircase thing going further to the right as we go down is called row echelon form."},{"Start":"01:59.960 ","End":"02:02.345","Text":"There is something called the column echelon form."},{"Start":"02:02.345 ","End":"02:04.640","Text":"But that\u0027s why there\u0027s the word row,"},{"Start":"02:04.640 ","End":"02:06.800","Text":"but we\u0027re just going to deal with row."},{"Start":"02:06.800 ","End":"02:10.170","Text":"We\u0027re not going to talk about the other kind."},{"Start":"02:10.870 ","End":"02:14.030","Text":"Here is yet another example."},{"Start":"02:14.030 ","End":"02:20.435","Text":"This time set of names of variables just indexed x1 through x5."},{"Start":"02:20.435 ","End":"02:24.830","Text":"But I gave this example so you can see that it could jump more than 1."},{"Start":"02:24.830 ","End":"02:29.570","Text":"It can jump all the way from X1 to X3 and X4 and so on."},{"Start":"02:29.570 ","End":"02:33.905","Text":"But we still get this staircase form."},{"Start":"02:33.905 ","End":"02:40.070","Text":"To conclude, I should give you an example of something that\u0027s not in row echelon form."},{"Start":"02:40.070 ","End":"02:42.500","Text":"I think you can see why this is not,"},{"Start":"02:42.500 ","End":"02:46.460","Text":"because the leading coefficient in the second row is"},{"Start":"02:46.460 ","End":"02:50.270","Text":"not to the right of the leading coefficient in the row above it."},{"Start":"02:50.270 ","End":"02:53.050","Text":"It\u0027s on the same level."},{"Start":"02:53.050 ","End":"02:57.130","Text":"This is not in row echelon form."},{"Start":"02:57.130 ","End":"03:00.230","Text":"I\u0027ll say more about row echelon form in the following clips."},{"Start":"03:00.230 ","End":"03:02.520","Text":"But meanwhile, we\u0027re done."}],"ID":9806},{"Watched":false,"Name":"Solution of the Row Echelon Form","Duration":"2m 23s","ChapterTopicVideoID":9463,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9463.jpeg","UploadDate":"2017-07-26T08:20:12.3070000","DurationForVideoObject":"PT2M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"Well, we\u0027re continuing with row-echelon form."},{"Start":"00:03.825 ","End":"00:08.610","Text":"In this clip, I\u0027m going to show you how to solve the system of equations"},{"Start":"00:08.610 ","End":"00:11.955","Text":"if it\u0027s already in row-echelon form."},{"Start":"00:11.955 ","End":"00:17.760","Text":"Later on we\u0027ll discuss how to bring it into row echelon form in a future clip."},{"Start":"00:17.760 ","End":"00:24.035","Text":"The technique we\u0027re going to use to solve such a system is called back substitution."},{"Start":"00:24.035 ","End":"00:27.750","Text":"When you see it, you\u0027ll see why it\u0027s called this."},{"Start":"00:27.750 ","End":"00:29.640","Text":"We\u0027ll start with an example."},{"Start":"00:29.640 ","End":"00:36.045","Text":"Notice that it really is in echelon form, the staircase form."},{"Start":"00:36.045 ","End":"00:42.800","Text":"What we do is we work our way backwards up the staircase from the bottom to the top."},{"Start":"00:42.800 ","End":"00:47.220","Text":"From this equation, we get that z equals 4"},{"Start":"00:47.220 ","End":"00:52.310","Text":"because you just divide this by 25 and we get z is 100 over 25, which is 4."},{"Start":"00:52.310 ","End":"00:56.095","Text":"Now that we have z, we plug it into here."},{"Start":"00:56.095 ","End":"00:58.470","Text":"If we do that, 2y plus 4 is 8,"},{"Start":"00:58.470 ","End":"01:02.355","Text":"2y is 4, and so y equals 2."},{"Start":"01:02.355 ","End":"01:04.940","Text":"Now that we have z and y,"},{"Start":"01:04.940 ","End":"01:07.375","Text":"we can plug them both in here."},{"Start":"01:07.375 ","End":"01:12.035","Text":"If you put y equals 2 and z equals 4 and do the computation,"},{"Start":"01:12.035 ","End":"01:15.660","Text":"you will see that x comes out to be 1."},{"Start":"01:17.630 ","End":"01:20.570","Text":"In this case, we found the solution"},{"Start":"01:20.570 ","End":"01:23.285","Text":"and there\u0027s only 1 solution because everything was forced,"},{"Start":"01:23.285 ","End":"01:27.095","Text":"and it\u0027s 1, 2, 4, x, y, and z."},{"Start":"01:27.095 ","End":"01:32.400","Text":"Another example, first note the echelon form."},{"Start":"01:33.620 ","End":"01:40.220","Text":"We\u0027re going to use the same technique again to work our way from the bottom to the top."},{"Start":"01:40.220 ","End":"01:41.570","Text":"That\u0027s the back substitution."},{"Start":"01:41.570 ","End":"01:44.000","Text":"You work our way up the staircase this way."},{"Start":"01:44.000 ","End":"01:46.760","Text":"From here, we get 2x_4 is 4,"},{"Start":"01:46.760 ","End":"01:50.380","Text":"so x_4 is 2, substitute it here."},{"Start":"01:50.380 ","End":"01:51.990","Text":"Put x_4 is 2,"},{"Start":"01:51.990 ","End":"01:54.195","Text":"so x_3 is 0."},{"Start":"01:54.195 ","End":"01:56.760","Text":"I\u0027ll just hurry through this."},{"Start":"01:56.760 ","End":"01:58.680","Text":"X_3 and x_4 we have,"},{"Start":"01:58.680 ","End":"02:01.305","Text":"we put them in here, and that gives us x_2."},{"Start":"02:01.305 ","End":"02:03.500","Text":"Now we have x_3, x_4, and x_2."},{"Start":"02:03.500 ","End":"02:05.869","Text":"Put all of those in the first equation,"},{"Start":"02:05.869 ","End":"02:07.780","Text":"and then we get x_1."},{"Start":"02:07.780 ","End":"02:09.840","Text":"We\u0027ve got all 4 of them."},{"Start":"02:09.840 ","End":"02:12.300","Text":"I\u0027ll just collect them into brackets,"},{"Start":"02:12.300 ","End":"02:14.810","Text":"so we have x_1, x_2, x_3, x_4."},{"Start":"02:14.810 ","End":"02:16.160","Text":"This is 10, 1, 0, 2."},{"Start":"02:16.160 ","End":"02:17.330","Text":"That\u0027s the solution."},{"Start":"02:17.330 ","End":"02:19.620","Text":"That\u0027s how easy it is."},{"Start":"02:20.230 ","End":"02:23.340","Text":"That\u0027s it for this clip."}],"ID":9807},{"Watched":false,"Name":"Transforming an SLE to Row Echelon Form","Duration":"8m 52s","ChapterTopicVideoID":9464,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9464.jpeg","UploadDate":"2017-07-26T08:20:48.2830000","DurationForVideoObject":"PT8M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In the previous clip, we saw how easy it is to"},{"Start":"00:03.480 ","End":"00:07.725","Text":"solve a system which is already in row-echelon form."},{"Start":"00:07.725 ","End":"00:10.905","Text":"The question is, can we somehow convert"},{"Start":"00:10.905 ","End":"00:17.490","Text":"a given system of linear equations in to this form without changing the solutions, if any?"},{"Start":"00:17.490 ","End":"00:19.305","Text":"The answer is yes,"},{"Start":"00:19.305 ","End":"00:24.820","Text":"and I\u0027m going to show you the method of how to do it in this clip."},{"Start":"00:24.820 ","End":"00:28.880","Text":"Yes, I want to have a few words written about this that such a system"},{"Start":"00:28.880 ","End":"00:33.410","Text":"can be converted to an equivalent system,"},{"Start":"00:33.410 ","End":"00:40.090","Text":"which is in row-echelon form and we could do this by a series of simple operations,"},{"Start":"00:40.090 ","End":"00:42.830","Text":"and this will all be explained and each"},{"Start":"00:42.830 ","End":"00:45.740","Text":"of which doesn\u0027t affect the solution set of the system."},{"Start":"00:45.740 ","End":"00:47.870","Text":"Why do I say solution set?"},{"Start":"00:47.870 ","End":"00:49.910","Text":"Because solution there may not be 1."},{"Start":"00:49.910 ","End":"00:51.080","Text":"It could be an empty set,"},{"Start":"00:51.080 ","End":"00:52.265","Text":"could be an infinite set,"},{"Start":"00:52.265 ","End":"00:53.600","Text":"could be 01 or infinity,"},{"Start":"00:53.600 ","End":"00:55.685","Text":"so the set of solutions."},{"Start":"00:55.685 ","End":"00:59.280","Text":"In other words, each operation doesn\u0027t change the solutions,"},{"Start":"00:59.280 ","End":"01:05.015","Text":"so we get an equivalent system which has the same solutions as the original system."},{"Start":"01:05.015 ","End":"01:09.950","Text":"Now these are simple operations are divided into 3 kinds"},{"Start":"01:09.950 ","End":"01:14.975","Text":"and they\u0027re known as elementary operations."},{"Start":"01:14.975 ","End":"01:17.540","Text":"But really elementary row operations"},{"Start":"01:17.540 ","End":"01:20.390","Text":"because there is such a thing as a column operation also,"},{"Start":"01:20.390 ","End":"01:23.345","Text":"but usually I\u0027ll omit the word row."},{"Start":"01:23.345 ","End":"01:25.820","Text":"There were no particular order,"},{"Start":"01:25.820 ","End":"01:27.710","Text":"1, 2, or 3,"},{"Start":"01:27.710 ","End":"01:32.345","Text":"so I\u0027ll just choose the orders that\u0027s called number 1,"},{"Start":"01:32.345 ","End":"01:34.490","Text":"the interchanging of 2 equations,"},{"Start":"01:34.490 ","End":"01:38.405","Text":"swapping them around and let me give you an example."},{"Start":"01:38.405 ","End":"01:41.420","Text":"We started off with this system."},{"Start":"01:41.420 ","End":"01:43.730","Text":"Then I might want to say, okay,"},{"Start":"01:43.730 ","End":"01:47.630","Text":"interchange or swap Row 1 and Row 2,"},{"Start":"01:47.630 ","End":"01:49.940","Text":"which means that this 1 goes to the first place,"},{"Start":"01:49.940 ","End":"01:53.450","Text":"and this 1 goes to second place and we get this."},{"Start":"01:53.450 ","End":"01:56.015","Text":"There\u0027s actually a shorthand notation."},{"Start":"01:56.015 ","End":"01:57.515","Text":"Instead of writing in words,"},{"Start":"01:57.515 ","End":"01:59.930","Text":"interchange Row 1 with Row 2,"},{"Start":"01:59.930 ","End":"02:03.575","Text":"we just write R_1 and R_2 with a double arrow."},{"Start":"02:03.575 ","End":"02:06.415","Text":"That means that we interchange them which will swap"},{"Start":"02:06.415 ","End":"02:11.695","Text":"their locations or however you want to express that, that\u0027s 1 kind."},{"Start":"02:11.695 ","End":"02:19.040","Text":"Another kind is to take an equation and multiply it by some number, scalar."},{"Start":"02:19.040 ","End":"02:22.130","Text":"Scalar as opposed to matrix or vector or whatever."},{"Start":"02:22.130 ","End":"02:25.220","Text":"But it mustn\u0027t be 0, I can\u0027t multiply by 0."},{"Start":"02:25.220 ","End":"02:26.495","Text":"That\u0027s no good for us."},{"Start":"02:26.495 ","End":"02:34.070","Text":"So an example would be to take this system and I\u0027m actually going to do 2 in 1."},{"Start":"02:34.070 ","End":"02:39.470","Text":"I\u0027m going to multiply the second row by 2,"},{"Start":"02:39.470 ","End":"02:43.735","Text":"and I\u0027m going to take half of the last row."},{"Start":"02:43.735 ","End":"02:46.860","Text":"It\u0027s really 2 operations."},{"Start":"02:46.860 ","End":"02:49.800","Text":"If we double this, we get this."},{"Start":"02:49.800 ","End":"02:51.240","Text":"If we have this,"},{"Start":"02:51.240 ","End":"02:53.340","Text":"then we get this."},{"Start":"02:53.340 ","End":"03:02.555","Text":"The shorthand notation for that says twice Row 2 goes into what was Row 2,"},{"Start":"03:02.555 ","End":"03:07.060","Text":"and half of Row 3,"},{"Start":"03:07.060 ","End":"03:11.860","Text":"goes to the new Row 3, and that\u0027s here."},{"Start":"03:12.110 ","End":"03:14.595","Text":"It\u0027s 2 out of 3."},{"Start":"03:14.595 ","End":"03:23.765","Text":"The third operation is to add a multiple of 1 equation to another equation."},{"Start":"03:23.765 ","End":"03:25.279","Text":"Could also be subtract,"},{"Start":"03:25.279 ","End":"03:31.115","Text":"although you could also multiply by minus 1 and add."},{"Start":"03:31.115 ","End":"03:34.749","Text":"I do subtract to or from another."},{"Start":"03:34.749 ","End":"03:37.070","Text":"Let\u0027s say this is our system,"},{"Start":"03:37.070 ","End":"03:40.100","Text":"and this is Row 1 and this is Row 2,"},{"Start":"03:40.100 ","End":"03:46.410","Text":"and let\u0027s say I want to subtract twice this row from this row."},{"Start":"03:46.410 ","End":"03:52.545","Text":"So I would say Row 2 minus twice Row 1,"},{"Start":"03:52.545 ","End":"03:56.655","Text":"the result of this goes into Row 2."},{"Start":"03:56.655 ","End":"03:58.785","Text":"Row 1 stays as is,"},{"Start":"03:58.785 ","End":"04:02.820","Text":"and then I take this minus twice this."},{"Start":"04:02.820 ","End":"04:05.955","Text":"So 4x minus twice 2x is nothing."},{"Start":"04:05.955 ","End":"04:08.685","Text":"10y minus twice 4y,"},{"Start":"04:08.685 ","End":"04:10.650","Text":"10 minus 8 is 2."},{"Start":"04:10.650 ","End":"04:13.950","Text":"20 minus twice 6 is 8,"},{"Start":"04:13.950 ","End":"04:17.055","Text":"and this is what we get."},{"Start":"04:17.055 ","End":"04:23.020","Text":"Notice that this just happens to be in row-echelon form."},{"Start":"04:23.090 ","End":"04:25.860","Text":"Here\u0027s the staircase."},{"Start":"04:25.860 ","End":"04:28.600","Text":"In fact, that\u0027s why I did this operation."},{"Start":"04:28.600 ","End":"04:35.730","Text":"You are starting to see how I might convert to echelon form,"},{"Start":"04:35.730 ","End":"04:38.480","Text":"but usually it takes a whole series of operations,"},{"Start":"04:38.480 ","End":"04:40.740","Text":"1, 2, and 3."},{"Start":"04:41.590 ","End":"04:44.600","Text":"That\u0027s basically what I wanted to say,"},{"Start":"04:44.600 ","End":"04:47.810","Text":"except that it\u0027s so close to the end that why don\u0027t we just go"},{"Start":"04:47.810 ","End":"04:51.965","Text":"ahead and solve this already."},{"Start":"04:51.965 ","End":"04:55.950","Text":"That is 2y is 8,"},{"Start":"04:55.950 ","End":"04:57.780","Text":"so divide that by 2,"},{"Start":"04:57.780 ","End":"05:02.205","Text":"we get y as 4, plug y equals 4 into here."},{"Start":"05:02.205 ","End":"05:05.190","Text":"2x plus 16 is 6,"},{"Start":"05:05.190 ","End":"05:06.570","Text":"2x is minus 10,"},{"Start":"05:06.570 ","End":"05:08.805","Text":"x is minus 5."},{"Start":"05:08.805 ","End":"05:18.995","Text":"That\u0027s the introduction to"},{"Start":"05:18.995 ","End":"05:22.260","Text":"how to get into echelon form."},{"Start":"05:23.060 ","End":"05:32.730","Text":"Let me give you another example of operation number 3."},{"Start":"05:34.880 ","End":"05:40.105","Text":"Here we are and I\u0027m going to use the same trick"},{"Start":"05:40.105 ","End":"05:46.385","Text":"as here to get it into echelon form in 1 operation."},{"Start":"05:46.385 ","End":"05:50.280","Text":"We\u0027re going to add Row 1 to Row 2,"},{"Start":"05:50.280 ","End":"05:53.550","Text":"and the result stays in Row 2."},{"Start":"05:53.550 ","End":"05:55.685","Text":"This 1 stays as is,"},{"Start":"05:55.685 ","End":"05:57.170","Text":"and then 4 plus,"},{"Start":"05:57.170 ","End":"05:59.915","Text":"minus 4 is 0, so there\u0027s nothing there."},{"Start":"05:59.915 ","End":"06:01.400","Text":"4 and 10 is 14,"},{"Start":"06:01.400 ","End":"06:02.660","Text":"4 and 10 is 14."},{"Start":"06:02.660 ","End":"06:07.780","Text":"Of course, when we get it to this form, we want to solve it."},{"Start":"06:07.780 ","End":"06:10.360","Text":"Might as well, it\u0027s so easy."},{"Start":"06:10.360 ","End":"06:13.265","Text":"Trick here was to show you how to get it into echelon form."},{"Start":"06:13.265 ","End":"06:16.265","Text":"But once we get it there, you see how easily the solution comes out."},{"Start":"06:16.265 ","End":"06:17.930","Text":"From here, we get y as 1,"},{"Start":"06:17.930 ","End":"06:21.625","Text":"plug it in here and we see that x is 0."},{"Start":"06:21.625 ","End":"06:23.720","Text":"These are the row operations."},{"Start":"06:23.720 ","End":"06:28.045","Text":"Now I\u0027ll do 1 big example of a 3 by 3."},{"Start":"06:28.045 ","End":"06:32.620","Text":"Here it is, our 3 equations, 3 unknowns example."},{"Start":"06:32.620 ","End":"06:35.775","Text":"We want to bring it into row-echelon form."},{"Start":"06:35.775 ","End":"06:41.640","Text":"What we want to do is we want to make it 0 below this 2."},{"Start":"06:41.640 ","End":"06:44.865","Text":"We want 0 out the rest of the column."},{"Start":"06:44.865 ","End":"06:49.160","Text":"What we\u0027ll do is we can subtract the first row from"},{"Start":"06:49.160 ","End":"06:54.035","Text":"the second row and if we add twice the first row to the last row,"},{"Start":"06:54.035 ","End":"06:56.405","Text":"that should give us 0s here."},{"Start":"06:56.405 ","End":"07:00.050","Text":"Here\u0027s what I just said in notation form,"},{"Start":"07:00.050 ","End":"07:04.640","Text":"in row notation and this is the results."},{"Start":"07:04.640 ","End":"07:08.010","Text":"I did 2 basic operations."},{"Start":"07:08.470 ","End":"07:13.169","Text":"This from this, 2 minus 2 is 0,"},{"Start":"07:13.169 ","End":"07:15.360","Text":"5 minus 4 is 1, that\u0027s the y."},{"Start":"07:15.360 ","End":"07:19.025","Text":"Minus 1 minus minus 2 is 1, 10 minus minus 1."},{"Start":"07:19.025 ","End":"07:21.215","Text":"You can check the last 1 also,"},{"Start":"07:21.215 ","End":"07:23.470","Text":"and that gives us this."},{"Start":"07:23.470 ","End":"07:26.720","Text":"Now we\u0027re starting with the echelon form,"},{"Start":"07:26.720 ","End":"07:29.195","Text":"but what bothers us now is this 2y,"},{"Start":"07:29.195 ","End":"07:30.710","Text":"want to get rid of that."},{"Start":"07:30.710 ","End":"07:34.210","Text":"The way to do that would be to add."},{"Start":"07:34.210 ","End":"07:39.590","Text":"No, sorry, to subtract twice this row from this row."},{"Start":"07:39.590 ","End":"07:41.375","Text":"In notation I write,"},{"Start":"07:41.375 ","End":"07:44.645","Text":"take Row 3 and subtract twice Row 2,"},{"Start":"07:44.645 ","End":"07:47.880","Text":"and that becomes the new Row 3."},{"Start":"07:49.040 ","End":"07:51.650","Text":"What we get is, well,"},{"Start":"07:51.650 ","End":"07:54.185","Text":"you can see this minus twice this is nothing."},{"Start":"07:54.185 ","End":"07:55.670","Text":"This minus twice this is 2,"},{"Start":"07:55.670 ","End":"07:57.125","Text":"this minus twice this is 18."},{"Start":"07:57.125 ","End":"08:03.320","Text":"Now, we already have it in echelon form."},{"Start":"08:03.320 ","End":"08:07.100","Text":"But I\u0027ll do 1 more step for aesthetics often when we see we can"},{"Start":"08:07.100 ","End":"08:11.580","Text":"divide a row by a common factor."},{"Start":"08:11.580 ","End":"08:14.870","Text":"Say 2, we can divide by 2 or if you like,"},{"Start":"08:14.870 ","End":"08:16.850","Text":"multiply by a half."},{"Start":"08:16.850 ","End":"08:19.190","Text":"So half of Row 2."},{"Start":"08:19.190 ","End":"08:24.030","Text":"Sorry, a half of Row 3 into Row 3,"},{"Start":"08:24.030 ","End":"08:25.710","Text":"that gives us z equals 9."},{"Start":"08:25.710 ","End":"08:30.025","Text":"This already is in row-echelon form."},{"Start":"08:30.025 ","End":"08:33.650","Text":"We\u0027re done, but once you get it to this form,"},{"Start":"08:33.650 ","End":"08:36.425","Text":"you really want to go the extra step and compute it."},{"Start":"08:36.425 ","End":"08:38.600","Text":"I already have z, put z in here,"},{"Start":"08:38.600 ","End":"08:39.995","Text":"we get y is minus 8."},{"Start":"08:39.995 ","End":"08:46.735","Text":"In short, this is the solution because the echelon form is so easy to solve."},{"Start":"08:46.735 ","End":"08:51.390","Text":"That\u0027s it for this clip. In the next clip we put it all together."}],"ID":9808},{"Watched":false,"Name":"Solution of a General SLE","Duration":"1m 19s","ChapterTopicVideoID":9465,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9465.jpeg","UploadDate":"2017-07-26T08:20:55.5830000","DurationForVideoObject":"PT1M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this clip, I\u0027m just going to put together what we did in"},{"Start":"00:02.970 ","End":"00:05.460","Text":"the last couple of clips and this will"},{"Start":"00:05.460 ","End":"00:10.860","Text":"show us how to actually solve an SLE in 2 steps."},{"Start":"00:10.860 ","End":"00:15.075","Text":"The method is due to the mathematician Gauss,"},{"Start":"00:15.075 ","End":"00:18.705","Text":"and it\u0027s called the Gaussian elimination."},{"Start":"00:18.705 ","End":"00:22.815","Text":"The first step which we learnt is how to bring"},{"Start":"00:22.815 ","End":"00:27.825","Text":"the system to row-echelon form"},{"Start":"00:27.825 ","End":"00:32.430","Text":"and then step 2 is to use back substitution on the echelon form."},{"Start":"00:32.430 ","End":"00:36.300","Text":"We\u0027ve learned both of them."},{"Start":"00:36.300 ","End":"00:38.520","Text":"I\u0027ll just as an example,"},{"Start":"00:38.520 ","End":"00:42.285","Text":"use the example from the previous clip."},{"Start":"00:42.285 ","End":"00:48.065","Text":"As I said, I just copied it in its entirety from the previous."},{"Start":"00:48.065 ","End":"00:54.530","Text":"We had this system and we brought it into row-echelon form."},{"Start":"00:54.530 ","End":"01:03.745","Text":"In fact, we even went the last step and solved this echelon form and we got the solution."},{"Start":"01:03.745 ","End":"01:05.540","Text":"Nothing really new here,"},{"Start":"01:05.540 ","End":"01:09.905","Text":"but just putting it altogether is to realize that there are these 2 main steps."},{"Start":"01:09.905 ","End":"01:13.640","Text":"First we do row-echelon and then use back substitution on"},{"Start":"01:13.640 ","End":"01:20.220","Text":"the echelon form and that\u0027s all there is to it. We\u0027re done."}],"ID":9809},{"Watched":false,"Name":"Using Matrices to Solve an SLE","Duration":"6m 22s","ChapterTopicVideoID":9466,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9466.jpeg","UploadDate":"2017-07-26T08:21:27.0630000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this clip, I\u0027m going to introduce"},{"Start":"00:02.250 ","End":"00:06.990","Text":"very informally the mathematical concept of a matrix."},{"Start":"00:06.990 ","End":"00:13.665","Text":"Mainly in regard to how we can use it to solve systems of linear equations."},{"Start":"00:13.665 ","End":"00:17.985","Text":"To that goal, I\u0027ll discuss what a matrix is,"},{"Start":"00:17.985 ","End":"00:20.085","Text":"and I\u0027ll write a few words first,"},{"Start":"00:20.085 ","End":"00:21.690","Text":"then I\u0027ll leave it for you to read that."},{"Start":"00:21.690 ","End":"00:29.910","Text":"Basically, it makes converting to echelon form a lot nicer and easier."},{"Start":"00:29.910 ","End":"00:34.015","Text":"Let\u0027s start with an example."},{"Start":"00:34.015 ","End":"00:37.855","Text":"Here we have the system of linear equations."},{"Start":"00:37.855 ","End":"00:40.990","Text":"This is what the matrix looks like."},{"Start":"00:40.990 ","End":"00:44.990","Text":"What we do is we just take the coefficients from here,"},{"Start":"00:44.990 ","End":"00:48.320","Text":"and stick them in these brackets."},{"Start":"00:48.320 ","End":"00:52.775","Text":"Because the process of converting to row echelon form,"},{"Start":"00:52.775 ","End":"00:56.090","Text":"we just keep repeating the x, x, x, y, y, y,"},{"Start":"00:56.090 ","End":"00:58.625","Text":"but we\u0027re really only working on the coefficients."},{"Start":"00:58.625 ","End":"01:02.570","Text":"What\u0027s the point of constantly repeating x, y, z?"},{"Start":"01:02.570 ","End":"01:11.915","Text":"Now, often we write a vertical bar here because in some sense,"},{"Start":"01:11.915 ","End":"01:14.905","Text":"these coefficients on the left-hand side of the equals,"},{"Start":"01:14.905 ","End":"01:18.740","Text":"are not quite the same as the 1s on the right."},{"Start":"01:18.740 ","End":"01:22.140","Text":"We do sometimes 1 to make the distinction."},{"Start":"01:22.210 ","End":"01:24.530","Text":"That\u0027s often what you\u0027ll see,"},{"Start":"01:24.530 ","End":"01:26.965","Text":"is the vertical line here."},{"Start":"01:26.965 ","End":"01:29.240","Text":"I wrote it again here with the BOD,"},{"Start":"01:29.240 ","End":"01:34.475","Text":"is actually a name for the matrix which is just this bit."},{"Start":"01:34.475 ","End":"01:39.550","Text":"The 3 columns here is called the restricted,"},{"Start":"01:39.550 ","End":"01:44.320","Text":"the matrix stripe that restricted,"},{"Start":"01:44.320 ","End":"01:46.610","Text":"or sometimes called the coefficient matrix,"},{"Start":"01:46.610 ","End":"01:48.185","Text":"because you use the coefficients."},{"Start":"01:48.185 ","End":"01:50.270","Text":"If we take the whole thing,"},{"Start":"01:50.270 ","End":"01:53.480","Text":"then it\u0027s called the augmented matrix."},{"Start":"01:53.480 ","End":"01:55.445","Text":"Augmented means increased."},{"Start":"01:55.445 ","End":"01:59.220","Text":"But you don\u0027t have to remember these words."},{"Start":"01:59.930 ","End":"02:03.710","Text":"This is an example we\u0027ve had before in a previous clip,"},{"Start":"02:03.710 ","End":"02:04.730","Text":"but I want to show you,"},{"Start":"02:04.730 ","End":"02:08.135","Text":"when we convert it to echelon form from a matrix,"},{"Start":"02:08.135 ","End":"02:10.590","Text":"it goes much easier."},{"Start":"02:10.590 ","End":"02:12.440","Text":"Here we have these numbers"},{"Start":"02:12.440 ","End":"02:17.060","Text":"and we have the same strategy that we want to get it into,"},{"Start":"02:17.060 ","End":"02:19.775","Text":"step form, echelon form."},{"Start":"02:19.775 ","End":"02:22.640","Text":"We see that there\u0027s a 2 here,"},{"Start":"02:22.640 ","End":"02:24.815","Text":"but we don\u0027t have 0s below it."},{"Start":"02:24.815 ","End":"02:26.990","Text":"We want to make the 0."},{"Start":"02:26.990 ","End":"02:31.205","Text":"We do the same operations as we did in the equation form."},{"Start":"02:31.205 ","End":"02:37.535","Text":"We say, \"Okay, we\u0027re going to subtract Row 1 from Row 2,"},{"Start":"02:37.535 ","End":"02:40.730","Text":"and we\u0027re going to add twice Row 1 to Row 3.\""},{"Start":"02:40.730 ","End":"02:43.190","Text":"That\u0027s what I said here."},{"Start":"02:43.190 ","End":"02:45.125","Text":"Now we have the 0s here."},{"Start":"02:45.125 ","End":"02:47.585","Text":"Next, we want to have a 0 here."},{"Start":"02:47.585 ","End":"02:49.915","Text":"We use this 1."},{"Start":"02:49.915 ","End":"02:55.580","Text":"What we\u0027re gonna do obviously is to subtract twice"},{"Start":"02:55.580 ","End":"02:58.465","Text":"the second row from the third row."},{"Start":"02:58.465 ","End":"03:01.710","Text":"That\u0027s how it\u0027s written in row notation."},{"Start":"03:01.710 ","End":"03:06.200","Text":"Now we have a 0 here and 4 minus twice 1 is 2,"},{"Start":"03:06.200 ","End":"03:08.865","Text":"and 20 minus twice 1 is 18, and so on."},{"Start":"03:08.865 ","End":"03:15.075","Text":"Now only this part needs to be in echelon form,"},{"Start":"03:15.075 ","End":"03:18.975","Text":"the restricted part and there\u0027s numbers here."},{"Start":"03:18.975 ","End":"03:23.460","Text":"Oh yeah, there\u0027s 1 more step which is optional."},{"Start":"03:23.460 ","End":"03:25.850","Text":"But when you see it divides by something,"},{"Start":"03:25.850 ","End":"03:29.040","Text":"that\u0027s what we did before with the equations that take 1/2 of this"},{"Start":"03:29.040 ","End":"03:32.045","Text":"or multiply it by 0.5 and we have this."},{"Start":"03:32.045 ","End":"03:37.870","Text":"Now, we have it in row-echelon form."},{"Start":"03:41.330 ","End":"03:44.065","Text":"Now that we\u0027ve done that,"},{"Start":"03:44.065 ","End":"03:48.970","Text":"now we go back to equation form and we get this."},{"Start":"03:48.970 ","End":"03:54.410","Text":"What\u0027s on the right side of the bar is what\u0027s behind the equals."},{"Start":"03:54.570 ","End":"03:59.350","Text":"These numbers here become coefficients here."},{"Start":"03:59.350 ","End":"04:03.790","Text":"Now all that remains is to do the back substitution"},{"Start":"04:03.790 ","End":"04:12.230","Text":"to put plugs, add into here to get y and then plug y and z."},{"Start":"04:12.770 ","End":"04:14.910","Text":"This is the result."},{"Start":"04:14.910 ","End":"04:16.915","Text":"As I said, we did this equation before."},{"Start":"04:16.915 ","End":"04:19.750","Text":"But you see that the process of converting to echelon form"},{"Start":"04:19.750 ","End":"04:21.730","Text":"is much easier in matrix form"},{"Start":"04:21.730 ","End":"04:28.025","Text":"and I\u0027m going to finish with an important remark."},{"Start":"04:28.025 ","End":"04:32.935","Text":"It says important because people sometimes make a mistake with this."},{"Start":"04:32.935 ","End":"04:36.900","Text":"Is that before you take a system of linear equations,"},{"Start":"04:36.900 ","End":"04:39.020","Text":"and convert it to a matrix,"},{"Start":"04:39.020 ","End":"04:41.120","Text":"it has to be properly arranged."},{"Start":"04:41.120 ","End":"04:42.890","Text":"It often is, but if it isn\u0027t,"},{"Start":"04:42.890 ","End":"04:45.535","Text":"you have to arrange it and I\u0027ll show you what I mean."},{"Start":"04:45.535 ","End":"04:49.250","Text":"I\u0027ll illustrate an example of a 3 by 3."},{"Start":"04:49.250 ","End":"04:52.555","Text":"I mean 3 equations and 3 unknowns."},{"Start":"04:52.555 ","End":"04:56.750","Text":"Let\u0027s take a specific example."},{"Start":"04:56.750 ","End":"04:59.330","Text":"We have to get this into this form."},{"Start":"04:59.330 ","End":"05:01.420","Text":"This is the general form."},{"Start":"05:01.420 ","End":"05:05.660","Text":"We could have written it with x1, x2, x3, or x, y, z."},{"Start":"05:05.660 ","End":"05:07.310","Text":"But it has to be that,"},{"Start":"05:07.310 ","End":"05:09.590","Text":"first is x, then y, then z,"},{"Start":"05:09.590 ","End":"05:12.260","Text":"and it\u0027s arranged in a certain manner."},{"Start":"05:12.260 ","End":"05:14.360","Text":"But if we look at this given 1,"},{"Start":"05:14.360 ","End":"05:20.125","Text":"it\u0027s not in this form because we have first y, then x, then z."},{"Start":"05:20.125 ","End":"05:23.580","Text":"Here, the z is on the right-hand side."},{"Start":"05:23.580 ","End":"05:26.325","Text":"Here the numbers on the left-hand side."},{"Start":"05:26.325 ","End":"05:31.730","Text":"We have a bit of arranging to do to get it into this format."},{"Start":"05:31.730 ","End":"05:35.000","Text":"Nothing very difficult just to bring stuff to the other side,"},{"Start":"05:35.000 ","End":"05:38.000","Text":"like the minus x goes before the y."},{"Start":"05:38.000 ","End":"05:41.465","Text":"Here I put the z on the left."},{"Start":"05:41.465 ","End":"05:43.805","Text":"Here I throw the 4 to the right,"},{"Start":"05:43.805 ","End":"05:46.970","Text":"and now I have x\u0027s and y\u0027s then z\u0027s."},{"Start":"05:46.970 ","End":"05:48.800","Text":"When I\u0027ve rearranged it,"},{"Start":"05:48.800 ","End":"05:52.030","Text":"now I can put it in matrix form,"},{"Start":"05:52.030 ","End":"05:54.510","Text":"and put the coefficients here."},{"Start":"05:54.510 ","End":"05:59.085","Text":"Minus 1, 1, 1 causes the 1s we don\u0027t write,"},{"Start":"05:59.085 ","End":"06:01.200","Text":"4 minus 5, minus 2"},{"Start":"06:01.200 ","End":"06:06.620","Text":"and then on the stuff beyond the equals goes on this column here,"},{"Start":"06:06.620 ","End":"06:10.470","Text":"the last column, and that\u0027s it."},{"Start":"06:10.470 ","End":"06:11.600","Text":"That\u0027s an important remark"},{"Start":"06:11.600 ","End":"06:16.970","Text":"and that\u0027s the introduction to the Concept of a matrix"},{"Start":"06:16.970 ","End":"06:20.660","Text":"and how it can help us to solve systems of linear equations"},{"Start":"06:20.660 ","End":"06:22.320","Text":"and I\u0027m done."}],"ID":9810},{"Watched":false,"Name":"Exercise 1","Duration":"2m 50s","ChapterTopicVideoID":9475,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9475.jpeg","UploadDate":"2017-07-26T08:24:51.5970000","DurationForVideoObject":"PT2M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we\u0027re going to practice the basic skill"},{"Start":"00:03.900 ","End":"00:09.540","Text":"of elementary row operations on an augmented matrix."},{"Start":"00:09.540 ","End":"00:17.350","Text":"It\u0027s going to be the same augmented matrix in each of the 4 sub exercises."},{"Start":"00:17.870 ","End":"00:21.090","Text":"I\u0027ll just say in words what they mean."},{"Start":"00:21.090 ","End":"00:25.785","Text":"This says take twice row 1 and put it into row 1."},{"Start":"00:25.785 ","End":"00:30.915","Text":"Here, exchange row 1 with row 2."},{"Start":"00:30.915 ","End":"00:36.100","Text":"Here, add row 2 to row 1 and put the result in row 2,"},{"Start":"00:36.100 ","End":"00:43.805","Text":"and here add row 2 to twice row 1 and put the answer in row 2."},{"Start":"00:43.805 ","End":"00:53.885","Text":"I\u0027ve copied these out and here we can see all 4 of them, hopefully here."},{"Start":"00:53.885 ","End":"01:00.425","Text":"Let\u0027s see. If we\u0027re putting twice row 1 into row 1,"},{"Start":"01:00.425 ","End":"01:03.710","Text":"certainly, row 2 is not affected,"},{"Start":"01:03.710 ","End":"01:08.350","Text":"so I can just leave row 2 as it is, just copy it."},{"Start":"01:08.350 ","End":"01:15.105","Text":"Now, twice row 1 would be 6 minus 4,"},{"Start":"01:15.105 ","End":"01:17.340","Text":"2, and that\u0027s what we have,"},{"Start":"01:17.340 ","End":"01:20.500","Text":"6 minus 4, 2."},{"Start":"01:20.500 ","End":"01:25.235","Text":"Here, we exchange row 1 with row 2."},{"Start":"01:25.235 ","End":"01:26.870","Text":"The 3 minus 2,"},{"Start":"01:26.870 ","End":"01:30.320","Text":"1 goes here, and the 2,"},{"Start":"01:30.320 ","End":"01:33.540","Text":"0 minus 1 goes up there."},{"Start":"01:34.790 ","End":"01:40.135","Text":"Here, we\u0027re adding the 2 rows,"},{"Start":"01:40.135 ","End":"01:42.620","Text":"and we\u0027re putting the answer in row 2."},{"Start":"01:42.620 ","End":"01:47.155","Text":"For 1 thing, it means that the first row is not changed."},{"Start":"01:47.155 ","End":"01:50.250","Text":"Now let\u0027s see, row 2 plus row 1,"},{"Start":"01:50.250 ","End":"01:52.005","Text":"that\u0027s 2 plus 3,"},{"Start":"01:52.005 ","End":"01:55.035","Text":"and I put it here, 5."},{"Start":"01:55.035 ","End":"01:57.420","Text":"0 plus negative 2,"},{"Start":"01:57.420 ","End":"02:02.835","Text":"negative 2, minus 1 plus 1 is 0."},{"Start":"02:02.835 ","End":"02:05.430","Text":"Now last 1,"},{"Start":"02:05.430 ","End":"02:07.415","Text":"we\u0027re taking row 2,"},{"Start":"02:07.415 ","End":"02:11.985","Text":"and we\u0027re adding twice row 1 and putting the answer in row 2."},{"Start":"02:11.985 ","End":"02:13.990","Text":"Row 1 is unchanged,"},{"Start":"02:13.990 ","End":"02:16.700","Text":"so let\u0027s leave that as is."},{"Start":"02:23.900 ","End":"02:26.985","Text":"Row 2 plus twice row 1,"},{"Start":"02:26.985 ","End":"02:33.260","Text":"2 plus twice 3 is 2 plus 6 is 8."},{"Start":"02:33.260 ","End":"02:42.540","Text":"0 plus twice minus 2 is 0 plus minus 4 is minus 4,"},{"Start":"02:42.540 ","End":"02:49.870","Text":"and minus 1 plus twice 1 is 1. That\u0027s it."}],"ID":9811},{"Watched":false,"Name":"Exercise 2","Duration":"3m 21s","ChapterTopicVideoID":9476,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9476.jpeg","UploadDate":"2017-07-26T08:25:06.1170000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.439","Text":"In this exercise, we\u0027re going to practice the basic skill"},{"Start":"00:04.439 ","End":"00:10.110","Text":"of elementary row operations on an augmented matrix."},{"Start":"00:10.110 ","End":"00:14.550","Text":"We\u0027re going to take this same augmented matrix"},{"Start":"00:14.550 ","End":"00:18.060","Text":"and do 4 different row operations on it."},{"Start":"00:18.060 ","End":"00:23.340","Text":"This one means put twice row 3 into row 3."},{"Start":"00:23.340 ","End":"00:26.515","Text":"Here, exchange row 1 with row 3."},{"Start":"00:26.515 ","End":"00:32.265","Text":"Here, subtract R_1 from R_3 and put the answer in row 3."},{"Start":"00:32.265 ","End":"00:40.620","Text":"Here, we take row 2 and subtract twice row 1 and put the answer in row 2."},{"Start":"00:40.620 ","End":"00:42.570","Text":"Let\u0027s get to it."},{"Start":"00:42.570 ","End":"00:44.820","Text":"First of all, scroll down a bit,"},{"Start":"00:44.820 ","End":"00:48.880","Text":"and let\u0027s see now."},{"Start":"00:49.580 ","End":"00:53.445","Text":"Twice row 3 into row 3."},{"Start":"00:53.445 ","End":"00:57.860","Text":"For one thing, it means that row 1 and row 2 are unchanged."},{"Start":"00:57.860 ","End":"01:04.150","Text":"Row 1, I\u0027ll just leave as is, same for row 2."},{"Start":"01:04.150 ","End":"01:08.220","Text":"Now let\u0027s see, twice row 3 into row 3,"},{"Start":"01:08.220 ","End":"01:10.355","Text":"so it means we double this,"},{"Start":"01:10.355 ","End":"01:17.142","Text":"like this part is going to be 6, 2, 4 if I double it, 6,2, 4,"},{"Start":"01:17.142 ","End":"01:23.045","Text":"and double this, minus 8, and it\u0027s in row 3."},{"Start":"01:23.045 ","End":"01:27.170","Text":"Here, we just exchange the positions of row 1 and row 3."},{"Start":"01:27.170 ","End":"01:30.355","Text":"Row 2 is unchanged,"},{"Start":"01:30.355 ","End":"01:38.980","Text":"and here, the 1, minus 2, 0, 3 goes down here,"},{"Start":"01:41.330 ","End":"01:44.700","Text":"and what was row 3 goes into row 1,"},{"Start":"01:44.700 ","End":"01:46.530","Text":"3, 1, 2 minus 4,"},{"Start":"01:46.530 ","End":"01:51.250","Text":"3, 1, 2, minus 4."},{"Start":"01:52.550 ","End":"02:01.210","Text":"Now here, we\u0027re subtracting row 1 from row 3 and putting it into row 3."},{"Start":"02:01.210 ","End":"02:04.350","Text":"Row 1 and row 2 would be unchanged."},{"Start":"02:04.350 ","End":"02:06.470","Text":"Here\u0027s row 1 unchanged,"},{"Start":"02:06.470 ","End":"02:09.295","Text":"and here\u0027s row 2 unchanged."},{"Start":"02:09.295 ","End":"02:13.910","Text":"Here we\u0027re subtracting row 1 from row 3 basically,"},{"Start":"02:13.910 ","End":"02:15.620","Text":"and it stays in row 3."},{"Start":"02:15.620 ","End":"02:19.415","Text":"Here, I want 3 minus 1."},{"Start":"02:19.415 ","End":"02:24.950","Text":"1 minus negative 2 is 1 plus 2 is 3."},{"Start":"02:24.950 ","End":"02:27.980","Text":"2 minus 0 is 2,"},{"Start":"02:27.980 ","End":"02:32.135","Text":"minus 4, minus 3, minus 7."},{"Start":"02:32.135 ","End":"02:35.390","Text":"That\u0027s that one. Now here."},{"Start":"02:35.390 ","End":"02:38.240","Text":"We\u0027re putting something into row 2,"},{"Start":"02:38.240 ","End":"02:45.955","Text":"so that already means that row 1 is unchanged and row 3 is unchanged."},{"Start":"02:45.955 ","End":"02:48.950","Text":"Now what is it that we\u0027re putting into row 2?"},{"Start":"02:48.950 ","End":"02:52.830","Text":"Row 2 itself minus twice the first row."},{"Start":"02:52.830 ","End":"02:57.190","Text":"I take each element here and subtract twice what\u0027s above it."},{"Start":"02:57.190 ","End":"03:00.900","Text":"4 minus twice 1 is 2,"},{"Start":"03:00.900 ","End":"03:04.170","Text":"0 minus twice negative 2 is 0,"},{"Start":"03:04.170 ","End":"03:06.555","Text":"plus 4 is 4,"},{"Start":"03:06.555 ","End":"03:11.280","Text":"minus 1 takeaway twice 0 is still minus 1,"},{"Start":"03:11.280 ","End":"03:17.430","Text":"and 2 minus twice 3 is 2 minus 6 is minus 4."},{"Start":"03:17.430 ","End":"03:20.950","Text":"That\u0027s it."}],"ID":9812},{"Watched":false,"Name":"Exercise 3","Duration":"3m 22s","ChapterTopicVideoID":9477,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9477.jpeg","UploadDate":"2017-07-26T08:25:32.7870000","DurationForVideoObject":"PT3M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this exercise, we\u0027re going to practice"},{"Start":"00:02.280 ","End":"00:08.370","Text":"the basic skill of elementary row operations on an augmented matrix."},{"Start":"00:08.370 ","End":"00:15.400","Text":"We\u0027re going to take this augmented matrix here and do 4 different row operations on it."},{"Start":"00:15.860 ","End":"00:19.500","Text":"Let\u0027s get started."},{"Start":"00:19.500 ","End":"00:22.090","Text":"Scroll down a bit."},{"Start":"00:22.340 ","End":"00:25.660","Text":"A bit more perhaps."},{"Start":"00:26.030 ","End":"00:34.930","Text":"This row operation means that we take 1/2 of row 3 and put it into the new row 3."},{"Start":"00:35.030 ","End":"00:39.665","Text":"It means that row 1 and row 2 are unchanged."},{"Start":"00:39.665 ","End":"00:44.775","Text":"I can certainly leave them as they were."},{"Start":"00:44.775 ","End":"00:48.410","Text":"Now I\u0027ll take half of this row and put it in here."},{"Start":"00:48.410 ","End":"00:51.460","Text":"Everything is divided by 2."},{"Start":"00:51.460 ","End":"00:54.300","Text":"I get minus 0.5."},{"Start":"00:54.300 ","End":"00:55.785","Text":"Here I\u0027ll get 1."},{"Start":"00:55.785 ","End":"00:58.635","Text":"Here it is just also 1."},{"Start":"00:58.635 ","End":"01:04.050","Text":"Here if I divide by 2 I get minus a 1/2 or minus 0.5."},{"Start":"01:04.050 ","End":"01:06.770","Text":"That\u0027s it. In this 1,"},{"Start":"01:06.770 ","End":"01:10.010","Text":"I need to exchange row 2 with row 3."},{"Start":"01:10.010 ","End":"01:12.350","Text":"Row 1 is unchanged,"},{"Start":"01:12.350 ","End":"01:13.790","Text":"just the way it was."},{"Start":"01:13.790 ","End":"01:16.705","Text":"I\u0027m exchanging these 2."},{"Start":"01:16.705 ","End":"01:22.590","Text":"Basically, what I can do is I can copy what was row 3 into row 2,"},{"Start":"01:22.590 ","End":"01:24.060","Text":"so minus 1, 2, 2,"},{"Start":"01:24.060 ","End":"01:26.100","Text":"minus 1, minus 1,"},{"Start":"01:26.100 ","End":"01:28.170","Text":"2, 2, minus 1."},{"Start":"01:28.170 ","End":"01:30.105","Text":"Then row 2 into row 3,"},{"Start":"01:30.105 ","End":"01:31.965","Text":"minus 2, 0, minus 1,"},{"Start":"01:31.965 ","End":"01:33.660","Text":"2, minus 2,"},{"Start":"01:33.660 ","End":"01:36.730","Text":"0, minus 1, 2."},{"Start":"01:36.800 ","End":"01:39.540","Text":"That\u0027s it for that 1."},{"Start":"01:39.540 ","End":"01:48.855","Text":"Here, I\u0027m taking row 2 and subtracting 4 times row 1,"},{"Start":"01:48.855 ","End":"01:51.520","Text":"and the result is in row 2,"},{"Start":"01:51.520 ","End":"01:55.085","Text":"which basically means that from this row,"},{"Start":"01:55.085 ","End":"01:58.280","Text":"I\u0027m subtracting 4 times the row above it."},{"Start":"01:58.280 ","End":"02:07.130","Text":"Let\u0027s see. Minus 2 minus 4 times 1 is minus 6."},{"Start":"02:07.130 ","End":"02:11.315","Text":"0 minus 4 times 2 is minus 8,"},{"Start":"02:11.315 ","End":"02:18.010","Text":"minus 1, less 4 times 4 minus 1 minus 16 is minus 17."},{"Start":"02:18.010 ","End":"02:22.765","Text":"2 less 4 times 0 is still just 2."},{"Start":"02:22.765 ","End":"02:25.130","Text":"As for row 1 and row 3,"},{"Start":"02:25.130 ","End":"02:28.085","Text":"they stay as they were."},{"Start":"02:28.085 ","End":"02:35.465","Text":"I can just copy them as they were."},{"Start":"02:35.465 ","End":"02:38.680","Text":"In this last 1,"},{"Start":"02:38.680 ","End":"02:43.965","Text":"I\u0027m taking row 3 and adding to it twice row 1,"},{"Start":"02:43.965 ","End":"02:46.960","Text":"and the result stays in row 3."},{"Start":"02:50.180 ","End":"02:54.730","Text":"Row 1 and row 2 stay unchanged."},{"Start":"02:54.730 ","End":"02:59.375","Text":"Now, I take this and add twice the first row,"},{"Start":"02:59.375 ","End":"03:03.560","Text":"minus 1 plus twice 1 is 1."},{"Start":"03:03.560 ","End":"03:07.394","Text":"2 plus twice 2 is 6."},{"Start":"03:07.394 ","End":"03:09.150","Text":"2 plus twice 4,"},{"Start":"03:09.150 ","End":"03:12.435","Text":"2 plus 8 is 10."},{"Start":"03:12.435 ","End":"03:21.370","Text":"Minus 1 plus twice 0 is still just minus 1. That\u0027s it."}],"ID":9813},{"Watched":false,"Name":"Exercise 4","Duration":"1m 46s","ChapterTopicVideoID":9478,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9478.jpeg","UploadDate":"2017-07-26T08:25:41.8770000","DurationForVideoObject":"PT1M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.670","Text":"In this exercise, we have a linear system of equations and we have to try and solve it."},{"Start":"00:08.670 ","End":"00:10.755","Text":"If we can\u0027t solve it,"},{"Start":"00:10.755 ","End":"00:13.770","Text":"then might mean that it\u0027s inconsistent."},{"Start":"00:13.770 ","End":"00:15.525","Text":"We\u0027ll say so."},{"Start":"00:15.525 ","End":"00:18.450","Text":"We\u0027re going to use augmented matrix techniques."},{"Start":"00:18.450 ","End":"00:22.140","Text":"The first thing is to get the augmented matrix for this."},{"Start":"00:22.140 ","End":"00:23.520","Text":"That looks like this,"},{"Start":"00:23.520 ","End":"00:26.505","Text":"which is essentially just the numbers from here."},{"Start":"00:26.505 ","End":"00:29.140","Text":"A vertical line where the equals are."},{"Start":"00:29.140 ","End":"00:32.215","Text":"Then we do what are called row operations."},{"Start":"00:32.215 ","End":"00:36.470","Text":"What we want to do is try and get a 0 in this position."},{"Start":"00:36.470 ","End":"00:43.240","Text":"The easiest way to do that is to subtract the top row from the bottom row."},{"Start":"00:43.240 ","End":"00:46.140","Text":"In shorthand, this says,"},{"Start":"00:46.140 ","End":"00:49.740","Text":"take Row 2, subtract Row 1,"},{"Start":"00:49.740 ","End":"00:52.140","Text":"and put that into the new row 2."},{"Start":"00:52.140 ","End":"00:53.945","Text":"If we do that,"},{"Start":"00:53.945 ","End":"00:58.400","Text":"then what we\u0027ll get is 2 minus 2 is 0,"},{"Start":"00:58.400 ","End":"01:00.320","Text":"5 minus 7 is negative 2."},{"Start":"01:00.320 ","End":"01:02.960","Text":"11 minus 13, minus 2,"},{"Start":"01:02.960 ","End":"01:05.700","Text":"and the first row is unchanged."},{"Start":"01:07.030 ","End":"01:14.165","Text":"We can now go back from the augmented matrix back to x and y."},{"Start":"01:14.165 ","End":"01:17.915","Text":"Possibly we could have divided by minus 2 here,"},{"Start":"01:17.915 ","End":"01:19.310","Text":"might have been a bit neater."},{"Start":"01:19.310 ","End":"01:22.220","Text":"It doesn\u0027t matter that division could be done here."},{"Start":"01:22.220 ","End":"01:25.285","Text":"We can get that y is equal to 1."},{"Start":"01:25.285 ","End":"01:31.850","Text":"If y equals 1, then we can substitute into this equation and get 2x plus 7 is 13,"},{"Start":"01:31.850 ","End":"01:34.865","Text":"which gives immediately x equals 3."},{"Start":"01:34.865 ","End":"01:37.910","Text":"We get 1 unique solution,"},{"Start":"01:37.910 ","End":"01:46.380","Text":"x equals 3 and y equals 1. That\u0027s it."}],"ID":9814},{"Watched":false,"Name":"Exercise 5","Duration":"1m 42s","ChapterTopicVideoID":9479,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9479.jpeg","UploadDate":"2017-07-26T08:25:50.7830000","DurationForVideoObject":"PT1M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"Here we have a system of linear equations,"},{"Start":"00:04.140 ","End":"00:05.820","Text":"2 equations and 2 unknowns."},{"Start":"00:05.820 ","End":"00:12.090","Text":"We\u0027re going to try to solve it using the augmented matrix techniques."},{"Start":"00:12.090 ","End":"00:15.540","Text":"If we somehow don\u0027t get a solution that it"},{"Start":"00:15.540 ","End":"00:19.390","Text":"might mean the system is inconsistent and we\u0027ll say so."},{"Start":"00:19.940 ","End":"00:24.520","Text":"First thing to do is to convert it into an augmented matrix,"},{"Start":"00:24.520 ","End":"00:27.410","Text":"which is just taking the numbers"},{"Start":"00:27.410 ","End":"00:32.330","Text":"from the original equation and putting them in this format."},{"Start":"00:32.330 ","End":"00:34.700","Text":"Then we start doing row operations."},{"Start":"00:34.700 ","End":"00:36.200","Text":"I want to get 0 here."},{"Start":"00:36.200 ","End":"00:40.490","Text":"If I subtract twice this row from this row,"},{"Start":"00:40.490 ","End":"00:44.930","Text":"in other words, if I do R_2, which is the second row,"},{"Start":"00:44.930 ","End":"00:48.530","Text":"minus twice the first row and put that into row 2,"},{"Start":"00:48.530 ","End":"00:56.230","Text":"then I\u0027ll get 4 minus twice 2 is 0 minus 5 minus twice 3 is minus 11, and so on."},{"Start":"00:56.330 ","End":"01:02.840","Text":"I can then divide the last row by minus 11,"},{"Start":"01:02.840 ","End":"01:05.910","Text":"get rid of that minus 11."},{"Start":"01:06.010 ","End":"01:11.840","Text":"Then I should get simple last row."},{"Start":"01:11.840 ","End":"01:14.840","Text":"This is basically in the form that I want."},{"Start":"01:14.840 ","End":"01:16.835","Text":"Mainly I want the 0 here."},{"Start":"01:16.835 ","End":"01:20.739","Text":"Let\u0027s go back from here to x and y."},{"Start":"01:20.739 ","End":"01:22.490","Text":"We get from here 0,"},{"Start":"01:22.490 ","End":"01:23.585","Text":"x, which I don\u0027t write,"},{"Start":"01:23.585 ","End":"01:27.260","Text":"plus y equals 1 and 2x plus 3y is 7."},{"Start":"01:27.260 ","End":"01:29.995","Text":"Once I have y, can plug that in here."},{"Start":"01:29.995 ","End":"01:31.935","Text":"2x plus 3 is 7,"},{"Start":"01:31.935 ","End":"01:34.245","Text":"and that gives me that x is 2."},{"Start":"01:34.245 ","End":"01:36.750","Text":"That\u0027s the only solution,"},{"Start":"01:36.750 ","End":"01:42.000","Text":"x equals 2, y equals 1, and we\u0027re done."}],"ID":9815},{"Watched":false,"Name":"Exercise 6","Duration":"1m 54s","ChapterTopicVideoID":9480,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9480.jpeg","UploadDate":"2017-07-26T08:26:01.9270000","DurationForVideoObject":"PT1M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Here we have a system of linear equations and we\u0027re going to try"},{"Start":"00:03.210 ","End":"00:06.570","Text":"and solve it using augmented matrix techniques,"},{"Start":"00:06.570 ","End":"00:11.160","Text":"and we will succeed if it\u0027s consistent and if not,"},{"Start":"00:11.160 ","End":"00:12.885","Text":"then we won\u0027t get a solution."},{"Start":"00:12.885 ","End":"00:15.930","Text":"Anyway, let\u0027s start by, first of all,"},{"Start":"00:15.930 ","End":"00:18.240","Text":"getting the augmented matrix from this,"},{"Start":"00:18.240 ","End":"00:23.025","Text":"which is essentially just taking the numbers and putting them in a kind of a brackets."},{"Start":"00:23.025 ","End":"00:30.105","Text":"Now, we start doing raw operations with the idea of getting a 0 here."},{"Start":"00:30.105 ","End":"00:35.040","Text":"What we can do is multiply this by 5 and this by 2,"},{"Start":"00:35.040 ","End":"00:38.400","Text":"and if we do that, in other words,"},{"Start":"00:38.400 ","End":"00:42.300","Text":"we\u0027ll put 5 times row 1 into row 1,"},{"Start":"00:42.300 ","End":"00:43.770","Text":"twice row 2 into row 2,"},{"Start":"00:43.770 ","End":"00:46.350","Text":"and then we should get 10 in both places,"},{"Start":"00:46.350 ","End":"00:48.210","Text":"in fact, we do."},{"Start":"00:48.210 ","End":"00:51.725","Text":"Then we can subtract the first from the second,"},{"Start":"00:51.725 ","End":"00:55.594","Text":"which means the second minus the first into the second."},{"Start":"00:55.594 ","End":"00:58.280","Text":"10 minus 10 would be 0 here, minus 8,"},{"Start":"00:58.280 ","End":"01:01.460","Text":"minus 15 would be minus 23,"},{"Start":"01:01.460 ","End":"01:09.410","Text":"and so on and now what we\u0027ll do is look both these are divisible by 23 or even minus 23,"},{"Start":"01:09.410 ","End":"01:11.180","Text":"so we can divide,"},{"Start":"01:11.180 ","End":"01:14.045","Text":"and these also we\u0027ll divide by 5."},{"Start":"01:14.045 ","End":"01:19.910","Text":"I\u0027ll take a fifth of row 1 and 1 over minus 23 of row 2,"},{"Start":"01:19.910 ","End":"01:23.050","Text":"and then we\u0027ll get something with smaller numbers."},{"Start":"01:23.050 ","End":"01:27.200","Text":"It\u0027s now the form we want with a 0 here."},{"Start":"01:27.200 ","End":"01:29.345","Text":"It\u0027s even nicer when there\u0027s a 1 here."},{"Start":"01:29.345 ","End":"01:32.420","Text":"Because that means when we go back to x and y,"},{"Start":"01:32.420 ","End":"01:35.420","Text":"we\u0027ve got y straight away as 2,"},{"Start":"01:35.420 ","End":"01:37.105","Text":"and then we plug that in here."},{"Start":"01:37.105 ","End":"01:38.595","Text":"3y becomes 6,"},{"Start":"01:38.595 ","End":"01:40.590","Text":"so 2x plus 6 is 8."},{"Start":"01:40.590 ","End":"01:43.430","Text":"If we solve that, that gives us x equals 1,"},{"Start":"01:43.430 ","End":"01:46.190","Text":"so y equals 2x equals 1,"},{"Start":"01:46.190 ","End":"01:51.590","Text":"and there\u0027s a single solution to the equation system."},{"Start":"01:51.590 ","End":"01:54.270","Text":"That\u0027s it."}],"ID":9816},{"Watched":false,"Name":"Exercise 7","Duration":"2m 18s","ChapterTopicVideoID":9481,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9481.jpeg","UploadDate":"2017-07-26T08:26:14.0830000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"Here we have a system of linear equations,"},{"Start":"00:03.450 ","End":"00:05.595","Text":"2 equations and 2 unknowns."},{"Start":"00:05.595 ","End":"00:10.575","Text":"We\u0027re going to try and solve it using augmented matrix techniques."},{"Start":"00:10.575 ","End":"00:11.910","Text":"Which means that we first take"},{"Start":"00:11.910 ","End":"00:17.295","Text":"the augmented matrix basically throughout the x\u0027s and the y\u0027s and the equals,"},{"Start":"00:17.295 ","End":"00:20.925","Text":"and get something like this with just the numbers."},{"Start":"00:20.925 ","End":"00:28.095","Text":"Then we do row operations on this to try and get to get 0 here,"},{"Start":"00:28.095 ","End":"00:30.300","Text":"and preferably a 1 here,"},{"Start":"00:30.300 ","End":"00:32.325","Text":"but that\u0027s less important."},{"Start":"00:32.325 ","End":"00:38.640","Text":"One way to do that would be to multiply the top 1 by 3 and the bottom 1 by 4."},{"Start":"00:38.640 ","End":"00:40.380","Text":"Then if we do that,"},{"Start":"00:40.380 ","End":"00:43.385","Text":"and we\u0027ll have 12 \u0027s in both places and indeed,"},{"Start":"00:43.385 ","End":"00:44.865","Text":"this is what we get,"},{"Start":"00:44.865 ","End":"00:48.535","Text":"but look they\u0027re both the same row."},{"Start":"00:48.535 ","End":"00:50.150","Text":"Trying to get 0 here,"},{"Start":"00:50.150 ","End":"00:52.025","Text":"I\u0027d subtract the first from the second,"},{"Start":"00:52.025 ","End":"00:55.205","Text":"in other words second minus the first into the second,"},{"Start":"00:55.205 ","End":"00:59.555","Text":"but then the whole second row is 0."},{"Start":"00:59.555 ","End":"01:05.210","Text":"When that happens, it just means that 1 of the equations"},{"Start":"01:05.210 ","End":"01:11.200","Text":"drops out because it just says that 0x plus 0y is 0."},{"Start":"01:11.200 ","End":"01:18.600","Text":"In any event we can divide by 12 first and get smaller numbers."},{"Start":"01:18.600 ","End":"01:21.910","Text":"We\u0027ve got 1, 2, and 5 here."},{"Start":"01:22.190 ","End":"01:27.530","Text":"If we go back to x\u0027s and y\u0027s,"},{"Start":"01:27.530 ","End":"01:34.590","Text":"then we have x plus 2y equals 5."},{"Start":"01:34.840 ","End":"01:39.345","Text":"We can then let y be anything."},{"Start":"01:39.345 ","End":"01:41.120","Text":"We have 2 equation and 2 unknowns,"},{"Start":"01:41.120 ","End":"01:42.590","Text":"y could be whatever we want."},{"Start":"01:42.590 ","End":"01:45.155","Text":"We could say, let y equal t,"},{"Start":"01:45.155 ","End":"01:47.570","Text":"some parameter, any number,"},{"Start":"01:47.570 ","End":"01:54.600","Text":"and if y is t, then we get that x is 5 minus 2y or 5 minus 2t."},{"Start":"01:54.600 ","End":"01:57.500","Text":"We have both x and y,"},{"Start":"01:57.500 ","End":"01:59.240","Text":"but in terms of a parameter,"},{"Start":"01:59.240 ","End":"02:01.610","Text":"it\u0027s infinite solutions for each t,"},{"Start":"02:01.610 ","End":"02:04.865","Text":"we get a solution, but there\u0027s an infinite number of t\u0027s."},{"Start":"02:04.865 ","End":"02:06.530","Text":"We can also write this."},{"Start":"02:06.530 ","End":"02:10.835","Text":"Some people prefer to write x comma y as the solution,"},{"Start":"02:10.835 ","End":"02:14.750","Text":"and then you would put the 5 minus 2t for x and t for y."},{"Start":"02:14.750 ","End":"02:18.630","Text":"That\u0027s the set of solutions. We\u0027re done."}],"ID":9817},{"Watched":false,"Name":"Exercise 8","Duration":"3m 46s","ChapterTopicVideoID":9482,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9482.jpeg","UploadDate":"2017-07-26T08:26:39.1200000","DurationForVideoObject":"PT3M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"We have here a system of linear equations,"},{"Start":"00:04.995 ","End":"00:06.705","Text":"2 equations, and 2 unknowns."},{"Start":"00:06.705 ","End":"00:13.410","Text":"I\u0027m going to try and solve it using augmented matrix techniques if possible."},{"Start":"00:13.410 ","End":"00:15.975","Text":"If we can\u0027t solve it, then it\u0027ll be inconsistent."},{"Start":"00:15.975 ","End":"00:18.720","Text":"But let\u0027s see what happens."},{"Start":"00:18.720 ","End":"00:21.600","Text":"First of all, the augmented matrix."},{"Start":"00:21.600 ","End":"00:25.920","Text":"Just throw out the x\u0027s and y\u0027s and package it and this is what it will look like,"},{"Start":"00:25.920 ","End":"00:30.495","Text":"minus 6, 3, 15, 10 minus 5 minus 25."},{"Start":"00:30.495 ","End":"00:35.430","Text":"Our goal is to get a 0 in this place."},{"Start":"00:35.430 ","End":"00:39.045","Text":"If we can get a 1 here also so much the better."},{"Start":"00:39.045 ","End":"00:47.630","Text":"At any rate what we can do is look at 6 and 10 and they have a common denominator of 30."},{"Start":"00:47.630 ","End":"00:51.590","Text":"If I multiply this by 5 and this by 3,"},{"Start":"00:51.590 ","End":"00:56.925","Text":"then we\u0027ll get equal coefficients, so negative."},{"Start":"00:56.925 ","End":"00:59.930","Text":"Even better if it\u0027s 1 of them\u0027s minus the"},{"Start":"00:59.930 ","End":"01:02.525","Text":"other because then we can do an addition, not a subtraction."},{"Start":"01:02.525 ","End":"01:05.765","Text":"If we add these 2 rows and put them into the second,"},{"Start":"01:05.765 ","End":"01:07.625","Text":"which means this is how we say it."},{"Start":"01:07.625 ","End":"01:10.205","Text":"R2 plus R1 put into R2,"},{"Start":"01:10.205 ","End":"01:12.185","Text":"then we\u0027ll get 0 here."},{"Start":"01:12.185 ","End":"01:15.125","Text":"Thing is, everything comes out 0."},{"Start":"01:15.125 ","End":"01:19.340","Text":"That second equation drops out."},{"Start":"01:19.340 ","End":"01:22.640","Text":"I mean, it tells us that 0 x plus 0 y is 0."},{"Start":"01:22.640 ","End":"01:26.465","Text":"Anyway, we keep going by simplifying,"},{"Start":"01:26.465 ","End":"01:30.705","Text":"because I can see that 15 goes into all these 3 numbers."},{"Start":"01:30.705 ","End":"01:36.750","Text":"Just doing 1/15th of the first row gives it in slightly smaller numbers."},{"Start":"01:38.020 ","End":"01:42.875","Text":"What we get then is just the first row,"},{"Start":"01:42.875 ","End":"01:48.125","Text":"which back in x and y gives us minus 2x plus y is 5."},{"Start":"01:48.125 ","End":"01:52.865","Text":"Then we can arbitrarily assign 1 of the variables."},{"Start":"01:52.865 ","End":"01:54.425","Text":"For example, we could say,"},{"Start":"01:54.425 ","End":"01:59.360","Text":"let y equal t and then we\u0027ve got"},{"Start":"01:59.360 ","End":"02:06.695","Text":"that 5 minus t is minus 2x and dividing by,"},{"Start":"02:06.695 ","End":"02:09.860","Text":"this is what we get and then just putting it as x,"},{"Start":"02:09.860 ","End":"02:13.880","Text":"y, this is what we get."},{"Start":"02:13.880 ","End":"02:17.000","Text":"I\u0027d just like to show you an alternative ending."},{"Start":"02:17.000 ","End":"02:18.320","Text":"I mean, this is it, we\u0027re done,"},{"Start":"02:18.320 ","End":"02:21.180","Text":"but I\u0027d just like to show you an alternative."},{"Start":"02:21.360 ","End":"02:24.970","Text":"At this point, I didn\u0027t have to substitute y,"},{"Start":"02:24.970 ","End":"02:26.995","Text":"I could have substituted x,"},{"Start":"02:26.995 ","End":"02:29.200","Text":"and I don\u0027t want to use the same letter twice."},{"Start":"02:29.200 ","End":"02:34.080","Text":"Let\u0027s take s,"},{"Start":"02:34.080 ","End":"02:41.025","Text":"it\u0027s right next to t and then we\u0027d get the minus 2s plus y is 5."},{"Start":"02:41.025 ","End":"02:49.950","Text":"We\u0027d get that y equals 5 plus 2s and then we could say that x,"},{"Start":"02:49.950 ","End":"02:59.190","Text":"y equals s and the 5 similar."},{"Start":"02:59.190 ","End":"03:01.410","Text":"Don\u0027t get confused. S,"},{"Start":"03:01.410 ","End":"03:03.570","Text":"5 plus 2, S,"},{"Start":"03:03.570 ","End":"03:07.780","Text":"or even 2s plus 5 might be nicer."},{"Start":"03:08.780 ","End":"03:12.140","Text":"This way, we keep using"},{"Start":"03:12.140 ","End":"03:16.430","Text":"whole numbers because I saw that there\u0027s a 2 here and I didn\u0027t want to divide by it."},{"Start":"03:16.430 ","End":"03:19.345","Text":"I actually assigned x."},{"Start":"03:19.345 ","End":"03:23.170","Text":"I forgot to mention that when we went from here to here, I didn\u0027t just copy this,"},{"Start":"03:23.170 ","End":"03:27.320","Text":"I actually did the division 5 over minus 2 is minus 2.5,"},{"Start":"03:27.320 ","End":"03:30.875","Text":"and minus 1 over minus 2 is plus 0.5."},{"Start":"03:30.875 ","End":"03:36.575","Text":"Anyway, you have a choice of which variable to substitute as a parameter."},{"Start":"03:36.575 ","End":"03:40.250","Text":"Whichever maybe comes out neater or whichever 1 you choose."},{"Start":"03:40.250 ","End":"03:46.110","Text":"That\u0027s supplementary alternative solution and we\u0027re done."}],"ID":9818},{"Watched":false,"Name":"Exercise 9","Duration":"2m 31s","ChapterTopicVideoID":9483,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9483.jpeg","UploadDate":"2017-07-26T08:27:02.1470000","DurationForVideoObject":"PT2M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this exercise, we have here a linear system of equations,"},{"Start":"00:05.040 ","End":"00:06.510","Text":"2 equations and 2 unknowns,"},{"Start":"00:06.510 ","End":"00:08.460","Text":"x and y, and we\u0027re going to use"},{"Start":"00:08.460 ","End":"00:13.200","Text":"augmented matrix techniques to try and solve it if possible."},{"Start":"00:13.200 ","End":"00:14.610","Text":"It might be inconsistent,"},{"Start":"00:14.610 ","End":"00:16.020","Text":"so we won\u0027t be able to."},{"Start":"00:16.020 ","End":"00:17.205","Text":"Let\u0027s see what happens."},{"Start":"00:17.205 ","End":"00:20.640","Text":"First of all, we need the augmented matrix,"},{"Start":"00:20.640 ","End":"00:26.175","Text":"which is just essentially copying the numbers in a certain format like this,"},{"Start":"00:26.175 ","End":"00:32.545","Text":"and then we start doing row operations on this to try and get a 0 here,"},{"Start":"00:32.545 ","End":"00:34.380","Text":"and ideally a 1 here,"},{"Start":"00:34.380 ","End":"00:35.730","Text":"but that\u0027s less important."},{"Start":"00:35.730 ","End":"00:42.015","Text":"Let\u0027s first of all see if we can get 2 rows to match,"},{"Start":"00:42.015 ","End":"00:46.855","Text":"and I can see that if I multiply this one by 3 and this one by 4,"},{"Start":"00:46.855 ","End":"00:51.780","Text":"then I\u0027ll get the same number here and here,"},{"Start":"00:51.780 ","End":"00:54.410","Text":"8 and 6 have a common denominator of 24,"},{"Start":"00:54.410 ","End":"01:00.340","Text":"so multiply this multiplied by 4 or minus 4."},{"Start":"01:00.340 ","End":"01:01.700","Text":"Some people like to add,"},{"Start":"01:01.700 ","End":"01:02.840","Text":"some people like to subtract."},{"Start":"01:02.840 ","End":"01:07.290","Text":"If it\u0027s minus 4 then I\u0027ll get them with the same sign, 24 and 24."},{"Start":"01:07.290 ","End":"01:10.030","Text":"Some people like to multiply by positive 4,"},{"Start":"01:10.030 ","End":"01:12.490","Text":"then they get a minus 24 here,"},{"Start":"01:12.490 ","End":"01:14.020","Text":"and then they add the equations here,"},{"Start":"01:14.020 ","End":"01:15.595","Text":"we subtract the equations."},{"Start":"01:15.595 ","End":"01:20.580","Text":"It\u0027s just slight variations in style."},{"Start":"01:20.580 ","End":"01:23.505","Text":"We have 24 and 24, so we subtract."},{"Start":"01:23.505 ","End":"01:26.515","Text":"I\u0027m going to subtract the first from the second."},{"Start":"01:26.515 ","End":"01:29.385","Text":"Other words, 2 minus 1 goes into 2,"},{"Start":"01:29.385 ","End":"01:31.655","Text":"so 24 minus 24 is 0,"},{"Start":"01:31.655 ","End":"01:37.750","Text":"minus this is 0, and minus 4 minus 30 is minus 34."},{"Start":"01:37.750 ","End":"01:41.570","Text":"That\u0027s the second row. The first row stays the same."},{"Start":"01:42.000 ","End":"01:45.610","Text":"Look, the second row has zeros here,"},{"Start":"01:45.610 ","End":"01:47.260","Text":"but not 0 here."},{"Start":"01:47.260 ","End":"01:54.400","Text":"What this means is that if you go back to x and y, first row, straightforward,"},{"Start":"01:54.400 ","End":"01:56.290","Text":"24x minus 12y is 30,"},{"Start":"01:56.290 ","End":"02:00.290","Text":"but the second row says 0x plus 0y,"},{"Start":"02:00.290 ","End":"02:04.065","Text":"and that\u0027s just 0 is equal to minus 34."},{"Start":"02:04.065 ","End":"02:06.950","Text":"There\u0027s no way 0 is going to equal minus 34."},{"Start":"02:06.950 ","End":"02:08.960","Text":"It doesn\u0027t matter what the first equation is,"},{"Start":"02:08.960 ","End":"02:11.105","Text":"0 will not be minus 34,"},{"Start":"02:11.105 ","End":"02:13.530","Text":"so there\u0027s no solution."},{"Start":"02:13.550 ","End":"02:15.600","Text":"In the original question,"},{"Start":"02:15.600 ","End":"02:20.555","Text":"we were asked to say if the system is inconsistent,"},{"Start":"02:20.555 ","End":"02:22.520","Text":"and in this case I would say yes,"},{"Start":"02:22.520 ","End":"02:24.830","Text":"the system is inconsistent."},{"Start":"02:24.830 ","End":"02:26.335","Text":"When you get no solution,"},{"Start":"02:26.335 ","End":"02:30.630","Text":"then the linear system is inconsistent. We\u0027re done."}],"ID":9819},{"Watched":false,"Name":"Exercise 10","Duration":"3m 5s","ChapterTopicVideoID":9484,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9484.jpeg","UploadDate":"2017-07-26T08:27:12.6570000","DurationForVideoObject":"PT3M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.210","Text":"We have here a linear system of equations or system of linear equations,"},{"Start":"00:06.210 ","End":"00:08.505","Text":"3 equations and 3 unknowns,"},{"Start":"00:08.505 ","End":"00:12.284","Text":"and we\u0027re going to try and solve it using augmented matrix techniques."},{"Start":"00:12.284 ","End":"00:15.180","Text":"The first step is to get the augmented matrix,"},{"Start":"00:15.180 ","End":"00:18.180","Text":"and that looks like this which is just"},{"Start":"00:18.180 ","End":"00:22.359","Text":"basically the numbers from here put in this format."},{"Start":"00:22.640 ","End":"00:29.180","Text":"The way we\u0027re going to solve it is by doing row operations and our goal is"},{"Start":"00:29.180 ","End":"00:35.015","Text":"to try and get 0s below the main diagonal here."},{"Start":"00:35.015 ","End":"00:38.975","Text":"First step is to get these 2 to be 0 and we do that by"},{"Start":"00:38.975 ","End":"00:43.715","Text":"multiplying rows and adding and subtracting appropriately."},{"Start":"00:43.715 ","End":"00:49.415","Text":"What I suggest is if we take this row and subtract twice this row,"},{"Start":"00:49.415 ","End":"00:51.230","Text":"then we\u0027ll get a 0 here."},{"Start":"00:51.230 ","End":"00:53.550","Text":"If we take this row minus 3 times this row,"},{"Start":"00:53.550 ","End":"00:54.995","Text":"we\u0027ll get a 0 here."},{"Start":"00:54.995 ","End":"00:57.160","Text":"This is the shorthand way of writing this,"},{"Start":"00:57.160 ","End":"00:59.030","Text":"row 2 minus twice row 1,"},{"Start":"00:59.030 ","End":"01:01.175","Text":"put into row 2 and so on."},{"Start":"01:01.175 ","End":"01:06.360","Text":"If we do that, we get 2 minus twice 1 is 0,"},{"Start":"01:06.360 ","End":"01:09.090","Text":"3 minus twice 2 is minus 1, and so on."},{"Start":"01:09.090 ","End":"01:11.930","Text":"Here, 3 minus 3 times 1 is 0,"},{"Start":"01:11.930 ","End":"01:14.765","Text":"1 minus 3 times 2 minus 5, and so on."},{"Start":"01:14.765 ","End":"01:16.520","Text":"We\u0027ve got the 2 0s here."},{"Start":"01:16.520 ","End":"01:20.060","Text":"Now I want to try and get the 0 here,"},{"Start":"01:20.060 ","End":"01:27.440","Text":"but I also notice that I can simplify by dividing by minus 5 here."},{"Start":"01:27.440 ","End":"01:30.980","Text":"If I take minus a 1/5 of the third row,"},{"Start":"01:30.980 ","End":"01:34.010","Text":"then we get the same first 2 rows,"},{"Start":"01:34.010 ","End":"01:41.070","Text":"but the last row becomes simpler smaller numbers 1,"},{"Start":"01:41.070 ","End":"01:43.210","Text":"2, and minus 7."},{"Start":"01:43.210 ","End":"01:49.420","Text":"At this point, if I add the second row to the third row, that will make this 0."},{"Start":"01:49.420 ","End":"01:57.275","Text":"Third plus the second put into the third and we get this matrix,"},{"Start":"01:57.275 ","End":"01:59.705","Text":"which is now good."},{"Start":"01:59.705 ","End":"02:01.775","Text":"We\u0027ve got 0s here."},{"Start":"02:01.775 ","End":"02:05.270","Text":"Let me just copy this and go on to the next page."},{"Start":"02:05.270 ","End":"02:06.920","Text":"Here we are just copied."},{"Start":"02:06.920 ","End":"02:08.570","Text":"It didn\u0027t change anything,"},{"Start":"02:08.570 ","End":"02:14.270","Text":"divide by minus 5 or even not really necessary."},{"Start":"02:14.270 ","End":"02:18.350","Text":"We could just straight away go back to x,"},{"Start":"02:18.350 ","End":"02:20.240","Text":"y, z from here."},{"Start":"02:20.240 ","End":"02:24.020","Text":"We\u0027ll have to do the division by minus 5 at some point and we can do it at"},{"Start":"02:24.020 ","End":"02:29.390","Text":"this stage already so that z becomes minus 2 and once we have z,"},{"Start":"02:29.390 ","End":"02:32.420","Text":"then everything else follows because if z is minus 2,"},{"Start":"02:32.420 ","End":"02:36.150","Text":"then we can plug that in here and get"},{"Start":"02:36.970 ","End":"02:45.410","Text":"minus y plus 14 is 17."},{"Start":"02:45.410 ","End":"02:51.680","Text":"Y has to be minus 3 and then plug z and y into here and we\u0027ll get x."},{"Start":"02:51.680 ","End":"02:54.400","Text":"That\u0027s just 1 unique solution."},{"Start":"02:54.400 ","End":"02:58.055","Text":"X is 1, y is minus 3, z is minus 2."},{"Start":"02:58.055 ","End":"02:59.840","Text":"Probably I should have just put them in order."},{"Start":"02:59.840 ","End":"03:02.210","Text":"Some people like to put x first, then y, then z."},{"Start":"03:02.210 ","End":"03:05.550","Text":"It doesn\u0027t really matter. We\u0027re done."}],"ID":9820},{"Watched":false,"Name":"Exercise 11","Duration":"3m 20s","ChapterTopicVideoID":9467,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9467.jpeg","UploadDate":"2017-07-26T08:21:44.6970000","DurationForVideoObject":"PT3M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:04.050","Text":"In this exercise, we have a system of linear equations,"},{"Start":"00:04.050 ","End":"00:06.930","Text":"3 equations and 3 unknowns, x, y, and z,"},{"Start":"00:06.930 ","End":"00:12.810","Text":"and we\u0027re going to try and solve it using augmented matrix techniques."},{"Start":"00:12.810 ","End":"00:14.970","Text":"If we can\u0027t solve it,"},{"Start":"00:14.970 ","End":"00:17.325","Text":"then we\u0027ll say it\u0027s inconsistent."},{"Start":"00:17.325 ","End":"00:20.010","Text":"The first step is to get the augmented matrix,"},{"Start":"00:20.010 ","End":"00:23.820","Text":"which is basically just copying the numbers here into this special form"},{"Start":"00:23.820 ","End":"00:25.920","Text":"and getting rid of the variables."},{"Start":"00:25.920 ","End":"00:30.285","Text":"Now we do row operations on this matrix."},{"Start":"00:30.285 ","End":"00:34.469","Text":"I want to try and get rid of these 2, make them 0."},{"Start":"00:34.469 ","End":"00:37.650","Text":"What I can do is I can get all these 3 to be equal"},{"Start":"00:37.650 ","End":"00:45.860","Text":"if I multiply by, well, first look at the common denominator, that would be 30."},{"Start":"00:45.860 ","End":"00:48.050","Text":"Then I can multiply this by 3, this by 10,"},{"Start":"00:48.050 ","End":"00:49.625","Text":"and this by 15,"},{"Start":"00:49.625 ","End":"00:51.470","Text":"and I write that in shorthand this way,"},{"Start":"00:51.470 ","End":"00:54.785","Text":"15 times row 1 into row 1 and so on,"},{"Start":"00:54.785 ","End":"00:57.760","Text":"and then I should get them all to be 30."},{"Start":"00:57.760 ","End":"00:59.750","Text":"After I do these row operations,"},{"Start":"00:59.750 ","End":"01:00.760","Text":"this is what I get,"},{"Start":"01:00.760 ","End":"01:06.710","Text":"and now I can subtract the first equation from the second"},{"Start":"01:06.710 ","End":"01:09.005","Text":"and then the first equation from the third,"},{"Start":"01:09.005 ","End":"01:12.260","Text":"and this is how we write that in terms of row operations,"},{"Start":"01:12.260 ","End":"01:15.770","Text":"and then I should get 0 both here and here."},{"Start":"01:15.770 ","End":"01:17.540","Text":"Let\u0027s see what we do get."},{"Start":"01:17.540 ","End":"01:25.865","Text":"Indeed, 30 minus 30 is 0, minus 20 minus minus 15 is minus 5, and so on."},{"Start":"01:25.865 ","End":"01:29.340","Text":"This minus this gives me this."},{"Start":"01:29.620 ","End":"01:34.670","Text":"Looking at this, I see I can do a bit of reduction."},{"Start":"01:34.670 ","End":"01:40.655","Text":"I can divide the top row by 15,"},{"Start":"01:40.655 ","End":"01:45.960","Text":"and this row can be divided by 5."},{"Start":"01:45.960 ","End":"01:47.000","Text":"Everything is divisible by 5,"},{"Start":"01:47.000 ","End":"01:49.670","Text":"and the last row, I can divide everything by 3."},{"Start":"01:49.670 ","End":"01:53.765","Text":"Actually, I could divide this by minus 3 and minus 5 and get 1s here."},{"Start":"01:53.765 ","End":"01:59.940","Text":"If I do that, then I\u0027ve got this matrix."},{"Start":"01:59.940 ","End":"02:02.370","Text":"I\u0027m not all the way there."},{"Start":"02:02.370 ","End":"02:04.005","Text":"I\u0027d like a 0 here as well."},{"Start":"02:04.005 ","End":"02:08.330","Text":"I\u0027ll just subtract the 2nd row from the 3rd row,"},{"Start":"02:08.330 ","End":"02:11.000","Text":"I need to move to a new page."},{"Start":"02:11.000 ","End":"02:13.880","Text":"Here we are, just the same."},{"Start":"02:13.880 ","End":"02:17.030","Text":"Just copied this last matrix here."},{"Start":"02:17.030 ","End":"02:20.630","Text":"As I was saying, we are going to subtract this one from this one,"},{"Start":"02:20.630 ","End":"02:24.800","Text":"and then we\u0027ll get a 0 here."},{"Start":"02:24.800 ","End":"02:26.660","Text":"This 1 minus 1 is 0."},{"Start":"02:26.660 ","End":"02:32.490","Text":"This minus this is 0, minus 7 minus 5 is minus 12,"},{"Start":"02:32.510 ","End":"02:37.850","Text":"and here we have a row where all these are 0, but this is not 0."},{"Start":"02:37.850 ","End":"02:41.675","Text":"If I translate this back into x, y, z,"},{"Start":"02:41.675 ","End":"02:45.050","Text":"the first two don\u0027t look problematic,"},{"Start":"02:45.050 ","End":"02:49.190","Text":"but the last one, 0x plus 0y plus 0z, all that is just 0."},{"Start":"02:49.190 ","End":"02:51.490","Text":"We\u0027ve got 0 is minus 12,"},{"Start":"02:51.490 ","End":"02:53.475","Text":"and it doesn\u0027t matter what these two say,"},{"Start":"02:53.475 ","End":"02:55.845","Text":"0 cannot be minus 12."},{"Start":"02:55.845 ","End":"02:57.565","Text":"When you get something like this,"},{"Start":"02:57.565 ","End":"02:59.010","Text":"there\u0027s no solution,"},{"Start":"02:59.010 ","End":"03:03.035","Text":"and as far as the original question went,"},{"Start":"03:03.035 ","End":"03:06.620","Text":"we can state that yes, this linear system of equations"},{"Start":"03:06.620 ","End":"03:09.965","Text":"or even the original one was inconsistent."},{"Start":"03:09.965 ","End":"03:11.825","Text":"I\u0027ll just write the word."},{"Start":"03:11.825 ","End":"03:17.750","Text":"The system of linear equations was inconsistent because we got no solutions."},{"Start":"03:17.750 ","End":"03:19.740","Text":"We\u0027re done."}],"ID":9821},{"Watched":false,"Name":"Exercise 12","Duration":"3m 21s","ChapterTopicVideoID":9468,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9468.jpeg","UploadDate":"2017-07-26T08:21:56.2430000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have here a system of linear equations,"},{"Start":"00:04.320 ","End":"00:08.010","Text":"3 equations with 3 unknowns, x, y, z."},{"Start":"00:08.010 ","End":"00:12.015","Text":"I\u0027m going to try and solve it using augmented matrix techniques."},{"Start":"00:12.015 ","End":"00:15.330","Text":"The first thing to do is to write the augmented matrix,"},{"Start":"00:15.330 ","End":"00:19.920","Text":"which is just basically the numbers copied from here in this special format."},{"Start":"00:19.920 ","End":"00:24.180","Text":"Now what we do is row operations on this matrix with"},{"Start":"00:24.180 ","End":"00:30.855","Text":"the idea of trying to get 0 here and here and later here also."},{"Start":"00:30.855 ","End":"00:34.410","Text":"The way to get 0 here and here in"},{"Start":"00:34.410 ","End":"00:38.760","Text":"row operations is to subtract 4 times this row from this row,"},{"Start":"00:38.760 ","End":"00:42.165","Text":"and then subtract 3 times this row from this row."},{"Start":"00:42.165 ","End":"00:45.170","Text":"This a special shorthand way of writing just what I"},{"Start":"00:45.170 ","End":"00:48.905","Text":"said and should be familiar with this."},{"Start":"00:48.905 ","End":"00:51.140","Text":"Anyway, after we do this operation,"},{"Start":"00:51.140 ","End":"00:55.220","Text":"we get the following matrix with 0s here and here."},{"Start":"00:55.220 ","End":"00:58.880","Text":"Just for example, 4 minus 4 times 1 is 0,"},{"Start":"00:58.880 ","End":"01:03.215","Text":"6 minus 4 times 2 is minus 2 and so on."},{"Start":"01:03.215 ","End":"01:11.165","Text":"17 minus 3 times 3 is 8, and so on."},{"Start":"01:11.165 ","End":"01:14.570","Text":"Now we can reduce the numbers a bit."},{"Start":"01:14.570 ","End":"01:21.350","Text":"If we divide this row by minus 2 and divide the last row by minus 4,"},{"Start":"01:21.350 ","End":"01:25.310","Text":"then what we get is a bit simpler."},{"Start":"01:25.310 ","End":"01:29.374","Text":"We got something very similar but in smaller numbers."},{"Start":"01:29.374 ","End":"01:32.270","Text":"Now we can easily get 0 here,"},{"Start":"01:32.270 ","End":"01:36.080","Text":"which is what we want by subtracting the second row from the third row,"},{"Start":"01:36.080 ","End":"01:38.450","Text":"which we write in shorthand this way."},{"Start":"01:38.450 ","End":"01:40.250","Text":"After we do that,"},{"Start":"01:40.250 ","End":"01:44.095","Text":"we get this augmented matrix."},{"Start":"01:44.095 ","End":"01:48.085","Text":"Note that all the last row is 0s."},{"Start":"01:48.085 ","End":"01:50.765","Text":"Moving to a new page."},{"Start":"01:50.765 ","End":"01:54.560","Text":"This is the same matrix just copied."},{"Start":"01:54.560 ","End":"02:00.049","Text":"If we now interpret this in terms of x and y,"},{"Start":"02:00.049 ","End":"02:02.390","Text":"the last row with 0s doesn\u0027t give anything."},{"Start":"02:02.390 ","End":"02:05.095","Text":"It basically says 0 equals 0."},{"Start":"02:05.095 ","End":"02:07.485","Text":"It doesn\u0027t help anything."},{"Start":"02:07.485 ","End":"02:13.555","Text":"We\u0027ve got x plus 2y plus 3z is 3 and y minus 2z is 2."},{"Start":"02:13.555 ","End":"02:18.545","Text":"Now, z is like a free variable."},{"Start":"02:18.545 ","End":"02:21.695","Text":"I can assign anything I want to Z."},{"Start":"02:21.695 ","End":"02:25.520","Text":"For example, I could let z be a parameter t where"},{"Start":"02:25.520 ","End":"02:29.480","Text":"it\u0027s understood that t is any arbitrary number."},{"Start":"02:29.480 ","End":"02:33.440","Text":"Once I have z, then I can compute y from"},{"Start":"02:33.440 ","End":"02:38.780","Text":"the 2nd equation because I get that y equals 2z plus 2 and z is t,"},{"Start":"02:38.780 ","End":"02:40.340","Text":"so it\u0027s 2t plus 2."},{"Start":"02:40.340 ","End":"02:44.335","Text":"Then I take z and y and substitute them here and here."},{"Start":"02:44.335 ","End":"02:49.710","Text":"I get that x equals 3 minus 2y minus 3z and so on and so on."},{"Start":"02:49.710 ","End":"02:51.660","Text":"I get what x equals."},{"Start":"02:51.660 ","End":"02:54.335","Text":"Then just to tidy things up,"},{"Start":"02:54.335 ","End":"02:55.610","Text":"I write what x, y,"},{"Start":"02:55.610 ","End":"02:59.450","Text":"and z are in terms of the right order,"},{"Start":"02:59.450 ","End":"03:04.220","Text":"minus 1 minus 7t 2t plus 2 and then t."},{"Start":"03:04.220 ","End":"03:11.420","Text":"This means that there is an infinite number of solutions"},{"Start":"03:11.420 ","End":"03:15.035","Text":"and just something to observe."},{"Start":"03:15.035 ","End":"03:21.570","Text":"Sometimes that happens. We\u0027re done."}],"ID":9822},{"Watched":false,"Name":"Exercise 13","Duration":"2m 26s","ChapterTopicVideoID":9469,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9469.jpeg","UploadDate":"2017-07-26T08:22:04.7230000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"Here we have an exercise,"},{"Start":"00:02.310 ","End":"00:06.390","Text":"and we\u0027re given a system of linear equations."},{"Start":"00:06.390 ","End":"00:10.770","Text":"As it turns out, it\u0027s 3 equations and 2 unknowns,"},{"Start":"00:10.770 ","End":"00:15.570","Text":"which is unusual, but it happens sometimes."},{"Start":"00:15.570 ","End":"00:19.565","Text":"Let\u0027s use augmented matrix techniques as asked."},{"Start":"00:19.565 ","End":"00:22.684","Text":"First thing to do is to get the augmented matrix,"},{"Start":"00:22.684 ","End":"00:27.335","Text":"which is just basically putting the numbers here in the right format,"},{"Start":"00:27.335 ","End":"00:32.420","Text":"1, 3, 2, 2, 1, minus 1, the vertical line here."},{"Start":"00:32.420 ","End":"00:42.930","Text":"Then we do row operations on this matrix to try and get 0s here and here at first."},{"Start":"00:42.930 ","End":"00:45.030","Text":"Then we\u0027ll stop by that."},{"Start":"00:45.030 ","End":"00:50.690","Text":"If we take the second row and subtract twice the first row,"},{"Start":"00:50.690 ","End":"00:52.460","Text":"that will give us 0 here."},{"Start":"00:52.460 ","End":"00:56.375","Text":"If we subtract the first row from the third row,"},{"Start":"00:56.375 ","End":"00:58.540","Text":"then we\u0027ll also get 0 here."},{"Start":"00:58.540 ","End":"00:59.930","Text":"If we do the computation,"},{"Start":"00:59.930 ","End":"01:02.130","Text":"this is what we get."},{"Start":"01:03.820 ","End":"01:06.430","Text":"When we see there\u0027s a common factor,"},{"Start":"01:06.430 ","End":"01:09.775","Text":"it\u0027s often a good idea to get smaller numbers."},{"Start":"01:09.775 ","End":"01:14.320","Text":"We\u0027ll take the second row and divide it by minus 5,"},{"Start":"01:14.320 ","End":"01:17.740","Text":"and we\u0027ll take the third row and divide it by minus 4."},{"Start":"01:17.740 ","End":"01:19.930","Text":"Then we get smaller numbers."},{"Start":"01:19.930 ","End":"01:22.345","Text":"This is what we get, in fact."},{"Start":"01:22.345 ","End":"01:29.160","Text":"Now I\u0027d like to get a 0 here also."},{"Start":"01:29.160 ","End":"01:32.275","Text":"I\u0027ll just subtract this row from this row."},{"Start":"01:32.275 ","End":"01:38.395","Text":"In other words, our row 3 minus row 2 put it into row 3."},{"Start":"01:38.395 ","End":"01:40.059","Text":"When we do that,"},{"Start":"01:40.059 ","End":"01:43.090","Text":"everything comes out 0 in the last row."},{"Start":"01:43.090 ","End":"01:47.075","Text":"This is like a third equation that we can throw out."},{"Start":"01:47.075 ","End":"01:49.670","Text":"When we go back into x and y,"},{"Start":"01:49.670 ","End":"01:53.720","Text":"we can just ignore it because it basically says 0x plus 0y equals 0,"},{"Start":"01:53.720 ","End":"01:57.970","Text":"which just doesn\u0027t tell us anything."},{"Start":"01:59.030 ","End":"02:01.080","Text":"Now we have this,"},{"Start":"02:01.080 ","End":"02:04.320","Text":"and this actually turns out to be 2 equations and 2 unknowns,"},{"Start":"02:04.320 ","End":"02:06.745","Text":"and we already have the value of y."},{"Start":"02:06.745 ","End":"02:11.915","Text":"All we have to do is substitute y in the first equation,"},{"Start":"02:11.915 ","End":"02:16.730","Text":"and then we get x plus 3 times 1 is 2,"},{"Start":"02:16.730 ","End":"02:19.220","Text":"which gives us that x is minus 1."},{"Start":"02:19.220 ","End":"02:20.660","Text":"We\u0027ve solved it,"},{"Start":"02:20.660 ","End":"02:22.340","Text":"and there is a single solution,"},{"Start":"02:22.340 ","End":"02:24.560","Text":"and x is minus 1, y is 1,"},{"Start":"02:24.560 ","End":"02:26.580","Text":"and we\u0027re done."}],"ID":9823},{"Watched":false,"Name":"Exercise 14","Duration":"1m 58s","ChapterTopicVideoID":9470,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9470.jpeg","UploadDate":"2017-07-26T08:22:18.2930000","DurationForVideoObject":"PT1M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.590 ","End":"00:04.620","Text":"We have here a system of linear equations."},{"Start":"00:04.620 ","End":"00:08.910","Text":"In fact, we have 3 equations in only 2 unknowns."},{"Start":"00:08.910 ","End":"00:10.860","Text":"Let\u0027s see what happens."},{"Start":"00:10.860 ","End":"00:14.640","Text":"We\u0027re going to use augmented matrix techniques as asked,"},{"Start":"00:14.640 ","End":"00:18.765","Text":"and see if we can find a solution."},{"Start":"00:18.765 ","End":"00:22.320","Text":"First thing to do is to write the augmented matrix,"},{"Start":"00:22.320 ","End":"00:26.685","Text":"which is just basically copying the numbers into this special format."},{"Start":"00:26.685 ","End":"00:30.870","Text":"Then we start doing row operations on this"},{"Start":"00:30.870 ","End":"00:35.430","Text":"to try and get first of all 0s here and here."},{"Start":"00:35.430 ","End":"00:44.575","Text":"If we subtract twice this from this and add 4 times this to this."},{"Start":"00:44.575 ","End":"00:51.050","Text":"In other words, this is the code for row 2 minus twice row 1,"},{"Start":"00:51.050 ","End":"00:53.780","Text":"put that into row 2, and so on."},{"Start":"00:53.780 ","End":"00:58.605","Text":"Then we get 0 here and 0 here."},{"Start":"00:58.605 ","End":"01:01.190","Text":"We also have 0 here, for example,"},{"Start":"01:01.190 ","End":"01:07.075","Text":"minus 14 minus twice, minus 7 is 0 and so on."},{"Start":"01:07.075 ","End":"01:09.330","Text":"Row 3 plus 4, row 1,"},{"Start":"01:09.330 ","End":"01:11.300","Text":"so this plus 4 times this is 0,"},{"Start":"01:11.300 ","End":"01:13.280","Text":"but also this plus 4 times this is 0."},{"Start":"01:13.280 ","End":"01:17.000","Text":"In fact we get all 0s in the last row."},{"Start":"01:17.000 ","End":"01:19.130","Text":"When we have that, we just throw it out"},{"Start":"01:19.130 ","End":"01:22.730","Text":"because it just says that 0x plus 0y is 0,"},{"Start":"01:22.730 ","End":"01:24.520","Text":"so we don\u0027t need that."},{"Start":"01:24.520 ","End":"01:27.979","Text":"When we go back to x and y,"},{"Start":"01:27.979 ","End":"01:32.759","Text":"we just take the top 2, 4x minus 7y is 0."},{"Start":"01:32.759 ","End":"01:34.415","Text":"But, more interestingly,"},{"Start":"01:34.415 ","End":"01:36.290","Text":"this 1 says 0 equals 2,"},{"Start":"01:36.290 ","End":"01:38.120","Text":"and that\u0027s an impossibility."},{"Start":"01:38.120 ","End":"01:40.160","Text":"It doesn\u0027t matter what the first equation is."},{"Start":"01:40.160 ","End":"01:42.140","Text":"We cannot have 0 equals 2,"},{"Start":"01:42.140 ","End":"01:44.150","Text":"so there is no solution."},{"Start":"01:44.150 ","End":"01:47.645","Text":"That\u0027s for the original system of equations."},{"Start":"01:47.645 ","End":"01:48.980","Text":"Then when this happens,"},{"Start":"01:48.980 ","End":"01:52.530","Text":"we say that it was inconsistent."},{"Start":"01:53.870 ","End":"01:58.360","Text":"No solution, and we\u0027re done."}],"ID":9824},{"Watched":false,"Name":"Exercise 15","Duration":"2m 11s","ChapterTopicVideoID":9471,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9471.jpeg","UploadDate":"2017-07-26T08:22:34.6700000","DurationForVideoObject":"PT2M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.485","Text":"In this exercise, we have a system of linear equations."},{"Start":"00:04.485 ","End":"00:07.775","Text":"In fact, 3 equations and 2 unknowns."},{"Start":"00:07.775 ","End":"00:10.650","Text":"We\u0027re going to use augmented matrix techniques"},{"Start":"00:10.650 ","End":"00:14.895","Text":"to try and solve this, if we can, if it\u0027s consistent."},{"Start":"00:14.895 ","End":"00:17.910","Text":"The first thing is to get the augmented matrix,"},{"Start":"00:17.910 ","End":"00:20.070","Text":"which is just copying the coefficients,"},{"Start":"00:20.070 ","End":"00:22.200","Text":"the numbers here, into this special form."},{"Start":"00:22.200 ","End":"00:25.215","Text":"Then we start to do row operations."},{"Start":"00:25.215 ","End":"00:28.290","Text":"First of all, we want to try and get 0 here and here."},{"Start":"00:28.290 ","End":"00:34.720","Text":"If I add 3 times this row to this row and subtract twice this row from this row,"},{"Start":"00:35.000 ","End":"00:44.835","Text":"what it says here, we take row 2 plus 3 times row 1 and put that into row 2."},{"Start":"00:44.835 ","End":"00:47.340","Text":"In short, you\u0027ve seen this thing before."},{"Start":"00:47.340 ","End":"00:52.560","Text":"What we get is 0 here."},{"Start":"00:52.560 ","End":"00:55.691","Text":"But this is interesting, we also get 0 here."},{"Start":"00:55.691 ","End":"00:56.540","Text":"Let\u0027s just check that."},{"Start":"00:56.540 ","End":"01:00.485","Text":"Suppose I want to take row 3 minus twice row 1."},{"Start":"01:00.485 ","End":"01:02.660","Text":"6 minus twice 3 is 0,"},{"Start":"01:02.660 ","End":"01:06.560","Text":"minus 4, minus twice, minus 2 is also 0"},{"Start":"01:06.560 ","End":"01:09.545","Text":"and 2 minus twice 1 is 0."},{"Start":"01:09.545 ","End":"01:14.050","Text":"Basically 2 equations drop out."},{"Start":"01:14.050 ","End":"01:20.550","Text":"All we\u0027re left with is the top one, 3x minus 2y is 1,"},{"Start":"01:20.550 ","End":"01:23.640","Text":"and we have 1 equation and 2 unknowns."},{"Start":"01:23.640 ","End":"01:27.455","Text":"Basically, we can let y be anything we want."},{"Start":"01:27.455 ","End":"01:30.290","Text":"Usually, we assign it to be a parameter,"},{"Start":"01:30.290 ","End":"01:32.450","Text":"t is a popular variable name."},{"Start":"01:32.450 ","End":"01:35.480","Text":"So we let y equal t."},{"Start":"01:35.480 ","End":"01:40.765","Text":"If y equals t, then we get 3x minus 2t is 1,"},{"Start":"01:40.765 ","End":"01:47.115","Text":"and then x is 1 plus 2t/3."},{"Start":"01:47.115 ","End":"01:51.110","Text":"Then we just package it by saying what is x, y."},{"Start":"01:51.110 ","End":"01:54.580","Text":"We copy x from here and y from here."},{"Start":"01:54.580 ","End":"01:56.970","Text":"This is the answer."},{"Start":"01:56.970 ","End":"02:01.025","Text":"But notice that we have an infinite number of solutions"},{"Start":"02:01.025 ","End":"02:04.250","Text":"because each value of t gives us a solution"},{"Start":"02:04.250 ","End":"02:08.555","Text":"and we can let t be anything, 0, 1, 2, whatever you want."},{"Start":"02:08.555 ","End":"02:12.000","Text":"Infinite solutions and we\u0027re done."}],"ID":9825},{"Watched":false,"Name":"Exercise 16","Duration":"4m 18s","ChapterTopicVideoID":9472,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9472.jpeg","UploadDate":"2017-07-26T08:22:53.6330000","DurationForVideoObject":"PT4M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have a system of linear equations."},{"Start":"00:04.890 ","End":"00:07.050","Text":"It\u0027s a bit longer than usual."},{"Start":"00:07.050 ","End":"00:10.830","Text":"It\u0027s 4 equations with 3 unknowns."},{"Start":"00:10.830 ","End":"00:13.950","Text":"We\u0027re going to use augmented matrix techniques."},{"Start":"00:13.950 ","End":"00:17.220","Text":"That way we can just work with numbers and not drag x,"},{"Start":"00:17.220 ","End":"00:19.180","Text":"y, and z around everywhere."},{"Start":"00:19.180 ","End":"00:22.380","Text":"The first thing we do is just to write the augmented matrix,"},{"Start":"00:22.380 ","End":"00:24.270","Text":"which is just essentially the coefficients,"},{"Start":"00:24.270 ","End":"00:25.725","Text":"the numbers from here."},{"Start":"00:25.725 ","End":"00:27.960","Text":"This is what it looks like."},{"Start":"00:27.960 ","End":"00:31.665","Text":"Then we\u0027re going to start doing row operations."},{"Start":"00:31.665 ","End":"00:35.480","Text":"The first goal is to try and get 0\u0027s here,"},{"Start":"00:35.480 ","End":"00:39.890","Text":"here and here by subtracting multiples of this row."},{"Start":"00:39.890 ","End":"00:43.490","Text":"If I take this row and subtract 3 times this row and"},{"Start":"00:43.490 ","End":"00:46.910","Text":"this row minus twice this row and so on and we"},{"Start":"00:46.910 ","End":"00:53.900","Text":"write that in this code like the fourth row minus twice the first row,"},{"Start":"00:53.900 ","End":"00:56.750","Text":"and put that into the fourth row."},{"Start":"00:56.750 ","End":"01:00.965","Text":"If we do all these computations and you should be familiar with this thing by now."},{"Start":"01:00.965 ","End":"01:02.360","Text":"Then this is what we get."},{"Start":"01:02.360 ","End":"01:06.060","Text":"Indeed we have 0\u0027s here, here and here."},{"Start":"01:06.160 ","End":"01:09.305","Text":"Now let\u0027s see what else we can do."},{"Start":"01:09.305 ","End":"01:18.295","Text":"Well, we want to try and get 0\u0027s also here."},{"Start":"01:18.295 ","End":"01:19.390","Text":"But first of all,"},{"Start":"01:19.390 ","End":"01:21.745","Text":"we can do a bit of reducing,"},{"Start":"01:21.745 ","End":"01:27.050","Text":"like dividing the last row by 4 just to make the numbers a bit smaller."},{"Start":"01:28.160 ","End":"01:31.840","Text":"We can get this last row like this,"},{"Start":"01:31.840 ","End":"01:35.635","Text":"but there\u0027s nothing much to do for the middle 2 rows."},{"Start":"01:35.635 ","End":"01:39.005","Text":"I\u0027m going to copy this onto a new page."},{"Start":"01:39.005 ","End":"01:47.770","Text":"Here we are. The next thing we\u0027re going to do is to exchange."},{"Start":"01:47.770 ","End":"01:52.060","Text":"This is something that is not commonly seen to exchange 2 rows."},{"Start":"01:52.060 ","End":"01:56.925","Text":"The reason I\u0027m exchanging row 2 and row 4 is when I have a 1 here."},{"Start":"01:56.925 ","End":"02:00.590","Text":"As we\u0027ve seen, it\u0027s much easier to make 0\u0027s below it."},{"Start":"02:00.590 ","End":"02:01.730","Text":"If I do that,"},{"Start":"02:01.730 ","End":"02:05.060","Text":"I\u0027ve just changed the order of these 2 rows."},{"Start":"02:05.060 ","End":"02:07.265","Text":"First and the third remain unchanged."},{"Start":"02:07.265 ","End":"02:09.350","Text":"Then I can start saying,"},{"Start":"02:09.350 ","End":"02:11.540","Text":"I\u0027ll add 9 times this row here,"},{"Start":"02:11.540 ","End":"02:13.595","Text":"and 8 times this row here,"},{"Start":"02:13.595 ","End":"02:17.000","Text":"which is how I write it in shorthand."},{"Start":"02:17.000 ","End":"02:18.680","Text":"After we do that,"},{"Start":"02:18.680 ","End":"02:20.490","Text":"we should get 0\u0027s here and here."},{"Start":"02:20.490 ","End":"02:23.375","Text":"In fact, this is what we get."},{"Start":"02:23.375 ","End":"02:27.530","Text":"At this point, I see that I can do some reduction."},{"Start":"02:27.530 ","End":"02:31.100","Text":"This can be divided by 17 and this by 9,"},{"Start":"02:31.100 ","End":"02:33.540","Text":"and if I do that,"},{"Start":"02:36.700 ","End":"02:43.295","Text":"I got the same matrix twice."},{"Start":"02:43.295 ","End":"02:44.915","Text":"Oh, that doesn\u0027t hurt."},{"Start":"02:44.915 ","End":"02:48.935","Text":"This one\u0027s redundant and let\u0027s just move it out the way."},{"Start":"02:48.935 ","End":"02:51.080","Text":"Yeah, as I was saying,"},{"Start":"02:51.080 ","End":"02:56.155","Text":"we\u0027re going to divide this by 17 and this by 9."},{"Start":"02:56.155 ","End":"02:58.755","Text":"In shorthand, this is how we write it."},{"Start":"02:58.755 ","End":"03:00.065","Text":"After we do that,"},{"Start":"03:00.065 ","End":"03:02.540","Text":"we get much simpler."},{"Start":"03:02.540 ","End":"03:05.355","Text":"We get a 1 here and a 1 here."},{"Start":"03:05.355 ","End":"03:09.045","Text":"Now, I\u0027d like to get a 0 here."},{"Start":"03:09.045 ","End":"03:12.800","Text":"If I just subtract this row from this row,"},{"Start":"03:12.800 ","End":"03:15.095","Text":"which we write like this,"},{"Start":"03:15.095 ","End":"03:20.115","Text":"then what we get is this matrix."},{"Start":"03:20.115 ","End":"03:23.650","Text":"I notice that the last row is all 0\u0027s,"},{"Start":"03:23.650 ","End":"03:26.870","Text":"which means that we can just throw it out and ignore it."},{"Start":"03:26.870 ","End":"03:29.450","Text":"I need to move on to a new page again."},{"Start":"03:29.450 ","End":"03:32.040","Text":"As I was saying, we have this row of 0\u0027s."},{"Start":"03:32.040 ","End":"03:34.745","Text":"When we go back to x, y, and z,"},{"Start":"03:34.745 ","End":"03:39.680","Text":"ignore that and we get x plus 2y plus 2z is 2 and so on."},{"Start":"03:39.680 ","End":"03:42.710","Text":"We have it in a nice triangular form."},{"Start":"03:42.710 ","End":"03:44.240","Text":"Once we have it in this form,"},{"Start":"03:44.240 ","End":"03:47.460","Text":"it\u0027s easy because we have what z is,"},{"Start":"03:47.460 ","End":"03:50.180","Text":"then we substitute in here and solve for y."},{"Start":"03:50.180 ","End":"03:52.190","Text":"Then we substitute y and z here,"},{"Start":"03:52.190 ","End":"03:54.085","Text":"and we substitute for x."},{"Start":"03:54.085 ","End":"03:58.100","Text":"What we get is z is minus 1 I just copied."},{"Start":"03:58.100 ","End":"04:00.710","Text":"Then here we have y minus 2,"},{"Start":"04:00.710 ","End":"04:03.920","Text":"is minus 1. so why is 1."},{"Start":"04:03.920 ","End":"04:08.180","Text":"If I put z is minus 1 and y equals 1 here,"},{"Start":"04:08.180 ","End":"04:10.760","Text":"and compute it, we get x equals 2,"},{"Start":"04:10.760 ","End":"04:12.800","Text":"and that\u0027s the answer."},{"Start":"04:12.800 ","End":"04:18.900","Text":"We got 1 single unique solution and this is it. We are done."}],"ID":9826},{"Watched":false,"Name":"Exercise 17 part a","Duration":"7m 15s","ChapterTopicVideoID":9473,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9473.jpeg","UploadDate":"2017-07-26T08:23:30.8330000","DurationForVideoObject":"PT7M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, I\u0027m going to assume you studied"},{"Start":"00:03.510 ","End":"00:09.060","Text":"the complex numbers which are denoted by this special C,"},{"Start":"00:09.060 ","End":"00:15.690","Text":"as opposed to the real numbers which are written as R. If you haven\u0027t,"},{"Start":"00:15.690 ","End":"00:18.600","Text":"you might want to skip this exercise or you\u0027re welcome to stay."},{"Start":"00:18.600 ","End":"00:25.950","Text":"The essential thing about complex numbers is it has the form a plus bi,"},{"Start":"00:25.950 ","End":"00:33.750","Text":"where a and b are real numbers and i has the special quality that i squared is minus 1."},{"Start":"00:33.750 ","End":"00:36.540","Text":"Actually it\u0027s i for imaginary."},{"Start":"00:36.540 ","End":"00:39.420","Text":"That\u0027s the main thing you need to know."},{"Start":"00:39.420 ","End":"00:45.360","Text":"Anyway, we have here 3 equations and 3 unknowns."},{"Start":"00:45.360 ","End":"00:50.240","Text":"We use the letter z usually for complex numbers,"},{"Start":"00:50.240 ","End":"00:53.750","Text":"whereas x is usually for real numbers."},{"Start":"00:53.750 ","End":"00:56.045","Text":"That\u0027s just a convention."},{"Start":"00:56.045 ","End":"00:59.784","Text":"Let\u0027s start and we\u0027ll use matrix form."},{"Start":"00:59.784 ","End":"01:02.765","Text":"Here\u0027s the augmented matrix."},{"Start":"01:02.765 ","End":"01:05.030","Text":"To the left of the separator,"},{"Start":"01:05.030 ","End":"01:11.220","Text":"the coefficients, and to the right or numbers on the right-hand side, the constants."},{"Start":"01:11.600 ","End":"01:14.420","Text":"We\u0027re going to use Gauss elimination,"},{"Start":"01:14.420 ","End":"01:17.680","Text":"which is bringing to row echelon form."},{"Start":"01:17.680 ","End":"01:20.355","Text":"It\u0027s nice when you have a 1 here,"},{"Start":"01:20.355 ","End":"01:23.700","Text":"It\u0027s easier to 0 out the rest of the column."},{"Start":"01:23.700 ","End":"01:27.420","Text":"Subtract i times the first row from the"},{"Start":"01:27.420 ","End":"01:33.565","Text":"second and minus 1 plus 3i times the first row from the 3rd."},{"Start":"01:33.565 ","End":"01:37.450","Text":"This is what I just said in condensed row notation form."},{"Start":"01:37.450 ","End":"01:41.790","Text":"This is the result, I\u0027m not going to go through the calculations 1 by 1."},{"Start":"01:41.790 ","End":"01:43.605","Text":"Let\u0027s just take an example."},{"Start":"01:43.605 ","End":"01:48.215","Text":"This entry here is gotten by"},{"Start":"01:48.215 ","End":"01:54.435","Text":"taking row 3 minus this times row 1."},{"Start":"01:54.435 ","End":"01:58.500","Text":"I\u0027ve got this, which is this part here,"},{"Start":"01:58.500 ","End":"02:03.345","Text":"minus the 1 plus 3i from here,"},{"Start":"02:03.345 ","End":"02:06.460","Text":"and this from the first row."},{"Start":"02:06.980 ","End":"02:12.055","Text":"Now, we\u0027ll do some simplification."},{"Start":"02:12.055 ","End":"02:14.365","Text":"Look how it shrunk."},{"Start":"02:14.365 ","End":"02:17.935","Text":"Let\u0027s just take a look at some examples, like here,"},{"Start":"02:17.935 ","End":"02:20.455","Text":"i squared is minus 1,"},{"Start":"02:20.455 ","End":"02:23.410","Text":"so 1 minus i squared is 2."},{"Start":"02:23.410 ","End":"02:28.600","Text":"Here we have 1 plus i minus i plus i"},{"Start":"02:28.600 ","End":"02:37.000","Text":"squared and 1 plus I squared is 1 minus 1 is 0, and everything cancels."},{"Start":"02:37.000 ","End":"02:47.020","Text":"Here, also everything cancels because we have a 3 and we have also a minus 3i squared."},{"Start":"02:50.000 ","End":"02:54.060","Text":"It\u0027s 3 plus 3 is 3 minus 3i squared,"},{"Start":"02:54.060 ","End":"02:56.790","Text":"which is 6 and the minus i."},{"Start":"02:56.790 ","End":"03:00.725","Text":"We also have a minus, minus plus i. Yeah,"},{"Start":"03:00.725 ","End":"03:02.510","Text":"here also the i\u0027s cancel,"},{"Start":"03:02.510 ","End":"03:11.900","Text":"it\u0027s 4i, and then you have to multiply this out."},{"Start":"03:11.900 ","End":"03:18.135","Text":"We get 3i plus i is 4i,"},{"Start":"03:18.135 ","End":"03:20.894","Text":"4i minus 4i is 0."},{"Start":"03:20.894 ","End":"03:25.425","Text":"It\u0027s enough. We get to this."},{"Start":"03:25.425 ","End":"03:30.699","Text":"Now, we still want to get it into echelon form."},{"Start":"03:30.699 ","End":"03:36.210","Text":"Remember, I said many times that I like to cancel if I can,"},{"Start":"03:36.210 ","End":"03:38.510","Text":"like this whole row is divisible by 2,"},{"Start":"03:38.510 ","End":"03:41.300","Text":"this whole row is divisible by 6."},{"Start":"03:41.300 ","End":"03:42.890","Text":"Let\u0027s do the division,"},{"Start":"03:42.890 ","End":"03:47.960","Text":"this is it in row notation that gives us some smaller numbers to work with."},{"Start":"03:47.960 ","End":"03:51.305","Text":"Now, the only thing that stops it from being echelon is we want a 0 here."},{"Start":"03:51.305 ","End":"03:54.605","Text":"Obviously we subtract the 2nd row from the 3rd row."},{"Start":"03:54.605 ","End":"03:57.050","Text":"If I write that in row notation,"},{"Start":"03:57.050 ","End":"04:01.550","Text":"row 3 minus row 2 and put the answer in row 3."},{"Start":"04:01.550 ","End":"04:03.260","Text":"This is what we get,"},{"Start":"04:03.260 ","End":"04:08.800","Text":"this really is in echelon form."},{"Start":"04:08.800 ","End":"04:11.145","Text":"Let\u0027s continue."},{"Start":"04:11.145 ","End":"04:12.850","Text":"Made some space,"},{"Start":"04:12.850 ","End":"04:17.090","Text":"time to go back from the world of matrices to the world of equations."},{"Start":"04:17.090 ","End":"04:20.240","Text":"Where the separators are the equals,"},{"Start":"04:20.240 ","End":"04:24.230","Text":"of course we do not need this last row."},{"Start":"04:24.230 ","End":"04:26.985","Text":"It\u0027s not there."},{"Start":"04:26.985 ","End":"04:33.080","Text":"We at this point in the restricted matrix have 2 rows and 3 columns."},{"Start":"04:33.080 ","End":"04:39.360","Text":"We know we\u0027re not going to get a unique solution or just 1 solution."},{"Start":"04:39.370 ","End":"04:41.825","Text":"Let\u0027s see how that works out."},{"Start":"04:41.825 ","End":"04:46.580","Text":"What we do is we write it z_1 plus iz_2 plus"},{"Start":"04:46.580 ","End":"04:53.455","Text":"1 minus iz_3 is 1 plus 4i and from here just z_2 equals 3."},{"Start":"04:53.455 ","End":"04:55.970","Text":"We\u0027re going to use back substitution."},{"Start":"04:55.970 ","End":"05:04.160","Text":"But before that, the leading term in each row, that\u0027s important."},{"Start":"05:04.160 ","End":"05:06.515","Text":"I put a little box around them."},{"Start":"05:06.515 ","End":"05:11.435","Text":"These depend on the rest of the variables which are free variables."},{"Start":"05:11.435 ","End":"05:14.765","Text":"In other words, z_1 and z_2 are determined,"},{"Start":"05:14.765 ","End":"05:19.190","Text":"but z_3 can be whatever we like it to be."},{"Start":"05:19.190 ","End":"05:23.820","Text":"We usually use the letters like t, u,"},{"Start":"05:23.820 ","End":"05:30.045","Text":"v. Let z_3 be t. As I said,"},{"Start":"05:30.045 ","End":"05:36.020","Text":"we know we can expect infinitely many solutions because we"},{"Start":"05:36.020 ","End":"05:43.980","Text":"have less rows than columns and no contradiction rows."},{"Start":"05:43.980 ","End":"05:47.000","Text":"A contradiction row would be all 0s here and non-zero here."},{"Start":"05:47.000 ","End":"05:49.085","Text":"We don\u0027t have that. It\u0027s infinitely many."},{"Start":"05:49.085 ","End":"05:51.070","Text":"You\u0027ll see why it\u0027s infinitely."},{"Start":"05:51.070 ","End":"05:54.885","Text":"Basically, this is because z_3 can be anything we like."},{"Start":"05:54.885 ","End":"06:00.910","Text":"Then we can find z_2 and z_1 from it again, using back substitution."},{"Start":"06:03.470 ","End":"06:07.620","Text":"Z_2, well there\u0027s no z_3 to substitute, so yeah,"},{"Start":"06:07.620 ","End":"06:14.505","Text":"it\u0027s 3 and then we can substitute z_2 and z_3 in the first 1."},{"Start":"06:14.505 ","End":"06:19.340","Text":"Z_1 is 1 plus 4i minus this, minus this."},{"Start":"06:19.340 ","End":"06:22.380","Text":"But yeah, I would have put z_2,"},{"Start":"06:22.380 ","End":"06:24.570","Text":"but z_2 is 3 and instead of z_3,"},{"Start":"06:24.570 ","End":"06:30.195","Text":"I\u0027m putting t. Which simplifies to this."},{"Start":"06:30.195 ","End":"06:35.555","Text":"I guess I forgot to print out the final solution."},{"Start":"06:35.555 ","End":"06:37.100","Text":"Could just leave it like that,"},{"Start":"06:37.100 ","End":"06:38.660","Text":"but it\u0027s nice to gather together,"},{"Start":"06:38.660 ","End":"06:45.570","Text":"say z_1 is 1 plus i minus 1 minus i,"},{"Start":"06:45.570 ","End":"06:50.805","Text":"t, and then z_2 is 3,"},{"Start":"06:50.805 ","End":"06:53.595","Text":"and then z_3 equals"},{"Start":"06:53.595 ","End":"07:02.194","Text":"t. There are an infinite number of solutions curves we can take t as anything we like,"},{"Start":"07:02.194 ","End":"07:06.360","Text":"and we\u0027ll get a Z_1, Z_2, Z_3."},{"Start":"07:06.920 ","End":"07:15.220","Text":"The last row is the actual answer. We\u0027re done."}],"ID":9827},{"Watched":false,"Name":"Exercise 17 part b","Duration":"8m 34s","ChapterTopicVideoID":9474,"CourseChapterTopicPlaylistID":7280,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9474.jpeg","UploadDate":"2017-07-26T08:24:43.0130000","DurationForVideoObject":"PT8M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Now, this exercise looks like the previous exercise"},{"Start":"00:03.600 ","End":"00:08.850","Text":"where we had to solve a system of linear equations over the complex numbers."},{"Start":"00:08.850 ","End":"00:12.285","Text":"This is a mixed problem,"},{"Start":"00:12.285 ","End":"00:15.420","Text":"meaning that there are complex numbers in it."},{"Start":"00:15.420 ","End":"00:20.070","Text":"Everywhere you see minus 1 plus 3i and all that,"},{"Start":"00:20.070 ","End":"00:21.990","Text":"they\u0027re all complex numbers i."},{"Start":"00:21.990 ","End":"00:25.560","Text":"But we want to solve it over the real numbers"},{"Start":"00:25.560 ","End":"00:28.110","Text":"because it says so explicitly,"},{"Start":"00:28.110 ","End":"00:29.100","Text":"we could have gone,"},{"Start":"00:29.100 ","End":"00:33.000","Text":"and solved it over the complex numbers on the previous question,"},{"Start":"00:33.000 ","End":"00:38.085","Text":"we did this but it says specifically i,"},{"Start":"00:38.085 ","End":"00:39.600","Text":"it doesn\u0027t say c."},{"Start":"00:39.600 ","End":"00:42.030","Text":"We tackle it differently."},{"Start":"00:42.030 ","End":"00:49.350","Text":"Now the z_1, z_2, z_3 are going to be real numbers,"},{"Start":"00:49.350 ","End":"00:52.670","Text":"not complex numbers even though z usually indicates"},{"Start":"00:52.670 ","End":"00:56.825","Text":"complex and x indicates real but that\u0027s just the latter."},{"Start":"00:56.825 ","End":"00:58.850","Text":"We want a real number solution."},{"Start":"00:58.850 ","End":"01:00.605","Text":"I think you get the point."},{"Start":"01:00.605 ","End":"01:02.180","Text":"Now, let\u0027s start."},{"Start":"01:02.180 ","End":"01:05.760","Text":"Let\u0027s convert it to matrix form."},{"Start":"01:05.780 ","End":"01:08.075","Text":"Oops, not yet."},{"Start":"01:08.075 ","End":"01:14.435","Text":"Let\u0027s first break it into real and imaginary parts like we break complex numbers up."},{"Start":"01:14.435 ","End":"01:18.110","Text":"In general, the theory behind this is that"},{"Start":"01:18.110 ","End":"01:20.915","Text":"if we have 2 complex numbers that are equal,"},{"Start":"01:20.915 ","End":"01:29.270","Text":"a plus bi, c plus di, then a equals c and b equals d."},{"Start":"01:29.270 ","End":"01:33.485","Text":"The real parts are equal and the imaginary parts are equal."},{"Start":"01:33.485 ","End":"01:35.800","Text":"Now, if we do this here, let\u0027s see."},{"Start":"01:35.800 ","End":"01:38.660","Text":"In the first step I\u0027m just opening brackets."},{"Start":"01:38.660 ","End":"01:40.370","Text":"That\u0027s all just bit of algebra,"},{"Start":"01:40.370 ","End":"01:45.730","Text":"like 1 minus iz_3, z_3 minus iz_3."},{"Start":"01:45.730 ","End":"01:47.760","Text":"Here I open the brackets,"},{"Start":"01:47.760 ","End":"01:50.145","Text":"I\u0027ve got z_3 plus iz_3."},{"Start":"01:50.145 ","End":"01:52.460","Text":"Here I have, well, again, a lot of terms."},{"Start":"01:52.460 ","End":"01:53.980","Text":"I get 2 terms, 2 terms, 2 terms,"},{"Start":"01:53.980 ","End":"01:55.550","Text":"there\u0027s 6 terms altogether,"},{"Start":"01:55.550 ","End":"02:00.500","Text":"like minus 1, z_1 which is just minus z_1 plus 3iz_1,"},{"Start":"02:00.500 ","End":"02:02.105","Text":"and so on and so on."},{"Start":"02:02.105 ","End":"02:04.385","Text":"That\u0027s the first step, just open brackets."},{"Start":"02:04.385 ","End":"02:11.705","Text":"Now we want to collect the real separately and the imaginary separately."},{"Start":"02:11.705 ","End":"02:14.420","Text":"Sheep to the left, goat to the right."},{"Start":"02:14.420 ","End":"02:16.040","Text":"That\u0027s what it says in the bible."},{"Start":"02:16.040 ","End":"02:20.090","Text":"Take care real as the sheep and imaginary is the goats."},{"Start":"02:20.090 ","End":"02:21.860","Text":"Let\u0027s see."},{"Start":"02:21.860 ","End":"02:25.835","Text":"Everywhere there\u0027s no i, z_1 plus z_3, that\u0027s here."},{"Start":"02:25.835 ","End":"02:27.170","Text":"The 1s with the i,"},{"Start":"02:27.170 ","End":"02:28.940","Text":"let\u0027s pull the i out of the bracket,"},{"Start":"02:28.940 ","End":"02:32.600","Text":"so z_2 minus z_3."},{"Start":"02:32.600 ","End":"02:35.860","Text":"Oops, 3 slipped over."},{"Start":"02:35.860 ","End":"02:44.045","Text":"It should be here and then that\u0027s z_2 minus z_3i from here,"},{"Start":"02:44.045 ","End":"02:47.570","Text":"and so on with the second row and the third row,"},{"Start":"02:47.570 ","End":"02:54.760","Text":"we put the reals first and then the imaginary part with i."},{"Start":"02:55.040 ","End":"03:01.790","Text":"The next thing I\u0027m going to do is just to add a bit of color."},{"Start":"03:01.790 ","End":"03:06.260","Text":"Not just to make it look nicer but to help me identify what\u0027s real"},{"Start":"03:06.260 ","End":"03:10.635","Text":"and what\u0027s imaginary in the mathematical sense of course."},{"Start":"03:10.635 ","End":"03:18.420","Text":"The blue is the real part and the red is the imaginary part without the i."},{"Start":"03:18.920 ","End":"03:20.960","Text":"Here we have this,"},{"Start":"03:20.960 ","End":"03:27.600","Text":"these are all just colored than this red, red, this is red"},{"Start":"03:27.600 ","End":"03:30.170","Text":"that goes with the i which I just left in black"},{"Start":"03:30.170 ","End":"03:32.665","Text":"and the equals i left in black."},{"Start":"03:32.665 ","End":"03:37.730","Text":"Because of what I said earlier about if 2 things are equal,"},{"Start":"03:37.730 ","End":"03:40.415","Text":"the real parts are equal and the imaginaries are equal,"},{"Start":"03:40.415 ","End":"03:43.370","Text":"we can split each 1 of these into 2 equations"},{"Start":"03:43.370 ","End":"03:48.850","Text":"by taking the blue bits equal the blue and the red equals the red."},{"Start":"03:48.850 ","End":"03:50.655","Text":"If I do all this,"},{"Start":"03:50.655 ","End":"03:56.190","Text":"then my 3 equations become 6 equations,"},{"Start":"03:56.190 ","End":"03:59.160","Text":"I get 3 blue 1s and 3 red 1s."},{"Start":"03:59.160 ","End":"04:00.450","Text":"Here it is."},{"Start":"04:00.450 ","End":"04:04.715","Text":"We\u0027ve got the blue and the red from the first equation, that\u0027s the first 2."},{"Start":"04:04.715 ","End":"04:08.135","Text":"Then these 2 are the blue and the red from the second equation,"},{"Start":"04:08.135 ","End":"04:13.480","Text":"and these 2 are the blue and the red from the last equation."},{"Start":"04:13.480 ","End":"04:17.220","Text":"Now I\u0027m going to just separate them."},{"Start":"04:18.160 ","End":"04:21.200","Text":"No need to do that, I change my mind."},{"Start":"04:21.200 ","End":"04:24.170","Text":"I just have to put it in matrix form."},{"Start":"04:24.170 ","End":"04:28.520","Text":"Now, notice that we don\u0027t need the color anymore"},{"Start":"04:28.520 ","End":"04:30.410","Text":"at this stage the i disappeared"},{"Start":"04:30.410 ","End":"04:34.580","Text":"and so it\u0027s all in black, just from here."},{"Start":"04:34.580 ","End":"04:37.670","Text":"Z_1 plus z_3 gives me 1, 0, 1."},{"Start":"04:37.670 ","End":"04:40.325","Text":"Don\u0027t forget the 0s for the missing."},{"Start":"04:40.325 ","End":"04:43.310","Text":"Like here there\u0027s no z_1, it\u0027s a 0 here,"},{"Start":"04:43.310 ","End":"04:47.630","Text":"6 equations and 4 unknowns,"},{"Start":"04:47.630 ","End":"04:50.610","Text":"and it\u0027s a right-hand side also,"},{"Start":"04:50.610 ","End":"04:55.235","Text":"that\u0027s why this is this augmented matrix with the partition,"},{"Start":"04:55.235 ","End":"04:57.630","Text":"the coefficients of z_1, z_2, z_3,"},{"Start":"04:57.630 ","End":"04:59.225","Text":"and the constants on the right."},{"Start":"04:59.225 ","End":"05:05.400","Text":"Now we want to bring this to echelon form, the row echelon."},{"Start":"05:06.140 ","End":"05:10.040","Text":"I\u0027ve already written down the row operations I intend to do."},{"Start":"05:10.040 ","End":"05:11.195","Text":"Let\u0027s go over it."},{"Start":"05:11.195 ","End":"05:13.010","Text":"I\u0027m looking at the first column"},{"Start":"05:13.010 ","End":"05:14.990","Text":"and I have a 1 here which is nice"},{"Start":"05:14.990 ","End":"05:20.120","Text":"and I already started off with 2 0s and then I\u0027ve got 3 non-zeros."},{"Start":"05:20.120 ","End":"05:22.580","Text":"I want to make all these 3 0s."},{"Start":"05:22.580 ","End":"05:24.170","Text":"The usual routine,"},{"Start":"05:24.170 ","End":"05:30.435","Text":"subtract 3 times the first row from the last row,"},{"Start":"05:30.435 ","End":"05:32.750","Text":"add this row to the fifth row,"},{"Start":"05:32.750 ","End":"05:35.150","Text":"and subtract this row from the fourth row,"},{"Start":"05:35.150 ","End":"05:37.640","Text":"and it\u0027s all written here."},{"Start":"05:37.640 ","End":"05:39.355","Text":"After we do that,"},{"Start":"05:39.355 ","End":"05:41.535","Text":"then we\u0027ve got this."},{"Start":"05:41.535 ","End":"05:46.780","Text":"We\u0027ve got a 1 with all 0s is good and then we have this 1."},{"Start":"05:46.780 ","End":"05:49.850","Text":"But now we want all 0s below it."},{"Start":"05:49.850 ","End":"05:51.970","Text":"We need to do,"},{"Start":"05:51.970 ","End":"05:55.455","Text":"well, it would be 4 but this is already at 0."},{"Start":"05:55.455 ","End":"06:00.945","Text":"Here, here, and here we need to change row 3, row 5, and row 6."},{"Start":"06:00.945 ","End":"06:04.140","Text":"In each case subtract 1 time,"},{"Start":"06:04.140 ","End":"06:05.520","Text":"we subtract R_2 1 time,"},{"Start":"06:05.520 ","End":"06:06.570","Text":"we subtract 3 R_2,"},{"Start":"06:06.570 ","End":"06:08.190","Text":"1 time we add R_2,"},{"Start":"06:08.190 ","End":"06:09.690","Text":"these are the row operations"},{"Start":"06:09.690 ","End":"06:12.840","Text":"and if we do those row operations,"},{"Start":"06:12.940 ","End":"06:16.670","Text":"and I\u0027m going to leave you to check the computations."},{"Start":"06:16.670 ","End":"06:19.040","Text":"You can always put this clip on pause."},{"Start":"06:19.040 ","End":"06:22.220","Text":"Then we get this all 0s here"},{"Start":"06:22.220 ","End":"06:28.230","Text":"and we\u0027re almost in echelon form except,"},{"Start":"06:28.230 ","End":"06:30.330","Text":"there\u0027s an I saw this 6,"},{"Start":"06:30.330 ","End":"06:32.230","Text":"I don\u0027t like that."},{"Start":"06:32.230 ","End":"06:33.950","Text":"Let\u0027s see what we\u0027ll do."},{"Start":"06:33.950 ","End":"06:38.700","Text":"But I also like to divide by common factors."},{"Start":"06:39.950 ","End":"06:43.545","Text":"We\u0027ll do 2 things in 1."},{"Start":"06:43.545 ","End":"06:46.565","Text":"That\u0027s what I wrote here."},{"Start":"06:46.565 ","End":"06:51.430","Text":"But what I wrote before I want to cross out,"},{"Start":"06:51.430 ","End":"06:53.560","Text":"I want to do that later."},{"Start":"06:53.560 ","End":"06:55.525","Text":"Because after I\u0027ve done this,"},{"Start":"06:55.525 ","End":"06:57.460","Text":"I\u0027ve got my 0s here,"},{"Start":"06:57.460 ","End":"06:59.635","Text":"I still have 2 minus 2."},{"Start":"06:59.635 ","End":"07:06.605","Text":"Then I want to do the half of row 3,"},{"Start":"07:06.605 ","End":"07:09.540","Text":"stays in row 3."},{"Start":"07:09.540 ","End":"07:16.639","Text":"I have a very nice echelon form with 1s on the diagonal which is nice."},{"Start":"07:18.440 ","End":"07:23.754","Text":"It\u0027s like these 3 rows don\u0027t exist when they\u0027re all 0s."},{"Start":"07:23.754 ","End":"07:27.250","Text":"We actually have for the restricted matrix,"},{"Start":"07:27.250 ","End":"07:29.750","Text":"remember just look at this part here."},{"Start":"07:29.750 ","End":"07:31.520","Text":"We have 3 rows and 3 columns."},{"Start":"07:31.520 ","End":"07:35.330","Text":"Already we know that we\u0027re going to get a single,"},{"Start":"07:35.330 ","End":"07:36.965","Text":"a unique solution."},{"Start":"07:36.965 ","End":"07:39.540","Text":"Let\u0027s continue."},{"Start":"07:39.670 ","End":"07:42.530","Text":"We go back from the world of matrices"},{"Start":"07:42.530 ","End":"07:45.890","Text":"to the world of systems of linear equations."},{"Start":"07:45.890 ","End":"07:50.180","Text":"This 1 gives me z_1 plus z_3 is 1 and so on,"},{"Start":"07:50.180 ","End":"07:53.065","Text":"3 equations and 3 unknowns."},{"Start":"07:53.065 ","End":"07:55.365","Text":"Now, the back substitution,"},{"Start":"07:55.365 ","End":"07:58.125","Text":"this 1 is already written, just copy it."},{"Start":"07:58.125 ","End":"07:59.895","Text":"We have z_3 already."},{"Start":"07:59.895 ","End":"08:02.210","Text":"Now, plug it into here at minus 1,"},{"Start":"08:02.210 ","End":"08:07.920","Text":"so it\u0027s z_2 plus 1 is 4, so z_2 is 3."},{"Start":"08:07.920 ","End":"08:11.685","Text":"Here once again, it should be z_3."},{"Start":"08:11.685 ","End":"08:16.620","Text":"This is z_1 minus 1 is 1."},{"Start":"08:16.620 ","End":"08:20.235","Text":"So z_1 equals 2."},{"Start":"08:20.235 ","End":"08:23.190","Text":"Then I write out z_1, z_2, z_3"},{"Start":"08:23.190 ","End":"08:26.210","Text":"and for some reason I wrote them in backwards order."},{"Start":"08:26.210 ","End":"08:30.185","Text":"Never mind that we used to write first z_1 then z_2, and z_3."},{"Start":"08:30.185 ","End":"08:33.930","Text":"Anyway, that\u0027s the answer and we\u0027re done."}],"ID":9828}],"Thumbnail":null,"ID":7280},{"Name":"SLE with Parameter","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Row-echelon Form with Parameter","Duration":"12m 37s","ChapterTopicVideoID":9513,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9513.jpeg","UploadDate":"2017-07-26T08:23:45.5030000","DurationForVideoObject":"PT12M37S","Description":null,"VideoComments":[],"Subtitles":[],"ID":9829},{"Watched":false,"Name":"Number of Solutions of SLE with Parameters I","Duration":"12m 6s","ChapterTopicVideoID":9511,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9511.jpeg","UploadDate":"2017-07-26T08:21:07.4170000","DurationForVideoObject":"PT12M6S","Description":null,"VideoComments":[],"Subtitles":[],"ID":9830},{"Watched":false,"Name":"Number of Solutions of SLE with Parameters II","Duration":"15m 41s","ChapterTopicVideoID":9512,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9512.jpeg","UploadDate":"2017-07-26T08:22:33.5400000","DurationForVideoObject":"PT15M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"Continuing from the previous clip,"},{"Start":"00:02.850 ","End":"00:05.070","Text":"we have another example here."},{"Start":"00:05.070 ","End":"00:10.710","Text":"By the way, the examples will get steadily more difficult,"},{"Start":"00:10.710 ","End":"00:13.859","Text":"probably beyond the point that you will need."},{"Start":"00:13.859 ","End":"00:19.480","Text":"Feel free to just drop out when you think the level is advanced enough."},{"Start":"00:19.970 ","End":"00:24.464","Text":"Back here, what we see,"},{"Start":"00:24.464 ","End":"00:31.470","Text":"and I\u0027m looking at the restricted matrix because that\u0027s what we do most of the analysis."},{"Start":"00:31.470 ","End":"00:34.860","Text":"It\u0027s not really 3 rows and 3 columns,"},{"Start":"00:34.860 ","End":"00:38.655","Text":"is really only 2 rows because a row of all 0\u0027s,"},{"Start":"00:38.655 ","End":"00:40.575","Text":"it\u0027s like it didn\u0027t exist."},{"Start":"00:40.575 ","End":"00:44.630","Text":"There\u0027s no way we\u0027re going to get a single solution."},{"Start":"00:44.630 ","End":"00:47.090","Text":"Remember, for single solution you have to have number of"},{"Start":"00:47.090 ","End":"00:50.570","Text":"rows equals number of columns in the restricted"},{"Start":"00:50.570 ","End":"00:53.225","Text":"but that\u0027s not the case here."},{"Start":"00:53.225 ","End":"00:56.915","Text":"You\u0027re never going to get a single solution."},{"Start":"00:56.915 ","End":"01:00.770","Text":"Next, let\u0027s check if we have any contradictory rows."},{"Start":"01:00.770 ","End":"01:02.405","Text":"Doesn\u0027t look like"},{"Start":"01:02.405 ","End":"01:07.925","Text":"but remember with parameters it\u0027s tricky because there can be 0\u0027s that you don\u0027t see."},{"Start":"01:07.925 ","End":"01:09.410","Text":"Look at the middle row."},{"Start":"01:09.410 ","End":"01:12.675","Text":"If k is equal to 1,"},{"Start":"01:12.675 ","End":"01:20.335","Text":"then here we get 0, 0, 0, 2."},{"Start":"01:20.335 ","End":"01:25.565","Text":"Let me emphasize that when k equals 1,"},{"Start":"01:25.565 ","End":"01:29.030","Text":"we have 0, 0, 0,"},{"Start":"01:29.030 ","End":"01:33.395","Text":"and then 2, which is definitely a contradictory row."},{"Start":"01:33.395 ","End":"01:36.875","Text":"When k is 1, no solution."},{"Start":"01:36.875 ","End":"01:39.590","Text":"Now what about the remaining values?"},{"Start":"01:39.590 ","End":"01:41.870","Text":"Let\u0027s now look at k not equal to 1."},{"Start":"01:41.870 ","End":"01:44.780","Text":"I\u0027ll just highlight that k equals 1,"},{"Start":"01:44.780 ","End":"01:47.525","Text":"no solution, k not equal to 1."},{"Start":"01:47.525 ","End":"01:51.965","Text":"We can actually probably guess because we\u0027ve ruled out single solution."},{"Start":"01:51.965 ","End":"01:54.110","Text":"We\u0027ve covered contradiction."},{"Start":"01:54.110 ","End":"01:55.835","Text":"This has to be infinite"},{"Start":"01:55.835 ","End":"01:58.685","Text":"but it also from other considerations,"},{"Start":"01:58.685 ","End":"02:00.695","Text":"look there\u0027s no contradictory rows,"},{"Start":"02:00.695 ","End":"02:03.365","Text":"number of rows is less than the number of columns,"},{"Start":"02:03.365 ","End":"02:05.120","Text":"2 is less than 3."},{"Start":"02:05.120 ","End":"02:08.190","Text":"Remember it\u0027s only 2 rows here really."},{"Start":"02:09.140 ","End":"02:13.910","Text":"That means that there are infinitely many solutions if you look at our table."},{"Start":"02:13.910 ","End":"02:16.030","Text":"That\u0027s it, that\u0027s this example."},{"Start":"02:16.030 ","End":"02:17.940","Text":"Let\u0027s move on to the next."},{"Start":"02:17.940 ","End":"02:20.570","Text":"Here it is. Like I said,"},{"Start":"02:20.570 ","End":"02:22.580","Text":"we only starting from the point."},{"Start":"02:22.580 ","End":"02:27.650","Text":"We\u0027ve already converted to matrix in echelon form."},{"Start":"02:27.650 ","End":"02:33.290","Text":"Also the analysis occurs for the restricted matrix,"},{"Start":"02:33.290 ","End":"02:36.650","Text":"we only need the augmented for contradictory rows,"},{"Start":"02:36.650 ","End":"02:39.955","Text":"which in fact is what we might have here"},{"Start":"02:39.955 ","End":"02:42.535","Text":"but let\u0027s do it systematically."},{"Start":"02:42.535 ","End":"02:45.400","Text":"Now single solution."},{"Start":"02:45.400 ","End":"02:50.060","Text":"We\u0027re not going to get for any value of k because we"},{"Start":"02:50.060 ","End":"02:54.350","Text":"have 3 columns here and at most 2 rows."},{"Start":"02:54.350 ","End":"02:56.870","Text":"I say at most looks like 2,"},{"Start":"02:56.870 ","End":"03:01.545","Text":"but this could be 0 also."},{"Start":"03:01.545 ","End":"03:03.165","Text":"Then in any event,"},{"Start":"03:03.165 ","End":"03:05.190","Text":"rows less than columns,"},{"Start":"03:05.190 ","End":"03:08.270","Text":"so we cannot have a single solution."},{"Start":"03:08.270 ","End":"03:10.940","Text":"Now let\u0027s check for contradictions."},{"Start":"03:10.940 ","End":"03:12.935","Text":"We see a 0, 0,"},{"Start":"03:12.935 ","End":"03:17.135","Text":"0, so it all depends on what happens here."},{"Start":"03:17.135 ","End":"03:21.200","Text":"If this is 0, then it\u0027s just another row we can cross out."},{"Start":"03:21.200 ","End":"03:24.460","Text":"If it\u0027s not 0, then it\u0027s a contradiction."},{"Start":"03:24.460 ","End":"03:26.400","Text":"If we do the check,"},{"Start":"03:26.400 ","End":"03:28.295","Text":"what we\u0027re really doing is looking,"},{"Start":"03:28.295 ","End":"03:31.700","Text":"when is k squared minus 1 equals 0,"},{"Start":"03:31.700 ","End":"03:36.095","Text":"k squared is 1, k is plus or minus 1."},{"Start":"03:36.095 ","End":"03:38.485","Text":"We have to check."},{"Start":"03:38.485 ","End":"03:40.725","Text":"There\u0027s 2 values,"},{"Start":"03:40.725 ","End":"03:44.815","Text":"when k is 1 or minus 1,"},{"Start":"03:44.815 ","End":"03:47.000","Text":"we get a contradiction."},{"Start":"03:47.000 ","End":"03:49.240","Text":"Sorry, the other way around."},{"Start":"03:49.240 ","End":"03:52.020","Text":"If k is 1 or minus 1,"},{"Start":"03:52.020 ","End":"03:54.140","Text":"that\u0027s 0, there\u0027s no contradiction."},{"Start":"03:54.140 ","End":"03:57.125","Text":"If k is not 1 or minus 1,"},{"Start":"03:57.125 ","End":"03:59.975","Text":"then we have no solution."},{"Start":"03:59.975 ","End":"04:02.000","Text":"Now really just by elimination,"},{"Start":"04:02.000 ","End":"04:08.065","Text":"we could do the other case because single solution is being ruled out."},{"Start":"04:08.065 ","End":"04:09.940","Text":"We\u0027ve taken care of contradictory."},{"Start":"04:09.940 ","End":"04:14.260","Text":"If k is equal to 1 or minus 1,"},{"Start":"04:14.260 ","End":"04:17.860","Text":"you must get infinite number of solutions just by elimination"},{"Start":"04:17.860 ","End":"04:24.520","Text":"but to follow the table of decisions that we had,"},{"Start":"04:24.520 ","End":"04:26.200","Text":"we notice that for k is 1,"},{"Start":"04:26.200 ","End":"04:28.915","Text":"or minus 1, because it\u0027s a 0 here."},{"Start":"04:28.915 ","End":"04:31.090","Text":"There\u0027s no contradictory rows."},{"Start":"04:31.090 ","End":"04:38.560","Text":"The number of rows is less than the number of columns."},{"Start":"04:38.560 ","End":"04:42.250","Text":"It\u0027s going to be at 2 rows,"},{"Start":"04:42.250 ","End":"04:44.275","Text":"even if k is 4,"},{"Start":"04:44.275 ","End":"04:49.395","Text":"this is still going to be not 0."},{"Start":"04:49.395 ","End":"04:51.995","Text":"There are 2 rows, but even if there was 1 row,"},{"Start":"04:51.995 ","End":"04:55.505","Text":"it will be less than the number of columns and the restricted."},{"Start":"04:55.505 ","End":"04:58.010","Text":"For these k, this and this,"},{"Start":"04:58.010 ","End":"04:59.855","Text":"there\u0027s infinitely many solutions."},{"Start":"04:59.855 ","End":"05:01.455","Text":"That ends this analysis."},{"Start":"05:01.455 ","End":"05:03.605","Text":"On to the next example,"},{"Start":"05:03.605 ","End":"05:08.300","Text":"and an important example, very instructive."},{"Start":"05:08.300 ","End":"05:14.195","Text":"As usual, we start after we\u0027ve represented as a matrix and reduced,"},{"Start":"05:14.195 ","End":"05:16.435","Text":"brought to echelon form."},{"Start":"05:16.435 ","End":"05:18.090","Text":"This is what we have."},{"Start":"05:18.090 ","End":"05:23.915","Text":"Now we want to look at the reduced matrix."},{"Start":"05:23.915 ","End":"05:28.550","Text":"When you see the case of all 0\u0027s in the restricted,"},{"Start":"05:28.550 ","End":"05:30.830","Text":"I recommend starting there,"},{"Start":"05:30.830 ","End":"05:34.610","Text":"that\u0027s the easiest because all we have to do is to"},{"Start":"05:34.610 ","End":"05:41.340","Text":"check about the right-hand side of the matrix,"},{"Start":"05:41.340 ","End":"05:43.500","Text":"if this is 0 or not."},{"Start":"05:43.500 ","End":"05:45.990","Text":"If it is 0, it\u0027s a redundant row,"},{"Start":"05:45.990 ","End":"05:49.355","Text":"we just throw it out and if it\u0027s not 0, it\u0027s a contradiction."},{"Start":"05:49.355 ","End":"05:51.110","Text":"Well, in the previous example,"},{"Start":"05:51.110 ","End":"05:56.150","Text":"we already solved this quadratic equation,"},{"Start":"05:56.150 ","End":"06:00.640","Text":"or the roots of this 1 and minus 1."},{"Start":"06:00.640 ","End":"06:05.470","Text":"If k is 1 or minus 1,"},{"Start":"06:05.470 ","End":"06:07.910","Text":"then that\u0027s 0 and it\u0027s a redundant row"},{"Start":"06:07.910 ","End":"06:13.840","Text":"but if k is not equal to 1 or minus 1,"},{"Start":"06:13.840 ","End":"06:17.055","Text":"then there\u0027s no solution."},{"Start":"06:17.055 ","End":"06:23.765","Text":"All we\u0027re left with logically is the case k equals 1 and the case k equals minus 1."},{"Start":"06:23.765 ","End":"06:28.225","Text":"We just check those 2 cases manually by substitution."},{"Start":"06:28.225 ","End":"06:30.020","Text":"Just pick one of them."},{"Start":"06:30.020 ","End":"06:34.340","Text":"Let\u0027s say I start with the minus 1 and if I plug it in, well,"},{"Start":"06:34.340 ","End":"06:37.145","Text":"this is 0 of course as expected,"},{"Start":"06:37.145 ","End":"06:41.870","Text":"which means that this really is a redundant row."},{"Start":"06:41.870 ","End":"06:47.400","Text":"K minus 1, k minus 2 comes out to be,"},{"Start":"06:47.400 ","End":"06:48.910","Text":"if we put k equals minus 1,"},{"Start":"06:48.910 ","End":"06:51.640","Text":"we get minus 2 times minus 3 is 6,"},{"Start":"06:51.640 ","End":"06:53.845","Text":"and this comes out minus 2."},{"Start":"06:53.845 ","End":"06:55.780","Text":"Now, at the side,"},{"Start":"06:55.780 ","End":"07:01.990","Text":"I\u0027ve put the equation form of this matrix just for illustration."},{"Start":"07:01.990 ","End":"07:05.080","Text":"If you look at the equation, you\u0027d immediately say, yeah,"},{"Start":"07:05.080 ","End":"07:07.900","Text":"it has a single solution,"},{"Start":"07:07.900 ","End":"07:12.235","Text":"just 1 because we can figure out why and back substitute here"},{"Start":"07:12.235 ","End":"07:14.140","Text":"but if we follow the decision table,"},{"Start":"07:14.140 ","End":"07:19.210","Text":"we also see it\u0027s a single solution because we have an equal number of rows and columns."},{"Start":"07:19.210 ","End":"07:21.310","Text":"See, this has been eliminated,"},{"Start":"07:21.310 ","End":"07:22.330","Text":"so we have 2 rows,"},{"Start":"07:22.330 ","End":"07:27.445","Text":"2 columns and non 0 diagonal,"},{"Start":"07:27.445 ","End":"07:32.689","Text":"and there\u0027s no contradictory rows, so single solution."},{"Start":"07:32.689 ","End":"07:34.820","Text":"Now let\u0027s continue."},{"Start":"07:34.820 ","End":"07:37.250","Text":"Remember we were checking out the 2 special values,"},{"Start":"07:37.250 ","End":"07:39.170","Text":"minus 1 and 1, that\u0027s minus 1."},{"Start":"07:39.170 ","End":"07:41.180","Text":"Now we check 1 substitute."},{"Start":"07:41.180 ","End":"07:42.230","Text":"I\u0027m not going to scroll back,"},{"Start":"07:42.230 ","End":"07:43.895","Text":"but if we substitute,"},{"Start":"07:43.895 ","End":"07:47.150","Text":"check that this is 0 as expected,"},{"Start":"07:47.150 ","End":"07:51.090","Text":"but these 2 entries both become 0."},{"Start":"07:51.090 ","End":"07:57.250","Text":"Like there was a k minus 1 here and here."},{"Start":"07:57.250 ","End":"07:58.825","Text":"That gives us k minus 1, k minus 2."},{"Start":"07:58.825 ","End":"08:00.985","Text":"Anyway, they\u0027re both 0."},{"Start":"08:00.985 ","End":"08:05.575","Text":"Like these 2 rows are completely redundant."},{"Start":"08:05.575 ","End":"08:07.930","Text":"Just for illustration again,"},{"Start":"08:07.930 ","End":"08:12.520","Text":"I put the system which reduces to just 1 equation,"},{"Start":"08:12.520 ","End":"08:14.185","Text":"x plus y equals 1."},{"Start":"08:14.185 ","End":"08:17.949","Text":"Now if I gave you this, you\u0027d immediately say I infinite solutions,"},{"Start":"08:17.949 ","End":"08:21.610","Text":"because any value of y I put,"},{"Start":"08:21.610 ","End":"08:24.820","Text":"I could figure out an x. I mean could be x is 0,"},{"Start":"08:24.820 ","End":"08:26.830","Text":"y is 1, x is 1, y is 0,"},{"Start":"08:26.830 ","End":"08:29.215","Text":"and x is 1/2, y is a 1/2, and so on"},{"Start":"08:29.215 ","End":"08:32.049","Text":"but even following our rules,"},{"Start":"08:32.049 ","End":"08:38.035","Text":"we know that there\u0027s infinite solutions because number of rows is 1,"},{"Start":"08:38.035 ","End":"08:41.635","Text":"which is less than the number of columns which is 2."},{"Start":"08:41.635 ","End":"08:45.070","Text":"The restricted matrix and there\u0027s no contradictions."},{"Start":"08:45.070 ","End":"08:48.430","Text":"There\u0027s no 00 non-zero."},{"Start":"08:48.430 ","End":"08:54.144","Text":"It makes sense from all angles and that\u0027s that example."},{"Start":"08:54.144 ","End":"08:56.425","Text":"Now we come to our last example,"},{"Start":"08:56.425 ","End":"09:00.430","Text":"but this is a long clip and it\u0027s going to be a long example."},{"Start":"09:00.430 ","End":"09:02.095","Text":"I recommend you take a break."},{"Start":"09:02.095 ","End":"09:05.920","Text":"Go stretch your legs or eat something and then come back."},{"Start":"09:05.920 ","End":"09:09.160","Text":"Anyway, I\u0027m going to continue."},{"Start":"09:09.160 ","End":"09:13.360","Text":"This example has 2 parameters."},{"Start":"09:13.360 ","End":"09:19.810","Text":"As you can see, there\u0027s a and there\u0027s b. I\u0027m going to check first for a single solution."},{"Start":"09:19.810 ","End":"09:24.700","Text":"Remember I mentioned that if it looks like same number of rows and columns here, 3 rows,"},{"Start":"09:24.700 ","End":"09:28.870","Text":"3 columns in the restricted matrix you have 3 rows,"},{"Start":"09:28.870 ","End":"09:30.715","Text":"3 columns looks like."},{"Start":"09:30.715 ","End":"09:35.350","Text":"Let\u0027s remember the conditions for single solution on the table."},{"Start":"09:35.350 ","End":"09:38.890","Text":"We said same number of rows and columns."},{"Start":"09:38.890 ","End":"09:41.740","Text":"No zeros on the diagonal."},{"Start":"09:41.740 ","End":"09:44.020","Text":"Diagonal makes sense when you have same number of rows and"},{"Start":"09:44.020 ","End":"09:46.780","Text":"columns and no contradictory row."},{"Start":"09:46.780 ","End":"09:50.275","Text":"The contradictory row relates to the whole augmented matrix,"},{"Start":"09:50.275 ","End":"09:53.930","Text":"whereas these just refer to the restricted matrix."},{"Start":"09:57.180 ","End":"10:00.654","Text":"If you just look at this no zeros on the diagonal."},{"Start":"10:00.654 ","End":"10:04.870","Text":"If a is not 0 and b is not 2,"},{"Start":"10:04.870 ","End":"10:07.360","Text":"then the elements on the diagonal are not going to be"},{"Start":"10:07.360 ","End":"10:11.050","Text":"0 and all the other conditions will be met."},{"Start":"10:11.050 ","End":"10:14.560","Text":"That\u0027s the single solution case."},{"Start":"10:14.560 ","End":"10:20.080","Text":"Then we manually check what happens if a is 0 or b is 2."},{"Start":"10:20.080 ","End":"10:21.430","Text":"Let\u0027s start with the second one."},{"Start":"10:21.430 ","End":"10:27.745","Text":"Let\u0027s say that b is equal to 2 and we plug that in here."},{"Start":"10:27.745 ","End":"10:31.765","Text":"Then what we\u0027ll get is this."},{"Start":"10:31.765 ","End":"10:33.790","Text":"What we see first of all,"},{"Start":"10:33.790 ","End":"10:37.825","Text":"is that the last row is all zeros."},{"Start":"10:37.825 ","End":"10:41.620","Text":"That\u0027s as if there is no row here."},{"Start":"10:41.620 ","End":"10:44.590","Text":"Now in this case, regardless of what a is,"},{"Start":"10:44.590 ","End":"10:50.035","Text":"we have 2 rows and 3 columns."},{"Start":"10:50.035 ","End":"10:53.530","Text":"Number of rows is less than number of columns."},{"Start":"10:53.530 ","End":"11:01.195","Text":"There can\u0027t be any contradictory rows because we don\u0027t have all zeros on the restricted."},{"Start":"11:01.195 ","End":"11:07.885","Text":"That\u0027s all the indications for infinitely many solutions."},{"Start":"11:07.885 ","End":"11:16.730","Text":"That\u0027s the b equals 2 and now we still have to check what happens if a equals 0."},{"Start":"11:17.520 ","End":"11:20.560","Text":"I copied what scrolled off the board."},{"Start":"11:20.560 ","End":"11:21.685","Text":"This is what we had."},{"Start":"11:21.685 ","End":"11:23.785","Text":"If you plug in a equals 0,"},{"Start":"11:23.785 ","End":"11:25.615","Text":"we\u0027re now working with this"},{"Start":"11:25.615 ","End":"11:27.775","Text":"but now look at this."},{"Start":"11:27.775 ","End":"11:32.170","Text":"This is not in row echelon form anymore."},{"Start":"11:32.170 ","End":"11:37.135","Text":"Let\u0027s keep going and bring it to echelon form."},{"Start":"11:37.135 ","End":"11:43.600","Text":"You might be able to get by without doing that with all hand-waving but not professional."},{"Start":"11:43.600 ","End":"11:46.490","Text":"Let\u0027s just do it by the book."},{"Start":"11:48.240 ","End":"11:51.670","Text":"I don\u0027t like the fact that there\u0027s a b here."},{"Start":"11:51.670 ","End":"11:53.230","Text":"I mean a parameter, another number,"},{"Start":"11:53.230 ","End":"11:55.075","Text":"I don\u0027t know if this is 0 or not."},{"Start":"11:55.075 ","End":"11:56.860","Text":"My strategy is as follows."},{"Start":"11:56.860 ","End":"11:59.455","Text":"Look, there\u0027s minus b and minus b here."},{"Start":"11:59.455 ","End":"12:04.075","Text":"Let me first of all add this row to this and this."},{"Start":"12:04.075 ","End":"12:11.335","Text":"This is what I mean in row notation and this is now what we get."},{"Start":"12:11.335 ","End":"12:15.415","Text":"What I want to do is swap rows,"},{"Start":"12:15.415 ","End":"12:18.565","Text":"but I\u0027d like this parameter to be as far away as possible."},{"Start":"12:18.565 ","End":"12:22.415","Text":"I\u0027m going to swap with the third row, not the second."},{"Start":"12:22.415 ","End":"12:25.265","Text":"This is what I mean."},{"Start":"12:25.265 ","End":"12:28.660","Text":"Now we\u0027ll add multiples of the top row,"},{"Start":"12:28.660 ","End":"12:31.390","Text":"add or subtract from the other 2."},{"Start":"12:31.390 ","End":"12:34.990","Text":"Like I\u0027ll subtract twice this from this."},{"Start":"12:34.990 ","End":"12:43.105","Text":"I\u0027ll subtract b over 2 times the first row from the last row."},{"Start":"12:43.105 ","End":"12:48.620","Text":"Here I\u0027ve written it out in row notation."},{"Start":"12:48.780 ","End":"12:53.840","Text":"Check the computations, but this is what we get."},{"Start":"12:55.260 ","End":"13:02.109","Text":"In the restricted matrix we only have 1 row"},{"Start":"13:02.109 ","End":"13:08.245","Text":"but really what we have to do is because of a 0s here,"},{"Start":"13:08.245 ","End":"13:10.090","Text":"we don\u0027t know if these are 0 or not."},{"Start":"13:10.090 ","End":"13:11.335","Text":"We have to check."},{"Start":"13:11.335 ","End":"13:14.920","Text":"Because if we have a contradiction, no solutions"},{"Start":"13:14.920 ","End":"13:16.495","Text":"but if there\u0027s no contradictions,"},{"Start":"13:16.495 ","End":"13:19.630","Text":"there would be infinite solutions because 1 row and 3 columns."},{"Start":"13:19.630 ","End":"13:24.700","Text":"Let\u0027s see what happens with this element and with this element."},{"Start":"13:24.700 ","End":"13:26.770","Text":"If either one of these is non-zero,"},{"Start":"13:26.770 ","End":"13:28.645","Text":"then it\u0027s a contradiction."},{"Start":"13:28.645 ","End":"13:32.680","Text":"Don\u0027t forget that we\u0027re still working under the case a equals 0."},{"Start":"13:32.680 ","End":"13:35.575","Text":"Now, for this to be 0,"},{"Start":"13:35.575 ","End":"13:39.745","Text":"let\u0027s see 2b equals 4, b equals 2."},{"Start":"13:39.745 ","End":"13:41.994","Text":"If we solve this, we can do it mentally."},{"Start":"13:41.994 ","End":"13:44.500","Text":"You see that b squared is 4,"},{"Start":"13:44.500 ","End":"13:48.820","Text":"so b is plus or minus 2."},{"Start":"13:48.820 ","End":"13:52.630","Text":"If b equals 2,"},{"Start":"13:52.630 ","End":"13:58.015","Text":"then there\u0027s not a contradiction because b equals 2 makes both of these 0."},{"Start":"13:58.015 ","End":"14:04.060","Text":"The only problem can be as if b is not equal to 2."},{"Start":"14:04.060 ","End":"14:06.400","Text":"For b not equal to 2,"},{"Start":"14:06.400 ","End":"14:10.780","Text":"we get a contradictory row and hence no solution."},{"Start":"14:10.780 ","End":"14:13.990","Text":"I\u0027ll just highlight that still under a equals 0,"},{"Start":"14:13.990 ","End":"14:15.880","Text":"b not equal to 2."},{"Start":"14:15.880 ","End":"14:17.485","Text":"Yeah, no solution."},{"Start":"14:17.485 ","End":"14:23.575","Text":"We already took care of the case where a is not equal to 0."},{"Start":"14:23.575 ","End":"14:28.210","Text":"The only possibility remaining is b equals 2,"},{"Start":"14:28.210 ","End":"14:31.700","Text":"but still a equals 0."},{"Start":"14:31.710 ","End":"14:37.420","Text":"When b equals 2, we get 0,"},{"Start":"14:37.420 ","End":"14:39.370","Text":"0 and this becomes 2."},{"Start":"14:39.370 ","End":"14:41.410","Text":"This is what we get."},{"Start":"14:41.410 ","End":"14:49.705","Text":"What you can see is that it\u0027s 1 row and 3 columns."},{"Start":"14:49.705 ","End":"14:52.060","Text":"We eliminate these."},{"Start":"14:52.060 ","End":"14:56.949","Text":"In this restricted, it\u0027s 1 row, 3 columns,"},{"Start":"14:56.949 ","End":"14:58.855","Text":"rows less than columns,"},{"Start":"14:58.855 ","End":"15:05.515","Text":"no contradiction and hence infinitely many solutions."},{"Start":"15:05.515 ","End":"15:08.020","Text":"What we really need now is a summary because there are so"},{"Start":"15:08.020 ","End":"15:12.655","Text":"many different cases and here it is."},{"Start":"15:12.655 ","End":"15:18.220","Text":"If a is not equal to 0 and b is not equal to 2,"},{"Start":"15:18.220 ","End":"15:20.185","Text":"we had a single solution."},{"Start":"15:20.185 ","End":"15:28.090","Text":"Then we said, let\u0027s see if a is 0 then we did subdivide it into 2 cases for a equals 0."},{"Start":"15:28.090 ","End":"15:30.925","Text":"Either b is not equal to 2 or b equals 2."},{"Start":"15:30.925 ","End":"15:36.650","Text":"In this case we had no solutions and this case we had infinitely many solutions."},{"Start":"15:37.680 ","End":"15:41.900","Text":"That is it. We\u0027re done with this clip."}],"ID":9831},{"Watched":false,"Name":"Exercise 1","Duration":"5m 20s","ChapterTopicVideoID":9524,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9524.jpeg","UploadDate":"2017-07-26T08:28:18.5670000","DurationForVideoObject":"PT5M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we have the system of linear equations"},{"Start":"00:03.960 ","End":"00:06.510","Text":"and it contains a parameter k."},{"Start":"00:06.510 ","End":"00:13.620","Text":"We have to decide for which values of k the system has no solutions,"},{"Start":"00:13.620 ","End":"00:18.120","Text":"exactly 1 solution or infinitely many solutions."},{"Start":"00:18.120 ","End":"00:22.560","Text":"We convert it to an augmented matrix, which is this,"},{"Start":"00:22.560 ","End":"00:24.615","Text":"just copying the coefficients,"},{"Start":"00:24.615 ","End":"00:26.790","Text":"and the right-hand side is this."},{"Start":"00:26.790 ","End":"00:32.805","Text":"Then we want to bring it into row-echelon form with row operations."},{"Start":"00:32.805 ","End":"00:38.640","Text":"I have this 1 here and I\u0027m going to use it to 0 out the rest of the column."},{"Start":"00:38.890 ","End":"00:43.610","Text":"We\u0027re going to subtract 5 times this row from the second row,"},{"Start":"00:43.610 ","End":"00:47.180","Text":"and that will make this 0 and 3 times this row takeaway"},{"Start":"00:47.180 ","End":"00:50.330","Text":"from the last row will give us 0 here."},{"Start":"00:50.330 ","End":"00:54.395","Text":"Here\u0027s what I just said but in a row notation."},{"Start":"00:54.395 ","End":"00:56.960","Text":"After we do that, we get this."},{"Start":"00:56.960 ","End":"01:00.200","Text":"We can check 5 minus 5 times 1 is 0,"},{"Start":"01:00.200 ","End":"01:02.765","Text":"3 minus 3 times 1 is 0."},{"Start":"01:02.765 ","End":"01:07.610","Text":"Minus 7, minus 5 times negative 1,"},{"Start":"01:07.610 ","End":"01:12.120","Text":"minus 7 plus 5 is minus 2,"},{"Start":"01:12.120 ","End":"01:14.355","Text":"and so on and so on."},{"Start":"01:14.355 ","End":"01:16.620","Text":"Well, let\u0027s take 1 more example,"},{"Start":"01:16.620 ","End":"01:20.220","Text":"k plus 3 minus 3 times this, is just k."},{"Start":"01:20.220 ","End":"01:23.280","Text":"Now we\u0027re okay,"},{"Start":"01:23.280 ","End":"01:24.635","Text":"we have 0s here."},{"Start":"01:24.635 ","End":"01:28.475","Text":"Now what we want is a 0 here also."},{"Start":"01:28.475 ","End":"01:33.505","Text":"Obviously we want to add this second row to the third row."},{"Start":"01:33.505 ","End":"01:36.825","Text":"Here\u0027s the notation for that."},{"Start":"01:36.825 ","End":"01:38.880","Text":"After we do that,"},{"Start":"01:38.880 ","End":"01:40.530","Text":"this is what we get,"},{"Start":"01:40.530 ","End":"01:43.005","Text":"we just add this and this,"},{"Start":"01:43.005 ","End":"01:45.090","Text":"did I say subtract, I meant add."},{"Start":"01:45.090 ","End":"01:46.635","Text":"This and this is 0."},{"Start":"01:46.635 ","End":"01:48.885","Text":"This plus this is this."},{"Start":"01:48.885 ","End":"01:52.010","Text":"K squared minus 4 plus 0 is k squared minus 4."},{"Start":"01:52.010 ","End":"01:56.615","Text":"Now, I examined the restricted matrix,"},{"Start":"01:56.615 ","End":"01:59.210","Text":"also called the coefficient matrix,"},{"Start":"01:59.210 ","End":"02:02.030","Text":"which is the part without this."},{"Start":"02:02.030 ","End":"02:04.490","Text":"Remember, we had different conditions"},{"Start":"02:04.490 ","End":"02:09.860","Text":"for a single solution or exactly 1 solution."},{"Start":"02:09.860 ","End":"02:12.005","Text":"There were 3 requirements."},{"Start":"02:12.005 ","End":"02:15.485","Text":"We have to have all non-zero along the diagonal."},{"Start":"02:15.485 ","End":"02:19.865","Text":"Well, before that, we have to have the same number of rows and columns."},{"Start":"02:19.865 ","End":"02:22.750","Text":"When I say rows, I mean non-zero rows"},{"Start":"02:22.750 ","End":"02:25.000","Text":"so this will have to be non-zero."},{"Start":"02:25.000 ","End":"02:27.380","Text":"Then we want all non-zeros on the diagonal"},{"Start":"02:27.380 ","End":"02:30.350","Text":"still brings us to this being non-zero."},{"Start":"02:30.350 ","End":"02:33.170","Text":"We also have to have no contradiction rows,"},{"Start":"02:33.170 ","End":"02:35.810","Text":"which means 0, 0, 0, not 0."},{"Start":"02:35.810 ","End":"02:39.980","Text":"In any event, if this is not 0,"},{"Start":"02:39.980 ","End":"02:42.875","Text":"will be fine from all the conditions."},{"Start":"02:42.875 ","End":"02:46.955","Text":"I want k plus k minus 2 not to be 0."},{"Start":"02:46.955 ","End":"02:52.870","Text":"If it is 0, I get k equals minus 2 or k equals 1,"},{"Start":"02:52.870 ","End":"02:57.910","Text":"but not 0 means that it has to be not minus 2 and not 1."},{"Start":"02:57.910 ","End":"03:00.460","Text":"In this case this will be non-zero"},{"Start":"03:00.460 ","End":"03:04.710","Text":"and we will have exactly 1 solution."},{"Start":"03:04.710 ","End":"03:07.000","Text":"Now we\u0027re going to flow the other cases"},{"Start":"03:07.000 ","End":"03:11.275","Text":"by just seeing what happens if k is minus 2 or 1."},{"Start":"03:11.275 ","End":"03:13.130","Text":"I\u0027ll go to the next page."},{"Start":"03:13.130 ","End":"03:15.450","Text":"Remember, we want to see what happens"},{"Start":"03:15.450 ","End":"03:18.675","Text":"when k is minus 2 or 1."},{"Start":"03:18.675 ","End":"03:21.670","Text":"Let\u0027s take the case of minus 2 first."},{"Start":"03:21.670 ","End":"03:23.290","Text":"It means everywhere I see k,"},{"Start":"03:23.290 ","End":"03:24.760","Text":"I put minus 2."},{"Start":"03:24.760 ","End":"03:28.795","Text":"For example here, minus 2 squared is 4, minus 2 is 2."},{"Start":"03:28.795 ","End":"03:31.850","Text":"Here if I put in minus 2, I get 0."},{"Start":"03:31.850 ","End":"03:34.200","Text":"Also I get 0 here and here,"},{"Start":"03:34.200 ","End":"03:38.205","Text":"and we end up with this augmented matrix."},{"Start":"03:38.205 ","End":"03:42.979","Text":"Once again, I look at the restricted matrix"},{"Start":"03:42.979 ","End":"03:45.410","Text":"and see what happens here."},{"Start":"03:45.410 ","End":"03:50.220","Text":"We see that the condition for number of rows"},{"Start":"03:50.220 ","End":"03:52.985","Text":"less than number of columns apply."},{"Start":"03:52.985 ","End":"03:54.665","Text":"There\u0027s only 2 rows."},{"Start":"03:54.665 ","End":"03:56.405","Text":"We don\u0027t count the 0 rows,"},{"Start":"03:56.405 ","End":"03:59.080","Text":"so there\u0027s 2 rows and 3 columns."},{"Start":"03:59.080 ","End":"04:02.400","Text":"That still doesn\u0027t tell us which possibility"},{"Start":"04:02.400 ","End":"04:05.270","Text":"but when we see there\u0027s no contradiction rows,"},{"Start":"04:05.270 ","End":"04:08.420","Text":"contradiction means 0, 0, 0, non-zero"},{"Start":"04:08.420 ","End":"04:10.250","Text":"and we don\u0027t have any of that."},{"Start":"04:10.250 ","End":"04:11.840","Text":"When that happens,"},{"Start":"04:11.840 ","End":"04:16.760","Text":"then we meet the condition for infinitely many solutions."},{"Start":"04:16.760 ","End":"04:20.269","Text":"Let\u0027s see what happens when k is 1."},{"Start":"04:20.269 ","End":"04:24.365","Text":"If I substitute it in what\u0027s just scrolled off the screen,"},{"Start":"04:24.365 ","End":"04:27.840","Text":"we get this augmented matrix."},{"Start":"04:27.840 ","End":"04:32.675","Text":"What immediately catches my eye is this last row,"},{"Start":"04:32.675 ","End":"04:34.775","Text":"which is a contradiction row,"},{"Start":"04:34.775 ","End":"04:36.965","Text":"we have 0s and a non-zero."},{"Start":"04:36.965 ","End":"04:38.720","Text":"What this really says,"},{"Start":"04:38.720 ","End":"04:41.585","Text":"in case you are wondering why it\u0027s a contradiction,"},{"Start":"04:41.585 ","End":"04:49.020","Text":"it stands for 0x plus 0y plus 0z equals minus 3."},{"Start":"04:49.020 ","End":"04:54.770","Text":"If you figure that it means that 0 is minus 3, which cannot be."},{"Start":"04:54.770 ","End":"04:57.410","Text":"That\u0027s a contradiction which means"},{"Start":"04:57.410 ","End":"05:02.210","Text":"that there are no solutions to the SLE."},{"Start":"05:02.210 ","End":"05:04.190","Text":"If we want to summarize,"},{"Start":"05:04.190 ","End":"05:08.605","Text":"just remember that k not minus 2 or 1,"},{"Start":"05:08.605 ","End":"05:11.810","Text":"we have a single solution, just 1."},{"Start":"05:11.810 ","End":"05:13.760","Text":"If we have minus 2,"},{"Start":"05:13.760 ","End":"05:16.430","Text":"it\u0027s infinitely many, and for k is 1,"},{"Start":"05:16.430 ","End":"05:17.960","Text":"we have no solutions at all."},{"Start":"05:17.960 ","End":"05:20.400","Text":"We\u0027re done."}],"ID":9832},{"Watched":false,"Name":"Exercise 2","Duration":"7m 47s","ChapterTopicVideoID":9525,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9525.jpeg","UploadDate":"2017-07-26T08:29:05.8730000","DurationForVideoObject":"PT7M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"In this exercise, we have a system of linear equations,"},{"Start":"00:04.755 ","End":"00:09.090","Text":"3 equations and 3 unknowns, x, y, z, and there is a parameter k."},{"Start":"00:09.090 ","End":"00:15.525","Text":"We have to determine the values of k for which the system has no solutions."},{"Start":"00:15.525 ","End":"00:22.620","Text":"A single solution or exactly 1 solution or infinitely many solutions."},{"Start":"00:22.620 ","End":"00:24.780","Text":"Those are the only 3 possibilities,"},{"Start":"00:24.780 ","End":"00:27.179","Text":"0, 1, or infinity."},{"Start":"00:27.179 ","End":"00:33.090","Text":"We tackle this by taking it\u0027s representative matrix, which is this,"},{"Start":"00:33.090 ","End":"00:34.965","Text":"where we just take the coefficients."},{"Start":"00:34.965 ","End":"00:37.470","Text":"It\u0027s an augmented matrix on the right-hand side"},{"Start":"00:37.470 ","End":"00:39.810","Text":"as the right-hand side here."},{"Start":"00:39.810 ","End":"00:43.430","Text":"We want to bring it to row echelon form."},{"Start":"00:43.430 ","End":"00:46.820","Text":"That\u0027s how we decide which of the 3 cases we\u0027re in."},{"Start":"00:46.820 ","End":"00:50.435","Text":"Now, notice that there\u0027s a 1 here."},{"Start":"00:50.435 ","End":"00:57.275","Text":"I can use this to zero out the rest of the column using elementary row operations."},{"Start":"00:57.275 ","End":"01:02.045","Text":"What I suggest is to subtract this row from this row,"},{"Start":"01:02.045 ","End":"01:06.320","Text":"the second minus the first and put that into the second."},{"Start":"01:06.320 ","End":"01:07.730","Text":"This is in the notation."},{"Start":"01:07.730 ","End":"01:09.395","Text":"Then I\u0027ll get a 0 here."},{"Start":"01:09.395 ","End":"01:12.995","Text":"If I subtract k times this row from the last row,"},{"Start":"01:12.995 ","End":"01:15.140","Text":"I\u0027ll get a 0 here."},{"Start":"01:15.140 ","End":"01:21.525","Text":"If you do the algebra,"},{"Start":"01:21.525 ","End":"01:22.460","Text":"you\u0027ll see that we get this."},{"Start":"01:22.460 ","End":"01:27.720","Text":"For example, this minus k times this gives us 1 minus k squared and so on."},{"Start":"01:27.720 ","End":"01:32.719","Text":"I suggest you pause this and just verify that all these calculations are okay."},{"Start":"01:32.719 ","End":"01:37.805","Text":"Now, we continue with trying to bring it into echelon form."},{"Start":"01:37.805 ","End":"01:43.210","Text":"What we need now is for a 0 here."},{"Start":"01:43.210 ","End":"01:47.300","Text":"Now, I can\u0027t help but notice that this is a difference of squares."},{"Start":"01:47.300 ","End":"01:50.900","Text":"I think it\u0027ll be good for us to factorize this"},{"Start":"01:50.900 ","End":"01:53.210","Text":"because 1 of the factors will be 1 minus k."},{"Start":"01:53.210 ","End":"01:56.400","Text":"Then we get this."},{"Start":"01:56.400 ","End":"01:59.080","Text":"We didn\u0027t change anything, just factorize this"},{"Start":"01:59.080 ","End":"02:02.480","Text":"because now I see I have a 1 minus k here and a 1 minus k here,"},{"Start":"02:02.480 ","End":"02:03.920","Text":"and that might be useful."},{"Start":"02:03.920 ","End":"02:07.025","Text":"Now, there are something that you might be tempted to do,"},{"Start":"02:07.025 ","End":"02:08.695","Text":"which you shouldn\u0027t do."},{"Start":"02:08.695 ","End":"02:13.205","Text":"That is to say 1 minus k is a factor here."},{"Start":"02:13.205 ","End":"02:18.770","Text":"This is also minus of 1 minus k. You might want to divide this row by 1 minus k,"},{"Start":"02:18.770 ","End":"02:20.560","Text":"or even for the third row,"},{"Start":"02:20.560 ","End":"02:21.770","Text":"you also might say,"},{"Start":"02:21.770 ","End":"02:23.660","Text":"I\u0027m going to divide it by 1 minus k"},{"Start":"02:23.660 ","End":"02:25.910","Text":"because I\u0027ve got y minus k here and here,"},{"Start":"02:25.910 ","End":"02:28.040","Text":"and this is minus of 1 minus k."},{"Start":"02:28.040 ","End":"02:31.640","Text":"Avoid this temptation as far as possible"},{"Start":"02:31.640 ","End":"02:35.780","Text":"because we\u0027re not multiplying or dividing by a non-zero number,"},{"Start":"02:35.780 ","End":"02:38.210","Text":"we have an algebraic expression, 1 minus k,"},{"Start":"02:38.210 ","End":"02:40.415","Text":"and it just might be 0."},{"Start":"02:40.415 ","End":"02:42.710","Text":"If k equals 1,"},{"Start":"02:42.710 ","End":"02:44.270","Text":"then this would be 0,"},{"Start":"02:44.270 ","End":"02:46.445","Text":"and you\u0027d be dividing by 0."},{"Start":"02:46.445 ","End":"02:48.365","Text":"Try to avoid doing this."},{"Start":"02:48.365 ","End":"02:50.090","Text":"If you have to,"},{"Start":"02:50.090 ","End":"02:55.160","Text":"then what you would do at this point would be to branch and to take 2 cases."},{"Start":"02:55.160 ","End":"02:58.925","Text":"One, where k is equal to 1 and then go that approach,"},{"Start":"02:58.925 ","End":"03:03.350","Text":"and then say, if k is not equal to 1 then we can divide and continue that way,"},{"Start":"03:03.350 ","End":"03:08.510","Text":"but I don\u0027t recommend that approach because there\u0027s an easy alternative."},{"Start":"03:08.510 ","End":"03:12.950","Text":"Remember, we talked about not using rule 2 if we can use rule 3."},{"Start":"03:12.950 ","End":"03:18.815","Text":"What we can do is multiply this row by 1 plus k,"},{"Start":"03:18.815 ","End":"03:21.050","Text":"and then this and this will be equal."},{"Start":"03:21.050 ","End":"03:23.665","Text":"Then we\u0027ll subtract this from this."},{"Start":"03:23.665 ","End":"03:26.250","Text":"In row notation,"},{"Start":"03:26.250 ","End":"03:28.220","Text":"this is what I\u0027m saying."},{"Start":"03:28.220 ","End":"03:34.180","Text":"Subtract 1 plus k times row 2 from row 3."},{"Start":"03:34.180 ","End":"03:35.670","Text":"This is what we get."},{"Start":"03:35.670 ","End":"03:37.815","Text":"There is only 2 changes, here and here."},{"Start":"03:37.815 ","End":"03:42.255","Text":"These 2 0s mean that the ones below it are not affected."},{"Start":"03:42.255 ","End":"03:45.030","Text":"This minus 1 plus k times this is 0,"},{"Start":"03:45.030 ","End":"03:46.770","Text":"that\u0027s the reason we did this."},{"Start":"03:46.770 ","End":"03:55.920","Text":"This, minus 1 plus k times this is what I wrote here. We\u0027ve checked that."},{"Start":"03:55.920 ","End":"03:58.020","Text":"This is a bit of a mess."},{"Start":"03:58.020 ","End":"04:00.919","Text":"I think we should simplify this element."},{"Start":"04:00.919 ","End":"04:02.974","Text":"Here\u0027s what I suggest."},{"Start":"04:02.974 ","End":"04:05.615","Text":"We have here a 1 minus k,"},{"Start":"04:05.615 ","End":"04:08.120","Text":"and here, we also have a 1 minus K."},{"Start":"04:08.120 ","End":"04:15.140","Text":"If I make this minus into a plus and I reverse the order and make this a 1 minus k,"},{"Start":"04:15.140 ","End":"04:18.245","Text":"then I can take 1 minus k out the brackets,"},{"Start":"04:18.245 ","End":"04:20.800","Text":"and we have 1 plus 1 plus k,"},{"Start":"04:20.800 ","End":"04:23.415","Text":"and show 2 plus k here."},{"Start":"04:23.415 ","End":"04:25.445","Text":"This is simplified now."},{"Start":"04:25.445 ","End":"04:29.490","Text":"It is already in echelon form,"},{"Start":"04:29.840 ","End":"04:32.790","Text":"means the restricted matrix,"},{"Start":"04:32.790 ","End":"04:36.275","Text":"the left of this is like a staircase."},{"Start":"04:36.275 ","End":"04:39.620","Text":"Now, we can start drawing conclusions."},{"Start":"04:39.620 ","End":"04:42.795","Text":"I\u0027m going to continue on the next page. Here we are."},{"Start":"04:42.795 ","End":"04:46.505","Text":"I\u0027ve highlighted the diagonal here because it\u0027s important."},{"Start":"04:46.505 ","End":"04:52.595","Text":"What we do is we examine the restricted matrix,"},{"Start":"04:52.595 ","End":"04:59.350","Text":"not the augmented, which is just to the left of this line separator."},{"Start":"04:59.350 ","End":"05:05.580","Text":"We typically look for the case of a single solution or exactly 1 solution."},{"Start":"05:05.910 ","End":"05:12.500","Text":"One of the requirements is that the entries on the diagonal have to all be non-."},{"Start":"05:12.500 ","End":"05:14.195","Text":"There\u0027s also other requirements."},{"Start":"05:14.195 ","End":"05:16.620","Text":"Let\u0027s first of all take care of that."},{"Start":"05:16.620 ","End":"05:21.245","Text":"We need for this and this to be non-zero."},{"Start":"05:21.245 ","End":"05:23.840","Text":"If we just look at this,"},{"Start":"05:23.840 ","End":"05:29.405","Text":"it tells us that k can be anything except 1 or minus 2."},{"Start":"05:29.405 ","End":"05:30.920","Text":"These are the 2 bad values."},{"Start":"05:30.920 ","End":"05:33.280","Text":"Everything else is okay."},{"Start":"05:33.280 ","End":"05:35.479","Text":"If these 3 are non-zero,"},{"Start":"05:35.479 ","End":"05:37.880","Text":"then all the other conditions are met"},{"Start":"05:37.880 ","End":"05:40.160","Text":"because we don\u0027t have any contradiction rows."},{"Start":"05:40.160 ","End":"05:44.815","Text":"Contradiction rows has to be all 0s in the restricted."},{"Start":"05:44.815 ","End":"05:48.200","Text":"Also, number of rows equals number of columns"},{"Start":"05:48.200 ","End":"05:50.880","Text":"because if this is non-zero and this,"},{"Start":"05:50.880 ","End":"05:58.155","Text":"then we do have 3 genuine non-zero rows and 3 columns, so we\u0027re okay."},{"Start":"05:58.155 ","End":"06:02.720","Text":"This gives us the condition for exactly 1 solution."},{"Start":"06:02.720 ","End":"06:08.080","Text":"Next, look what happens when k is equal to 1 or minus 2."},{"Start":"06:08.080 ","End":"06:10.500","Text":"Let\u0027s start with k equals 1,"},{"Start":"06:10.500 ","End":"06:12.915","Text":"then we put everywhere in here."},{"Start":"06:12.915 ","End":"06:15.540","Text":"Where we see k, we put 1."},{"Start":"06:15.540 ","End":"06:17.370","Text":"This is what comes out."},{"Start":"06:17.370 ","End":"06:20.445","Text":"This and this becomes 0 and this becomes 0."},{"Start":"06:20.445 ","End":"06:22.040","Text":"The top row is all 1."},{"Start":"06:22.040 ","End":"06:25.130","Text":"Anyway, we have 2 rows of all 0s."},{"Start":"06:25.130 ","End":"06:29.380","Text":"Now, we have only 1 row but 3 columns."},{"Start":"06:29.380 ","End":"06:32.160","Text":"We don\u0027t count the 0 as rows, of course."},{"Start":"06:32.160 ","End":"06:34.670","Text":"There\u0027s no contradiction rows"},{"Start":"06:34.670 ","End":"06:38.810","Text":"because contradiction would mean all 0s but something other than 0 here."},{"Start":"06:38.810 ","End":"06:43.910","Text":"This is all the conditions for infinitely many solutions."},{"Start":"06:43.910 ","End":"06:49.860","Text":"Finally, to check what happens when k is minus 2,"},{"Start":"06:49.860 ","End":"06:55.115","Text":"and if you go back to our matrix with k and substitute this,"},{"Start":"06:55.115 ","End":"06:56.660","Text":"then this is what we get."},{"Start":"06:56.660 ","End":"07:01.165","Text":"I\u0027ll scroll back and just show you because the last row is important there."},{"Start":"07:01.165 ","End":"07:05.810","Text":"If I put k equals minus 2,"},{"Start":"07:05.810 ","End":"07:07.920","Text":"then here 0, here 0."},{"Start":"07:07.920 ","End":"07:09.225","Text":"This becomes 0,"},{"Start":"07:09.225 ","End":"07:13.395","Text":"but 1 minus minus 2 is 3."},{"Start":"07:13.395 ","End":"07:17.135","Text":"Regardless of what happens anywhere else,"},{"Start":"07:17.135 ","End":"07:22.160","Text":"we get a contradiction row, all 0s,"},{"Start":"07:22.160 ","End":"07:27.465","Text":"and then a non-zero and that indicates no solutions."},{"Start":"07:27.465 ","End":"07:30.945","Text":"We\u0027ve got basically everything,"},{"Start":"07:30.945 ","End":"07:35.010","Text":"k not equal to 1 or 2 is 1 solution."},{"Start":"07:35.010 ","End":"07:37.185","Text":"It\u0027s just off the screen."},{"Start":"07:37.185 ","End":"07:39.240","Text":"Maybe I can fit it all in."},{"Start":"07:39.240 ","End":"07:42.890","Text":"Just about k equals 1, infinitely many"},{"Start":"07:42.890 ","End":"07:46.040","Text":"and k equals minus 2, no solutions."},{"Start":"07:46.040 ","End":"07:48.210","Text":"We\u0027re done."}],"ID":9833},{"Watched":false,"Name":"Exercise 3","Duration":"7m 43s","ChapterTopicVideoID":9526,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9526.jpeg","UploadDate":"2017-07-26T08:29:37.4930000","DurationForVideoObject":"PT7M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we have our familiar setup of the system of linear equations."},{"Start":"00:05.130 ","End":"00:08.310","Text":"In this case 3 equations and 3 unknowns, x, y, z,"},{"Start":"00:08.310 ","End":"00:11.010","Text":"and the coefficients contain the parameter k."},{"Start":"00:11.010 ","End":"00:16.455","Text":"Our job is to determine which values of k give us the 3 scenarios,"},{"Start":"00:16.455 ","End":"00:22.845","Text":"no solution, a single solution, or infinite solutions."},{"Start":"00:22.845 ","End":"00:30.900","Text":"As usual, we represent this system by an augmented matrix,"},{"Start":"00:30.900 ","End":"00:32.340","Text":"here are the coefficients"},{"Start":"00:32.340 ","End":"00:37.375","Text":"and here are the free constants from the right-hand side with the separator."},{"Start":"00:37.375 ","End":"00:41.670","Text":"What we have to do is bring it to row echelon form"},{"Start":"00:41.670 ","End":"00:45.445","Text":"so that we can diagnose what\u0027s going on."},{"Start":"00:45.445 ","End":"00:52.370","Text":"The first thing to do, I mean, what sticks out is that we have a 1 here."},{"Start":"00:52.370 ","End":"00:56.810","Text":"Using this 1, we can make zeros here and here"},{"Start":"00:56.810 ","End":"01:03.755","Text":"by subtracting 3 times this row from this row as it says here,"},{"Start":"01:03.755 ","End":"01:07.910","Text":"and subtracting this row as is from the last row,"},{"Start":"01:07.910 ","End":"01:10.240","Text":"which is what it says here."},{"Start":"01:10.240 ","End":"01:13.820","Text":"Here\u0027s the result, and I\u0027ll leave you to check the calculations."},{"Start":"01:13.820 ","End":"01:17.975","Text":"You can always put this clip on pause and just check the details."},{"Start":"01:17.975 ","End":"01:21.485","Text":"Now, the second column is more challenging."},{"Start":"01:21.485 ","End":"01:26.030","Text":"If you want to continue to bring into row echelon form, we need a 0 here,"},{"Start":"01:26.030 ","End":"01:28.730","Text":"but we look for a non-zero element,"},{"Start":"01:28.730 ","End":"01:31.715","Text":"and notice that this k is everywhere."},{"Start":"01:31.715 ","End":"01:35.980","Text":"We had an example like this in the tutorial."},{"Start":"01:35.980 ","End":"01:38.220","Text":"Doesn\u0027t matter that this is k,"},{"Start":"01:38.220 ","End":"01:39.630","Text":"it\u0027s these 2 that I\u0027m worried about."},{"Start":"01:39.630 ","End":"01:41.910","Text":"I don\u0027t have just a number."},{"Start":"01:41.910 ","End":"01:46.035","Text":"What I\u0027m going to do is the following trick."},{"Start":"01:46.035 ","End":"01:50.275","Text":"If I multiply this by 7 and this by 6,"},{"Start":"01:50.275 ","End":"01:54.290","Text":"then in both cases, I\u0027ll have the common multiple 42k,"},{"Start":"01:54.290 ","End":"01:57.980","Text":"and then I can subtract and get rid of k."},{"Start":"01:57.980 ","End":"02:05.340","Text":"Specifically, I\u0027m going to multiply this by 7 and add that to 6 times this,"},{"Start":"02:05.340 ","End":"02:08.150","Text":"that\u0027s going to be in the last row as it says here."},{"Start":"02:08.150 ","End":"02:13.440","Text":"Notice that this coefficient of R_3 is non-zero."},{"Start":"02:13.440 ","End":"02:17.010","Text":"That\u0027s important. It should be non-zero."},{"Start":"02:17.010 ","End":"02:20.790","Text":"This didn\u0027t matter if it was 0 or not."},{"Start":"02:20.790 ","End":"02:23.960","Text":"If we do that computation,"},{"Start":"02:23.960 ","End":"02:27.335","Text":"then we get this mess in the last row."},{"Start":"02:27.335 ","End":"02:28.970","Text":"We\u0027re going to tidy it up,"},{"Start":"02:28.970 ","End":"02:31.955","Text":"but the main thing to notice is that already we can see"},{"Start":"02:31.955 ","End":"02:35.750","Text":"that here is going to be 42k and minus 42k"},{"Start":"02:35.750 ","End":"02:40.220","Text":"and indeed, after a bit of simplification,"},{"Start":"02:40.220 ","End":"02:42.335","Text":"some basic algebra, I\u0027ll leave you to verify,"},{"Start":"02:42.335 ","End":"02:46.080","Text":"we got rid of k from here."},{"Start":"02:46.080 ","End":"02:49.210","Text":"But I want the non-zero element to be here,"},{"Start":"02:49.210 ","End":"02:52.300","Text":"so just to swap row 2 with row 3."},{"Start":"02:52.300 ","End":"02:58.030","Text":"We\u0027ve seen this in the tutorial about swapping rows."},{"Start":"02:58.030 ","End":"03:04.510","Text":"After we do that, this is now the situation, which is an improvement"},{"Start":"03:04.510 ","End":"03:07.470","Text":"because we have a non-zero element in here."},{"Start":"03:07.470 ","End":"03:12.460","Text":"Now, I can add a multiple of this to this and 0 out this element."},{"Start":"03:12.460 ","End":"03:14.725","Text":"I\u0027m going to go to the next page."},{"Start":"03:14.725 ","End":"03:19.300","Text":"Now I could do 1 row operation to get it 0 here by taking"},{"Start":"03:19.300 ","End":"03:23.875","Text":"this plus this over this times this row."},{"Start":"03:23.875 ","End":"03:25.945","Text":"I\u0027d rather do it in 2 steps."},{"Start":"03:25.945 ","End":"03:31.010","Text":"Let\u0027s first of all divide by 7 because it\u0027s easier if there\u0027s a 1 here, like so."},{"Start":"03:31.010 ","End":"03:35.150","Text":"So there\u0027s a 1 here, and these are divided by 7."},{"Start":"03:35.150 ","End":"03:42.180","Text":"Now, I can multiply this second row by this 1 minus 6k and subtract."},{"Start":"03:42.180 ","End":"03:45.345","Text":"In row notation, this is what I mean."},{"Start":"03:45.345 ","End":"03:48.325","Text":"If we perform that calculation,"},{"Start":"03:48.325 ","End":"03:50.840","Text":"then we have to get the 0 here,"},{"Start":"03:50.840 ","End":"03:52.985","Text":"which was important for us."},{"Start":"03:52.985 ","End":"03:59.360","Text":"Here, it\u0027s a bit of a mess algebraically, but we can tidy that up."},{"Start":"03:59.360 ","End":"04:02.150","Text":"It\u0027s not too bad."},{"Start":"04:02.150 ","End":"04:04.520","Text":"I\u0027ll leave you to check the calculations."},{"Start":"04:04.520 ","End":"04:09.170","Text":"What we did is here and here, we put a common denominator 7"},{"Start":"04:09.170 ","End":"04:15.500","Text":"and just simplified the numerator by collecting like terms."},{"Start":"04:15.500 ","End":"04:18.335","Text":"I\u0027m going to go to another page."},{"Start":"04:18.335 ","End":"04:25.338","Text":"Here we are, and I\u0027ve colored the diagonal of the restricted matrix,"},{"Start":"04:25.338 ","End":"04:28.410","Text":"which is this part of it."},{"Start":"04:28.960 ","End":"04:37.170","Text":"The usual thing we do is to look for the ks of exactly 1 solution or single solution."},{"Start":"04:37.880 ","End":"04:41.310","Text":"We need the diagonal to be non-zero."},{"Start":"04:41.310 ","End":"04:43.185","Text":"Here and here, we\u0027re okay."},{"Start":"04:43.185 ","End":"04:45.030","Text":"We need this to be non-zero."},{"Start":"04:45.030 ","End":"04:47.355","Text":"6, 7, we can ignore,"},{"Start":"04:47.355 ","End":"04:53.015","Text":"so we\u0027re down to this quadratic polynomial being non-zero."},{"Start":"04:53.015 ","End":"04:59.825","Text":"We solve the polynomial equal to 0, and then say not those values."},{"Start":"04:59.825 ","End":"05:03.545","Text":"I\u0027ll leave you to solve the quadratic."},{"Start":"05:03.545 ","End":"05:06.290","Text":"You know how to find the roots."},{"Start":"05:06.290 ","End":"05:11.690","Text":"If you like, I could give you a hint and say that this also factorizes as"},{"Start":"05:11.690 ","End":"05:15.960","Text":"k plus 1, 7k minus 4."},{"Start":"05:15.960 ","End":"05:19.280","Text":"You could check that, and then from here, we can easily see,"},{"Start":"05:19.280 ","End":"05:21.380","Text":"or you can just solve it using the formula."},{"Start":"05:21.380 ","End":"05:27.020","Text":"In any event, we meet the conditions for exactly 1 solution."},{"Start":"05:27.020 ","End":"05:30.650","Text":"I mean, the fact that we have a diagonal which is non-zero,"},{"Start":"05:30.650 ","End":"05:33.335","Text":"means the same number of rows and columns,"},{"Start":"05:33.335 ","End":"05:39.135","Text":"and there can\u0027t be any contradiction rows because a contradiction row has to be all 0,"},{"Start":"05:39.135 ","End":"05:42.890","Text":"so this is precisely the condition for exactly 1 solution."},{"Start":"05:42.890 ","End":"05:46.310","Text":"In the remainder, all we have to do is check what happens if"},{"Start":"05:46.310 ","End":"05:52.810","Text":"k is equal to minus 1 or is equal to 4/7."},{"Start":"05:52.810 ","End":"05:55.335","Text":"Let\u0027s take this 1 first."},{"Start":"05:55.335 ","End":"05:59.480","Text":"If k is minus 1 and we substitute, we get this."},{"Start":"05:59.480 ","End":"06:04.115","Text":"Let me just go back and show you that this last row is important."},{"Start":"06:04.115 ","End":"06:06.230","Text":"That if k is minus 1,"},{"Start":"06:06.230 ","End":"06:07.985","Text":"we know this is going to be 0,"},{"Start":"06:07.985 ","End":"06:12.090","Text":"but notice also that 12 plus 12k is also 0."},{"Start":"06:12.090 ","End":"06:14.775","Text":"We get a row with all zeros,"},{"Start":"06:14.775 ","End":"06:23.404","Text":"which means that we get the restricted matrix."},{"Start":"06:23.404 ","End":"06:29.570","Text":"I mean this 1 here has only 2 rows because we don\u0027t count the zeros"},{"Start":"06:29.570 ","End":"06:33.590","Text":"and 3 columns, left rows then columns."},{"Start":"06:33.590 ","End":"06:37.850","Text":"Also, there are no contradictions in the augmented"},{"Start":"06:37.850 ","End":"06:41.330","Text":"because all 0s is not a contradiction."},{"Start":"06:41.330 ","End":"06:46.220","Text":"If you had 3 zeros and add a 5 here or something, then it would be."},{"Start":"06:46.220 ","End":"06:51.470","Text":"So we\u0027ve met the condition for infinitely many solutions."},{"Start":"06:51.470 ","End":"06:56.180","Text":"We\u0027ve taken care of k not equal to minus 1 and 4/7."},{"Start":"06:56.180 ","End":"06:58.340","Text":"We\u0027ve taken k equals minus 1."},{"Start":"06:58.340 ","End":"07:01.505","Text":"What remains is k equals 4/7."},{"Start":"07:01.505 ","End":"07:06.290","Text":"You can probably guess that this corresponds to the no solutions."},{"Start":"07:06.290 ","End":"07:08.080","Text":"Let\u0027s see if we\u0027re right."},{"Start":"07:08.080 ","End":"07:11.080","Text":"Let me just scroll a bit."},{"Start":"07:11.420 ","End":"07:17.220","Text":"Substitute 4/7 when you are going to get a 0 here."},{"Start":"07:17.220 ","End":"07:21.795","Text":"Here, remember it was 12 plus 12k over 7, that\u0027s what we get."},{"Start":"07:21.795 ","End":"07:24.425","Text":"At any event without even actually computing it,"},{"Start":"07:24.425 ","End":"07:26.450","Text":"I know that this is non-zero."},{"Start":"07:26.450 ","End":"07:29.355","Text":"I mean, these are all little positive computations,"},{"Start":"07:29.355 ","End":"07:32.310","Text":"so we get a contradiction row here."},{"Start":"07:32.310 ","End":"07:33.650","Text":"You don\u0027t have to look anywhere else."},{"Start":"07:33.650 ","End":"07:35.870","Text":"Once I see a contradiction row,"},{"Start":"07:35.870 ","End":"07:43.350","Text":"then I can say that the SLE has no solutions and we\u0027re done."}],"ID":9834},{"Watched":false,"Name":"Exercise 4","Duration":"5m 22s","ChapterTopicVideoID":9527,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9527.jpeg","UploadDate":"2017-07-26T08:30:01.2830000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"In this exercise, we have a system of linear equations."},{"Start":"00:03.330 ","End":"00:07.725","Text":"There\u0027s 3 equations and 3 unknowns, x, y, z."},{"Start":"00:07.725 ","End":"00:11.445","Text":"But the coefficients contain a parameter k,"},{"Start":"00:11.445 ","End":"00:16.140","Text":"and we have to sort out which values of k give which scenario."},{"Start":"00:16.140 ","End":"00:19.395","Text":"We can have 3 possibilities, no solution,"},{"Start":"00:19.395 ","End":"00:24.150","Text":"exactly 1 solution, or infinitely many solutions."},{"Start":"00:24.150 ","End":"00:28.785","Text":"As usual, we begin by representing the system by a matrix,"},{"Start":"00:28.785 ","End":"00:32.775","Text":"and our first task is to bring this to row-echelon form."},{"Start":"00:32.775 ","End":"00:35.340","Text":"Because as is it, I can\u0027t really say much."},{"Start":"00:35.340 ","End":"00:39.195","Text":"What we\u0027re going to do is some basic row operations."},{"Start":"00:39.195 ","End":"00:44.230","Text":"Now, here I have something which is a number and it\u0027s non-zero,"},{"Start":"00:44.230 ","End":"00:47.720","Text":"and I could add multiples of this to these rows,"},{"Start":"00:47.720 ","End":"00:49.370","Text":"but look, there\u0027s a 1 here."},{"Start":"00:49.370 ","End":"00:51.680","Text":"It\u0027s so much easier when it\u0027s a 1,"},{"Start":"00:51.680 ","End":"00:55.290","Text":"so I recommend to have too many fractions,"},{"Start":"00:55.290 ","End":"00:57.425","Text":"swap these 2 around."},{"Start":"00:57.425 ","End":"01:02.840","Text":"This is the notation and we end up clearly this row here,"},{"Start":"01:02.840 ","End":"01:05.870","Text":"no big deal to see that."},{"Start":"01:05.870 ","End":"01:10.565","Text":"Now we can subtract multiples of this from these,"},{"Start":"01:10.565 ","End":"01:12.305","Text":"twice this from this,"},{"Start":"01:12.305 ","End":"01:14.525","Text":"and 5 times this from this."},{"Start":"01:14.525 ","End":"01:16.950","Text":"This is it in row notation,"},{"Start":"01:16.950 ","End":"01:18.315","Text":"and if we do this,"},{"Start":"01:18.315 ","End":"01:22.895","Text":"we get the same first row as above,"},{"Start":"01:22.895 ","End":"01:26.620","Text":"but now we have a 0 here and here."},{"Start":"01:26.620 ","End":"01:29.275","Text":"Check the computations."},{"Start":"01:29.275 ","End":"01:31.010","Text":"I suggest if you want to check,"},{"Start":"01:31.010 ","End":"01:35.585","Text":"put it on pause and see for example that minus 1,"},{"Start":"01:35.585 ","End":"01:39.380","Text":"minus twice, this is minus 5."},{"Start":"01:39.380 ","End":"01:47.225","Text":"This minus 5 times this is this minus 10, and so on."},{"Start":"01:47.225 ","End":"01:49.130","Text":"Now we\u0027re up to here."},{"Start":"01:49.130 ","End":"01:52.040","Text":"Now we\u0027re still not in row-echelon form."},{"Start":"01:52.040 ","End":"01:54.575","Text":"We need a 0 here."},{"Start":"01:54.575 ","End":"01:56.300","Text":"If we had a 1 here,"},{"Start":"01:56.300 ","End":"01:58.165","Text":"it would be a bit easier."},{"Start":"01:58.165 ","End":"02:01.250","Text":"Let\u0027s just divide this row by minus 5,"},{"Start":"02:01.250 ","End":"02:02.780","Text":"which is non-zero,"},{"Start":"02:02.780 ","End":"02:07.160","Text":"and then we get this and they use decimals instead of fractions."},{"Start":"02:07.160 ","End":"02:09.905","Text":"I also took the minus out here."},{"Start":"02:09.905 ","End":"02:12.535","Text":"Just suits me more."},{"Start":"02:12.535 ","End":"02:14.550","Text":"Let\u0027s continue on the next page."},{"Start":"02:14.550 ","End":"02:16.665","Text":"Here we are."},{"Start":"02:16.665 ","End":"02:20.840","Text":"Now we can easily get this to be 0 if we multiply"},{"Start":"02:20.840 ","End":"02:26.800","Text":"this row by 9 plus k and then add to here."},{"Start":"02:26.800 ","End":"02:30.055","Text":"This is it in row notation and the result."},{"Start":"02:30.055 ","End":"02:34.025","Text":"This we did get a 0 here,"},{"Start":"02:34.025 ","End":"02:37.190","Text":"but we also get a little bit of a mess here."},{"Start":"02:37.190 ","End":"02:40.060","Text":"Let\u0027s tidy this up a bit."},{"Start":"02:40.060 ","End":"02:42.890","Text":"I went back from my decimal to fractions"},{"Start":"02:42.890 ","End":"02:46.190","Text":"and wrote this as minus 3/5."},{"Start":"02:46.190 ","End":"02:50.300","Text":"Then if you do the simplification and put it all over 5,"},{"Start":"02:50.300 ","End":"02:52.075","Text":"this is what you get."},{"Start":"02:52.075 ","End":"02:54.740","Text":"Technically actually you should have put this as a fraction too,"},{"Start":"02:54.740 ","End":"02:57.180","Text":"doesn\u0027t matter, we can mix."},{"Start":"02:57.580 ","End":"03:01.535","Text":"These entries are highlighted in blue"},{"Start":"03:01.535 ","End":"03:09.575","Text":"because we\u0027re examining the restricted matrix and not the augmented."},{"Start":"03:09.575 ","End":"03:16.430","Text":"We\u0027re looking for a single solution or exactly 1 solution."},{"Start":"03:16.430 ","End":"03:24.080","Text":"We\u0027re looking for a number of rows equals number of columns and non-zero on the diagonal."},{"Start":"03:24.080 ","End":"03:29.390","Text":"Essentially it boils down to saying that this is non-zero."},{"Start":"03:29.390 ","End":"03:32.020","Text":"We already have non-zero here and here."},{"Start":"03:32.020 ","End":"03:33.490","Text":"If this is non-zero,"},{"Start":"03:33.490 ","End":"03:34.840","Text":"then our diagonals fine,"},{"Start":"03:34.840 ","End":"03:38.630","Text":"and also we have 3 rows and 3 columns."},{"Start":"03:40.070 ","End":"03:43.515","Text":"We only count non-zero rows, of course."},{"Start":"03:43.515 ","End":"03:47.740","Text":"We have to tackle the quadratic."},{"Start":"03:47.740 ","End":"03:50.740","Text":"We want this to be non-zero,"},{"Start":"03:50.740 ","End":"03:54.410","Text":"so we solve equals 0 and say everything about that."},{"Start":"03:54.410 ","End":"03:56.985","Text":"Now, for the equation,"},{"Start":"03:56.985 ","End":"04:03.730","Text":"we get 2 roots, 1, and depending on whether you like fractions or decimals,"},{"Start":"04:03.730 ","End":"04:07.135","Text":"minus 0.4, or minus 2/5, take your pick."},{"Start":"04:07.135 ","End":"04:14.625","Text":"Of course we want not these values in order to get exactly 1 solution."},{"Start":"04:14.625 ","End":"04:16.800","Text":"That\u0027s taken care of."},{"Start":"04:16.800 ","End":"04:22.820","Text":"Now all we have left to do is to see what happens when k equals 1,"},{"Start":"04:22.820 ","End":"04:26.990","Text":"and what happens when k equals minus 2/5."},{"Start":"04:26.990 ","End":"04:33.260","Text":"Now notice that there are no k\u0027s anywhere except for this entry here."},{"Start":"04:33.260 ","End":"04:38.900","Text":"I can combine these 2 cases of 1 or minus 0.4"},{"Start":"04:38.900 ","End":"04:41.090","Text":"because they both make this 0."},{"Start":"04:41.090 ","End":"04:45.395","Text":"In both cases, we end up with this here."},{"Start":"04:45.395 ","End":"04:49.220","Text":"Just basically copy this except to that put a 0 here."},{"Start":"04:49.220 ","End":"04:57.420","Text":"But now look, this last row of the augmented matrix is a contradiction row."},{"Start":"04:58.430 ","End":"05:02.685","Text":"For these values of k, there are no solutions."},{"Start":"05:02.685 ","End":"05:06.253","Text":"This exercise is a bit unusual in the sense"},{"Start":"05:06.253 ","End":"05:11.899","Text":"that we didn\u0027t get the third scenario of infinitely many solutions."},{"Start":"05:11.899 ","End":"05:14.450","Text":"We got the case exactly 1 solution,"},{"Start":"05:14.450 ","End":"05:16.085","Text":"we got 0 solutions,"},{"Start":"05:16.085 ","End":"05:19.355","Text":"but no value of k gave us infinite solutions,"},{"Start":"05:19.355 ","End":"05:20.510","Text":"and that\u0027s okay."},{"Start":"05:20.510 ","End":"05:23.130","Text":"We\u0027re done."}],"ID":9835},{"Watched":false,"Name":"Exercise 5","Duration":"7m 30s","ChapterTopicVideoID":9528,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9528.jpeg","UploadDate":"2017-07-26T08:30:31.0370000","DurationForVideoObject":"PT7M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise, we have to determine"},{"Start":"00:03.030 ","End":"00:07.635","Text":"the values of k for which the system below this here."},{"Start":"00:07.635 ","End":"00:09.900","Text":"Linear equations in x, y,"},{"Start":"00:09.900 ","End":"00:13.170","Text":"and z, 3 equations, 3 unknowns,"},{"Start":"00:13.170 ","End":"00:15.480","Text":"for which it has no solutions,"},{"Start":"00:15.480 ","End":"00:17.070","Text":"exactly 1 solution,"},{"Start":"00:17.070 ","End":"00:19.260","Text":"or infinitely many solutions."},{"Start":"00:19.260 ","End":"00:26.790","Text":"As usual, we start by taking the representative matrix for this system,"},{"Start":"00:26.790 ","End":"00:30.480","Text":"which is this augmented matrix."},{"Start":"00:30.480 ","End":"00:33.820","Text":"The idea is to bring it to row echelon form"},{"Start":"00:33.820 ","End":"00:37.305","Text":"and then we can easily analyze what\u0027s going on."},{"Start":"00:37.305 ","End":"00:42.085","Text":"The thing is that this column,"},{"Start":"00:42.085 ","End":"00:47.220","Text":"it doesn\u0027t have any non zero numbers without K."},{"Start":"00:47.220 ","End":"00:50.360","Text":"In the tutorial,"},{"Start":"00:50.360 ","End":"00:54.200","Text":"I gave you some tools of how to deal with this."},{"Start":"00:54.200 ","End":"01:00.035","Text":"For example, we could subtract the second row from the first row,"},{"Start":"01:00.035 ","End":"01:02.755","Text":"and then we would get a 2 here."},{"Start":"01:02.755 ","End":"01:05.740","Text":"We could use that."},{"Start":"01:06.110 ","End":"01:14.645","Text":"I remember warning you about multiplying a row by a constant that could be"},{"Start":"01:14.645 ","End":"01:18.140","Text":"0 or dividing by 0 because what we want to"},{"Start":"01:18.140 ","End":"01:20.870","Text":"do really is divide this by k."},{"Start":"01:20.870 ","End":"01:23.720","Text":"I\u0027ve warned you against this, but this time,"},{"Start":"01:23.720 ","End":"01:27.440","Text":"I thought it might be instructive for ones to go ahead"},{"Start":"01:27.440 ","End":"01:30.475","Text":"and do it and see what happens."},{"Start":"01:30.475 ","End":"01:35.260","Text":"Of course, dividing by k,"},{"Start":"01:35.260 ","End":"01:39.320","Text":"I have to take into account that k might be 0."},{"Start":"01:39.320 ","End":"01:41.840","Text":"In other words, I have to forbid k to be 0,"},{"Start":"01:41.840 ","End":"01:43.850","Text":"which means that at the end,"},{"Start":"01:43.850 ","End":"01:49.535","Text":"we have to come back here and check what happens if k is 0."},{"Start":"01:49.535 ","End":"01:56.270","Text":"Meanwhile, we\u0027ll take the branch where k is not 0."},{"Start":"01:56.270 ","End":"01:59.035","Text":"Let\u0027s see what that gives us."},{"Start":"01:59.035 ","End":"02:02.520","Text":"Straightforward division of the first row."},{"Start":"02:02.520 ","End":"02:05.510","Text":"Here, we have a 1 and we like that because now,"},{"Start":"02:05.510 ","End":"02:08.300","Text":"we can 0 out the rest of the column."},{"Start":"02:08.300 ","End":"02:09.680","Text":"Well, this is already 0."},{"Start":"02:09.680 ","End":"02:16.235","Text":"All I\u0027ll have to do is to subtract k minus 2 times this row from this row,"},{"Start":"02:16.235 ","End":"02:18.295","Text":"and that\u0027s what I wrote here."},{"Start":"02:18.295 ","End":"02:20.190","Text":"If we do that,"},{"Start":"02:20.190 ","End":"02:23.535","Text":"the result is this."},{"Start":"02:23.535 ","End":"02:25.370","Text":"We\u0027ve got 0s here,"},{"Start":"02:25.370 ","End":"02:28.775","Text":"and luckily, we also have a 0 here."},{"Start":"02:28.775 ","End":"02:33.480","Text":"This is already in echelon form,"},{"Start":"02:36.350 ","End":"02:40.040","Text":"but we have to tidy up a bit."},{"Start":"02:40.040 ","End":"02:44.285","Text":"It\u0027s messy, basically, this entry."},{"Start":"02:44.285 ","End":"02:47.690","Text":"A little bit of algebra, common denominator k."},{"Start":"02:47.690 ","End":"02:50.485","Text":"I\u0027m going to continue on the next page."},{"Start":"02:50.485 ","End":"02:52.700","Text":"Let\u0027s start analyzing."},{"Start":"02:52.700 ","End":"02:58.160","Text":"We usually go for the case of exactly 1 solution, the single solution."},{"Start":"02:58.160 ","End":"03:04.195","Text":"For that, we need the number of rows equals number of columns,"},{"Start":"03:04.195 ","End":"03:09.815","Text":"but we\u0027re talking about the restricted matrix, not the augmented."},{"Start":"03:09.815 ","End":"03:11.360","Text":"Now, you might say, of course,"},{"Start":"03:11.360 ","End":"03:12.830","Text":"there\u0027s 3 rows and 3 columns,"},{"Start":"03:12.830 ","End":"03:15.020","Text":"we don\u0027t count 0 rows."},{"Start":"03:15.020 ","End":"03:22.655","Text":"Essentially, what we have to do is make sure that these 2 entries are not 0."},{"Start":"03:22.655 ","End":"03:25.220","Text":"And then we\u0027ll be okay, and we\u0027ll have 3 rows,"},{"Start":"03:25.220 ","End":"03:29.130","Text":"3 columns, no zeros on the diagonal."},{"Start":"03:29.510 ","End":"03:34.375","Text":"If they\u0027re not 0, then we don\u0027t have any contradiction rows."},{"Start":"03:34.375 ","End":"03:37.490","Text":"What we\u0027re looking for is this condition,"},{"Start":"03:37.490 ","End":"03:39.395","Text":"not 0, this not 0,"},{"Start":"03:39.395 ","End":"03:45.925","Text":"and we\u0027re still carrying this k not equal to 0 condition."},{"Start":"03:45.925 ","End":"03:49.550","Text":"Now, we have quadratics here and we need the roots,"},{"Start":"03:49.550 ","End":"03:51.680","Text":"this k squared plus k minus 2,"},{"Start":"03:51.680 ","End":"03:54.740","Text":"the roots turned out to be 1 and minus 2."},{"Start":"03:54.740 ","End":"03:56.735","Text":"This k squared minus 1,"},{"Start":"03:56.735 ","End":"03:59.480","Text":"the roots are 1 and minus 1."},{"Start":"03:59.480 ","End":"04:03.100","Text":"We want to avoid these values and we want to avoid 0."},{"Start":"04:03.100 ","End":"04:07.280","Text":"So k can\u0027t be the 3 values here, 1 minus 2,"},{"Start":"04:07.280 ","End":"04:12.245","Text":"and this is 1 again and minus 1, and the 0."},{"Start":"04:12.245 ","End":"04:17.155","Text":"There\u0027s 4 values, these 4 that we have to avoid."},{"Start":"04:17.155 ","End":"04:20.130","Text":"For all k, except for these 4 values,"},{"Start":"04:20.130 ","End":"04:23.235","Text":"we have a single solution."},{"Start":"04:23.235 ","End":"04:30.035","Text":"What remains to do is to check each 1 of these values and see what happens."},{"Start":"04:30.035 ","End":"04:32.990","Text":"Let\u0027s start with k equals 0,"},{"Start":"04:32.990 ","End":"04:36.920","Text":"even though this is an odd 1 out in the sense that for the other 3 values,"},{"Start":"04:36.920 ","End":"04:39.695","Text":"we can substitute in this matrix,"},{"Start":"04:39.695 ","End":"04:41.630","Text":"but for k equals 0,"},{"Start":"04:41.630 ","End":"04:44.710","Text":"we\u0027ll have to go back to the original matrix."},{"Start":"04:44.710 ","End":"04:49.415","Text":"Scrolling, I copied the original matrix again."},{"Start":"04:49.415 ","End":"04:53.155","Text":"If we put in k equals 0,"},{"Start":"04:53.155 ","End":"04:57.180","Text":"then we get this 0 here minus 2,"},{"Start":"04:57.180 ","End":"05:00.585","Text":"0, 0 squared minus 1."},{"Start":"05:00.585 ","End":"05:02.925","Text":"This is what we get."},{"Start":"05:02.925 ","End":"05:05.690","Text":"Of course, we don\u0027t have to go back at the end and do"},{"Start":"05:05.690 ","End":"05:09.150","Text":"the case k equals 0 because we\u0027re doing it now."},{"Start":"05:09.520 ","End":"05:19.485","Text":"Exchange row 2 with row 1 and get this and it\u0027s already in row echelon form."},{"Start":"05:19.485 ","End":"05:21.780","Text":"We see that there are 3 rows,"},{"Start":"05:21.780 ","End":"05:24.440","Text":"3 columns, non zeros on the diagonal."},{"Start":"05:24.440 ","End":"05:25.925","Text":"Everything is fine."},{"Start":"05:25.925 ","End":"05:28.235","Text":"Exactly 1 solution."},{"Start":"05:28.235 ","End":"05:32.660","Text":"The next k, I\u0027m going to do the case k equals 1"},{"Start":"05:32.660 ","End":"05:38.975","Text":"and I copied what we just scrolled off the screen after the echelon form."},{"Start":"05:38.975 ","End":"05:43.830","Text":"This is it. Plug in k equals 1, we get this."},{"Start":"05:43.830 ","End":"05:45.920","Text":"Look, we have a contradiction row."},{"Start":"05:45.920 ","End":"05:47.795","Text":"In fact, we have 2 contradiction rows,"},{"Start":"05:47.795 ","End":"05:53.070","Text":"but 1 is bad enough to conclude that there are no solutions."},{"Start":"05:53.070 ","End":"05:55.365","Text":"Now, the next k,"},{"Start":"05:55.365 ","End":"05:58.350","Text":"I\u0027ll go for k equals minus 1."},{"Start":"05:58.350 ","End":"06:04.760","Text":"Again, I\u0027ll plug it into this matrix and check the calculations,"},{"Start":"06:04.760 ","End":"06:06.805","Text":"this is what we get."},{"Start":"06:06.805 ","End":"06:10.895","Text":"Once again, we have a contradiction row."},{"Start":"06:10.895 ","End":"06:13.895","Text":"Again, no solutions."},{"Start":"06:13.895 ","End":"06:17.920","Text":"We have 1 more value of k to try."},{"Start":"06:17.920 ","End":"06:22.065","Text":"That last case is k equals minus 2."},{"Start":"06:22.065 ","End":"06:25.440","Text":"Once again, I plugin to"},{"Start":"06:25.440 ","End":"06:33.990","Text":"this echelon form and we get yet again,"},{"Start":"06:33.990 ","End":"06:39.455","Text":"a contradiction row and once again, no solutions."},{"Start":"06:39.455 ","End":"06:46.955","Text":"I suppose we should really summarize and say that the or k equals minus 2 minus 1 and 1,"},{"Start":"06:46.955 ","End":"06:49.715","Text":"we have no solutions."},{"Start":"06:49.715 ","End":"06:53.255","Text":"For k equals 0,"},{"Start":"06:53.255 ","End":"06:55.280","Text":"we have 1 solution."},{"Start":"06:55.280 ","End":"06:59.490","Text":"Also, for all the k\u0027s."},{"Start":"06:59.570 ","End":"07:08.330","Text":"In short, if it\u0027s minus 1 or 1 or minus 2,"},{"Start":"07:08.330 ","End":"07:13.050","Text":"then we have no solutions."},{"Start":"07:17.150 ","End":"07:23.330","Text":"Otherwise, we have 1 solution or write it as"},{"Start":"07:23.330 ","End":"07:26.270","Text":"just a single solution."},{"Start":"07:26.270 ","End":"07:30.630","Text":"We are done."}],"ID":9836},{"Watched":false,"Name":"Exercise 6","Duration":"7m 38s","ChapterTopicVideoID":9514,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9514.jpeg","UploadDate":"2017-07-26T08:24:09.4730000","DurationForVideoObject":"PT7M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.405","Text":"Here we have this system of linear equations and it has a parameter k,"},{"Start":"00:06.405 ","End":"00:11.760","Text":"and we have to determine which values of k make the system have no solutions,"},{"Start":"00:11.760 ","End":"00:16.335","Text":"exactly 1 solution, or infinitely many solutions."},{"Start":"00:16.335 ","End":"00:19.500","Text":"As usual, we do it with matrices."},{"Start":"00:19.500 ","End":"00:26.940","Text":"The representative matrix for this SLE is this augmented matrix,"},{"Start":"00:26.940 ","End":"00:30.705","Text":"and before we can analyze anything,"},{"Start":"00:30.705 ","End":"00:33.390","Text":"we have to bring it into row-echelon form"},{"Start":"00:33.390 ","End":"00:37.065","Text":"using elementary row operations."},{"Start":"00:37.065 ","End":"00:41.070","Text":"I see we have this 1 here,"},{"Start":"00:41.070 ","End":"00:45.230","Text":"and we can use it to 0 out the rest of the column"},{"Start":"00:45.230 ","End":"00:49.925","Text":"by subtracting multiples of this row from the other 2."},{"Start":"00:49.925 ","End":"00:57.620","Text":"To be precise, I subtract k times the first row from the second row,"},{"Start":"00:57.620 ","End":"01:03.865","Text":"and we subtract 3 times the first row from the last row."},{"Start":"01:03.865 ","End":"01:06.960","Text":"If we do that, of course we get 0s here,"},{"Start":"01:06.960 ","End":"01:09.920","Text":"and I\u0027ll leave you to check the computations."},{"Start":"01:09.920 ","End":"01:13.300","Text":"You can always pause the clip and verify."},{"Start":"01:13.300 ","End":"01:15.810","Text":"It is still not in echelon form,"},{"Start":"01:15.810 ","End":"01:18.180","Text":"I need a 0 here also,"},{"Start":"01:18.180 ","End":"01:20.150","Text":"but this is an awkward expression."},{"Start":"01:20.150 ","End":"01:24.164","Text":"I\u0027d like it better if it was a 1 or a number."},{"Start":"01:24.164 ","End":"01:27.640","Text":"But notice if I just negate this row,"},{"Start":"01:27.640 ","End":"01:30.350","Text":"meaning multiply it by minus 1,"},{"Start":"01:30.350 ","End":"01:32.905","Text":"then I see that what I have here,"},{"Start":"01:32.905 ","End":"01:38.545","Text":"1 plus k squared is always positive and it\u0027s never 0."},{"Start":"01:38.545 ","End":"01:41.360","Text":"In fact, it\u0027s bigger than 1 always."},{"Start":"01:41.360 ","End":"01:45.790","Text":"Normally we don\u0027t multiply or divide a row"},{"Start":"01:45.790 ","End":"01:48.025","Text":"by something containing the parameter,"},{"Start":"01:48.025 ","End":"01:49.225","Text":"but in this case,"},{"Start":"01:49.225 ","End":"01:55.595","Text":"it\u0027s safe because we can see that this is never 0."},{"Start":"01:55.595 ","End":"01:59.530","Text":"This is an exceptions to the general advice I gave you about"},{"Start":"01:59.530 ","End":"02:04.980","Text":"not using elementary row operation number 2,"},{"Start":"02:04.980 ","End":"02:07.485","Text":"but here it\u0027s safe."},{"Start":"02:07.485 ","End":"02:10.950","Text":"After doing that, we get a 1 here,"},{"Start":"02:10.950 ","End":"02:17.690","Text":"and here it\u0027s a little bit starting to get messy,"},{"Start":"02:17.690 ","End":"02:19.460","Text":"but we\u0027re still okay."},{"Start":"02:19.460 ","End":"02:21.985","Text":"Of course, next thing we\u0027re going to do,"},{"Start":"02:21.985 ","End":"02:24.495","Text":"and I\u0027ll move to another page,"},{"Start":"02:24.495 ","End":"02:29.300","Text":"is to subtract a multiple of this row from here."},{"Start":"02:29.300 ","End":"02:35.825","Text":"To be precise, I\u0027m going to take 1 minus 3k times this row, which is row 2,"},{"Start":"02:35.825 ","End":"02:40.735","Text":"and subtract it from row 3 and the answer in row 3,"},{"Start":"02:40.735 ","End":"02:44.375","Text":"and here\u0027s where it really starts to get messy."},{"Start":"02:44.375 ","End":"02:47.435","Text":"Now, it is an echelon form."},{"Start":"02:47.435 ","End":"02:54.745","Text":"I mean it is, but we really have to tidy this element up."},{"Start":"02:54.745 ","End":"02:56.514","Text":"This is less important,"},{"Start":"02:56.514 ","End":"03:00.520","Text":"but this is in the restricted matrix and it\u0027s on the diagonal,"},{"Start":"03:00.520 ","End":"03:04.495","Text":"so we\u0027re going to do a bit of algebra to tidy that up."},{"Start":"03:04.495 ","End":"03:07.750","Text":"Now I\u0027m putting the calculations here for reference,"},{"Start":"03:07.750 ","End":"03:10.825","Text":"but I don\u0027t want to dwell on this."},{"Start":"03:10.825 ","End":"03:13.225","Text":"I\u0027ll just give you the result."},{"Start":"03:13.225 ","End":"03:16.750","Text":"Just the last bit of the computation I put it here."},{"Start":"03:16.750 ","End":"03:19.190","Text":"I only worked on this element."},{"Start":"03:20.390 ","End":"03:29.290","Text":"This is what I get, and the first thing I want to do is look for a single solution,"},{"Start":"03:29.290 ","End":"03:32.125","Text":"the case of exactly 1 solution."},{"Start":"03:32.125 ","End":"03:35.580","Text":"For that, I need 3 columns,"},{"Start":"03:35.580 ","End":"03:39.965","Text":"but I need 3 rows by which I mean that I want this to be non-zero."},{"Start":"03:39.965 ","End":"03:42.710","Text":"The denominator here is never 0,"},{"Start":"03:42.710 ","End":"03:43.890","Text":"as we mentioned before,"},{"Start":"03:43.890 ","End":"03:48.005","Text":"so we just need the numerator to be not 0."},{"Start":"03:48.005 ","End":"03:50.105","Text":"Now this is a cubic,"},{"Start":"03:50.105 ","End":"03:55.235","Text":"but we have studied or I hope you have,"},{"Start":"03:55.235 ","End":"03:58.895","Text":"in any event, I\u0027ll tell you the result of how we go about this."},{"Start":"03:58.895 ","End":"04:06.335","Text":"That if we\u0027re looking for a whole number solution or even a rational solution fraction,"},{"Start":"04:06.335 ","End":"04:14.370","Text":"it has to be a whole number that divides the last element, the free co-efficient."},{"Start":"04:14.960 ","End":"04:18.470","Text":"We have 8 possibilities to check"},{"Start":"04:18.470 ","End":"04:23.930","Text":"because the divisors of 6 are plus or minus 1, 2, 3, or 6,"},{"Start":"04:23.930 ","End":"04:27.035","Text":"and just trial and error."},{"Start":"04:27.035 ","End":"04:30.860","Text":"Take each of these and see if it\u0027s a root of this."},{"Start":"04:30.860 ","End":"04:33.050","Text":"I\u0027ve done the work for you."},{"Start":"04:33.050 ","End":"04:36.200","Text":"The roots come out to be minus 1,"},{"Start":"04:36.200 ","End":"04:37.580","Text":"2, and minus 3."},{"Start":"04:37.580 ","End":"04:39.470","Text":"For it not to be 0,"},{"Start":"04:39.470 ","End":"04:43.825","Text":"we want k not to be any 1 of these 3 values,"},{"Start":"04:43.825 ","End":"04:48.900","Text":"and then we\u0027ll have exactly 1 solution for the SLE."},{"Start":"04:48.900 ","End":"04:57.900","Text":"Well, what remains now is to substitute each 1 of these into the matrix here."},{"Start":"04:59.060 ","End":"05:04.010","Text":"Check what happens with each 1 of these separately."},{"Start":"05:04.010 ","End":"05:08.310","Text":"Just want to note that I didn\u0027t really care what this is equal to,"},{"Start":"05:08.310 ","End":"05:10.400","Text":"if this is not 0,"},{"Start":"05:10.400 ","End":"05:14.105","Text":"then this will not give me a contradiction row,"},{"Start":"05:14.105 ","End":"05:18.350","Text":"and it doesn\u0027t really determine how many solutions I have."},{"Start":"05:18.350 ","End":"05:21.260","Text":"It will come in to effect, I mean,"},{"Start":"05:21.260 ","End":"05:26.840","Text":"it will be important when we start substituting values because when this is 0,"},{"Start":"05:26.840 ","End":"05:28.970","Text":"we\u0027ll need to know if this is 0 or not."},{"Start":"05:28.970 ","End":"05:31.270","Text":"Meanwhile, we didn\u0027t care about this."},{"Start":"05:31.270 ","End":"05:35.390","Text":"I\u0027m going to move on. I need to substitute 3 different values."},{"Start":"05:35.390 ","End":"05:39.455","Text":"I\u0027ll take the case of minus 1 first,"},{"Start":"05:39.455 ","End":"05:41.930","Text":"plug it in here."},{"Start":"05:41.930 ","End":"05:45.425","Text":"This comes out 0 of course,"},{"Start":"05:45.425 ","End":"05:48.905","Text":"but this comes out to be non-zero,"},{"Start":"05:48.905 ","End":"05:52.925","Text":"so immediately we see that we have a contradiction row,"},{"Start":"05:52.925 ","End":"05:56.060","Text":"it doesn\u0027t matter what else do we observe."},{"Start":"05:56.060 ","End":"06:02.920","Text":"This means that there are no solutions to the SLE."},{"Start":"06:02.920 ","End":"06:08.870","Text":"The other 2 values we\u0027re also going to plug in that they\u0027re all going to give me 0 here."},{"Start":"06:08.870 ","End":"06:12.395","Text":"The question is whether I get here a 0 or not."},{"Start":"06:12.395 ","End":"06:14.554","Text":"Let\u0027s go for the next value."},{"Start":"06:14.554 ","End":"06:17.545","Text":"I\u0027ll go for k equals 2."},{"Start":"06:17.545 ","End":"06:19.580","Text":"Check the calculations."},{"Start":"06:19.580 ","End":"06:22.640","Text":"This is what we get, but the important thing is that"},{"Start":"06:22.640 ","End":"06:27.230","Text":"we again got non-zero here so this is a contradiction row,"},{"Start":"06:27.230 ","End":"06:29.585","Text":"and again, no solutions."},{"Start":"06:29.585 ","End":"06:32.270","Text":"We\u0027ve done minus 1 and 2,"},{"Start":"06:32.270 ","End":"06:34.365","Text":"the last 1 is,"},{"Start":"06:34.365 ","End":"06:36.540","Text":"if I recall, minus 3."},{"Start":"06:36.540 ","End":"06:39.350","Text":"If we substitute that in, well,"},{"Start":"06:39.350 ","End":"06:40.550","Text":"it\u0027s scrolled off the screen,"},{"Start":"06:40.550 ","End":"06:45.710","Text":"but yet again, we get a non-zero here,"},{"Start":"06:45.710 ","End":"06:47.720","Text":"and of course still all 0s here."},{"Start":"06:47.720 ","End":"06:53.135","Text":"So once again, contradiction row and no solutions."},{"Start":"06:53.135 ","End":"06:55.985","Text":"I suppose I should have really written a summary."},{"Start":"06:55.985 ","End":"06:57.949","Text":"Let\u0027s see if I remember."},{"Start":"06:57.949 ","End":"07:08.740","Text":"We had minus 1, 2, and minus 3 that gave us exactly 1 solution."},{"Start":"07:10.550 ","End":"07:15.590","Text":"In other words, all k that are not equal to these then"},{"Start":"07:15.590 ","End":"07:21.840","Text":"we had no solution, 0 solutions."},{"Start":"07:21.840 ","End":"07:24.080","Text":"I might also remark that"},{"Start":"07:24.080 ","End":"07:30.635","Text":"we never got the case of infinite solutions, infinite number."},{"Start":"07:30.635 ","End":"07:33.280","Text":"No value of k gives infinite."},{"Start":"07:33.280 ","End":"07:36.630","Text":"I know you might want to mark that or not."},{"Start":"07:36.630 ","End":"07:38.530","Text":"Anyway."}],"ID":9837},{"Watched":false,"Name":"Exercise 7","Duration":"5m 21s","ChapterTopicVideoID":9515,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9515.jpeg","UploadDate":"2017-07-26T08:24:25.1430000","DurationForVideoObject":"PT5M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"System of linear equations,"},{"Start":"00:02.010 ","End":"00:05.730","Text":"actually 3 equations in 2 variables, x and y."},{"Start":"00:05.730 ","End":"00:09.480","Text":"There\u0027s a parameter k, and we have to decide which values of"},{"Start":"00:09.480 ","End":"00:13.230","Text":"k give the system 1 solution,"},{"Start":"00:13.230 ","End":"00:17.385","Text":"infinite solutions, or 0 solutions."},{"Start":"00:17.385 ","End":"00:22.455","Text":"Let\u0027s start by converting it to a matrix."},{"Start":"00:22.455 ","End":"00:25.920","Text":"Here\u0027s the representation as an augmented matrix."},{"Start":"00:25.920 ","End":"00:27.240","Text":"Notice the bar here,"},{"Start":"00:27.240 ","End":"00:30.495","Text":"that\u0027s a separate left-hand side from the right-hand side."},{"Start":"00:30.495 ","End":"00:34.800","Text":"The thing to do is to start trying to bring it to row"},{"Start":"00:34.800 ","End":"00:39.390","Text":"echelon form with the elementary row operations,"},{"Start":"00:39.390 ","End":"00:45.395","Text":"we have this leading coefficient 2 and we want to 0 out what\u0027s below it."},{"Start":"00:45.395 ","End":"00:49.790","Text":"But it will be simpler if we make it into a 1."},{"Start":"00:49.790 ","End":"00:54.570","Text":"Let\u0027s divide the top row by 2 or multiply by half."},{"Start":"00:54.680 ","End":"00:58.910","Text":"That gives us these, only the top row is changed."},{"Start":"00:58.910 ","End":"01:04.550","Text":"Now we can subtract multiples of the top row from these rows to get zeros here."},{"Start":"01:04.550 ","End":"01:07.910","Text":"K plus 3 times this row subtracted from here,"},{"Start":"01:07.910 ","End":"01:10.760","Text":"and 6 times this row subtracted from here."},{"Start":"01:10.760 ","End":"01:17.095","Text":"This is what it is in row operation notation."},{"Start":"01:17.095 ","End":"01:19.760","Text":"Here we are, I won\u0027t go into all the details,"},{"Start":"01:19.760 ","End":"01:22.610","Text":"but notice that we have 0 and 0 here,"},{"Start":"01:22.610 ","End":"01:26.720","Text":"and we have to do a little bit more work than with numbers because we have also letters,"},{"Start":"01:26.720 ","End":"01:28.955","Text":"k, little bit of algebra."},{"Start":"01:28.955 ","End":"01:37.520","Text":"Just give you maybe an example that this minus 6 times this is this minus 9,"},{"Start":"01:37.520 ","End":"01:41.330","Text":"which is 7k squared minus 7x plus 2 minus 9,"},{"Start":"01:41.330 ","End":"01:43.460","Text":"and so on and so on."},{"Start":"01:43.460 ","End":"01:49.445","Text":"Look, we got lucky because there\u0027s a 0 here already."},{"Start":"01:49.445 ","End":"01:54.380","Text":"It already is in echelon form."},{"Start":"01:54.380 ","End":"01:57.710","Text":"The restricted part, this part here,"},{"Start":"01:57.710 ","End":"02:01.130","Text":"we don\u0027t care about this part of this,"},{"Start":"02:01.130 ","End":"02:03.270","Text":"I mean for echelon."},{"Start":"02:05.390 ","End":"02:09.680","Text":"Well, before that, just want to tidy it up a bit."},{"Start":"02:09.680 ","End":"02:12.590","Text":"Let\u0027s put these 2 entries."},{"Start":"02:12.590 ","End":"02:15.755","Text":"Sorry, just this one with a common denominator,"},{"Start":"02:15.755 ","End":"02:18.855","Text":"let\u0027s put it over 2."},{"Start":"02:18.855 ","End":"02:21.405","Text":"Then we can see it better."},{"Start":"02:21.405 ","End":"02:27.530","Text":"Look, the main thing is happening here is in the last row."},{"Start":"02:27.530 ","End":"02:31.415","Text":"Most important thing we could know is if there\u0027s a contradiction row or not,"},{"Start":"02:31.415 ","End":"02:33.575","Text":"and we have 0 and we have 0."},{"Start":"02:33.575 ","End":"02:35.060","Text":"Now if we have 0 here,"},{"Start":"02:35.060 ","End":"02:37.025","Text":"it\u0027s just a 0 row, we throw it out."},{"Start":"02:37.025 ","End":"02:39.575","Text":"But if this is not 0,"},{"Start":"02:39.575 ","End":"02:42.155","Text":"then it means we have a contradiction,"},{"Start":"02:42.155 ","End":"02:44.020","Text":"and it means that there\u0027s no solution."},{"Start":"02:44.020 ","End":"02:47.325","Text":"Let\u0027s check that condition first."},{"Start":"02:47.325 ","End":"02:49.880","Text":"Well obviously this is 0,"},{"Start":"02:49.880 ","End":"02:51.020","Text":"divide it by 7,"},{"Start":"02:51.020 ","End":"02:53.120","Text":"then k squared minus 1 is 0,"},{"Start":"02:53.120 ","End":"02:54.620","Text":"k squared is 1,"},{"Start":"02:54.620 ","End":"02:57.005","Text":"k is plus or minus 1."},{"Start":"02:57.005 ","End":"03:05.180","Text":"For plus or minus 1, it\u0027s 0 and we\u0027re okay but other than that,"},{"Start":"03:05.180 ","End":"03:08.045","Text":"then we don\u0027t have any solutions."},{"Start":"03:08.045 ","End":"03:09.470","Text":"For most k,"},{"Start":"03:09.470 ","End":"03:11.150","Text":"all except 2, we have no solution."},{"Start":"03:11.150 ","End":"03:15.740","Text":"Now let\u0027s just check what happens when k is 1 and when k is minus 1."},{"Start":"03:15.740 ","End":"03:17.765","Text":"Move to a new page."},{"Start":"03:17.765 ","End":"03:20.090","Text":"Let\u0027s take the k equals 1 first."},{"Start":"03:20.090 ","End":"03:21.500","Text":"Later we\u0027ll do minus 1."},{"Start":"03:21.500 ","End":"03:25.180","Text":"Just plug everywhere you see k plug-in 1 and of course,"},{"Start":"03:25.180 ","End":"03:27.170","Text":"we get 0 here as we were expecting,"},{"Start":"03:27.170 ","End":"03:28.880","Text":"but as a surprise,"},{"Start":"03:28.880 ","End":"03:32.330","Text":"we also get a 0 here because look,"},{"Start":"03:32.330 ","End":"03:34.145","Text":"1 squared is 1,"},{"Start":"03:34.145 ","End":"03:38.195","Text":"1 minus 1.5 plus 0.5 is also 0."},{"Start":"03:38.195 ","End":"03:40.970","Text":"We\u0027ve got another row of zeros and that\u0027s no problem."},{"Start":"03:40.970 ","End":"03:42.620","Text":"It\u0027s not a contradiction."},{"Start":"03:42.620 ","End":"03:45.320","Text":"But if we look at the restricted matrix,"},{"Start":"03:45.320 ","End":"03:49.770","Text":"this part, we see that there are 3."},{"Start":"03:50.030 ","End":"03:53.930","Text":"Sorry, we don\u0027t count these as rows."},{"Start":"03:53.930 ","End":"03:56.645","Text":"We have 1 row and 2 columns."},{"Start":"03:56.645 ","End":"04:01.420","Text":"Let me just write that, 1 row and 2 columns."},{"Start":"04:01.420 ","End":"04:05.915","Text":"We don\u0027t count the 0 rows and 1 is less than 2."},{"Start":"04:05.915 ","End":"04:08.990","Text":"When the number of rows is less than number of columns,"},{"Start":"04:08.990 ","End":"04:12.170","Text":"that\u0027s an indication that there are infinitely many solutions."},{"Start":"04:12.170 ","End":"04:14.305","Text":"So that\u0027s that case."},{"Start":"04:14.305 ","End":"04:15.890","Text":"Now we\u0027ll take the other one,"},{"Start":"04:15.890 ","End":"04:17.465","Text":"k is minus 1."},{"Start":"04:17.465 ","End":"04:20.300","Text":"This is what we get, or I guess it\u0027s scrolled off-screen."},{"Start":"04:20.300 ","End":"04:25.175","Text":"But look, if we plug in minus 1 here,"},{"Start":"04:25.175 ","End":"04:30.510","Text":"what we get minus 1 squared is 1,"},{"Start":"04:30.510 ","End":"04:33.000","Text":"then plus 1.5 plus 0.5 is 3,"},{"Start":"04:33.000 ","End":"04:36.500","Text":"and you know what, I\u0027ll leave you to check the other entries."},{"Start":"04:36.500 ","End":"04:44.270","Text":"But this time if we look at the restricted matrix or coefficient matrix,"},{"Start":"04:44.270 ","End":"04:48.785","Text":"we see that remove this row because it\u0027s not a row it\u0027s a zeros."},{"Start":"04:48.785 ","End":"04:51.305","Text":"We have 2 rows and 2 columns."},{"Start":"04:51.305 ","End":"04:55.040","Text":"I\u0027ll just write that 2 rows, 2 columns."},{"Start":"04:55.040 ","End":"04:57.380","Text":"But that\u0027s the same 2 equals 2."},{"Start":"04:57.380 ","End":"05:00.280","Text":"When we have the same number of rows and columns,"},{"Start":"05:00.280 ","End":"05:03.360","Text":"then there\u0027s exactly one solution."},{"Start":"05:03.360 ","End":"05:05.840","Text":"We\u0027re basically done although if you wanted to,"},{"Start":"05:05.840 ","End":"05:08.075","Text":"you could summarize and say, okay,"},{"Start":"05:08.075 ","End":"05:10.175","Text":"k is 1 infinite solutions,"},{"Start":"05:10.175 ","End":"05:12.670","Text":"k is minus 1, just one solution."},{"Start":"05:12.670 ","End":"05:15.260","Text":"When k is neither 1 nor minus 1,"},{"Start":"05:15.260 ","End":"05:16.625","Text":"as we saw before,"},{"Start":"05:16.625 ","End":"05:21.540","Text":"then there are no solutions and we\u0027re done."}],"ID":9838},{"Watched":false,"Name":"Exercise 8","Duration":"4m 51s","ChapterTopicVideoID":9516,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9516.jpeg","UploadDate":"2017-07-26T08:24:40.5570000","DurationForVideoObject":"PT4M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have a system of linear equations."},{"Start":"00:03.660 ","End":"00:08.250","Text":"Note that it\u0027s 2 equations and 3 unknowns, x, y, and z."},{"Start":"00:08.250 ","End":"00:10.410","Text":"There\u0027s also a parameter k."},{"Start":"00:10.410 ","End":"00:14.280","Text":"As usual, we have to figure out which values of k"},{"Start":"00:14.280 ","End":"00:16.380","Text":"give which number of solutions."},{"Start":"00:16.380 ","End":"00:22.110","Text":"There\u0027s only 3 possibilities 0, 1 or infinity for the number of solutions."},{"Start":"00:22.110 ","End":"00:23.970","Text":"We\u0027ll do it with a matrix,"},{"Start":"00:23.970 ","End":"00:25.560","Text":"it\u0027s easier that way."},{"Start":"00:25.560 ","End":"00:28.515","Text":"This is the matrix that represents this."},{"Start":"00:28.515 ","End":"00:32.850","Text":"It\u0027s an augmented matrix with this bar here and here\u0027s the right-hand side,"},{"Start":"00:32.850 ","End":"00:35.625","Text":"and here are the coefficients of the left-hand side."},{"Start":"00:35.625 ","End":"00:39.680","Text":"Let\u0027s try and bring it to row,"},{"Start":"00:39.680 ","End":"00:41.660","Text":"echelon form as we usually do."},{"Start":"00:41.660 ","End":"00:43.850","Text":"There\u0027s 2 here and 4 here,"},{"Start":"00:43.850 ","End":"00:46.310","Text":"though no problem to get rid of this 4,"},{"Start":"00:46.310 ","End":"00:50.705","Text":"by subtracting twice the top row from the bottom row."},{"Start":"00:50.705 ","End":"00:53.780","Text":"This is the notation for what I just said."},{"Start":"00:53.780 ","End":"00:58.650","Text":"This actually is already in echelon form"},{"Start":"00:58.650 ","End":"01:04.785","Text":"for 2 equation though 2 rows of the matrix."},{"Start":"01:04.785 ","End":"01:08.335","Text":"This is the leading coefficient here."},{"Start":"01:08.335 ","End":"01:13.775","Text":"The thing is that there could be some surprise."},{"Start":"01:13.775 ","End":"01:15.665","Text":"Well, that\u0027s not the way to express it."},{"Start":"01:15.665 ","End":"01:17.585","Text":"This could be 0."},{"Start":"01:17.585 ","End":"01:20.570","Text":"If this is 0, then that changes everything"},{"Start":"01:20.570 ","End":"01:23.035","Text":"because then we have all 3 0s here."},{"Start":"01:23.035 ","End":"01:27.200","Text":"Then the only question is whether it\u0027s all 0s"},{"Start":"01:27.200 ","End":"01:30.875","Text":"when we can throw it out or not 0 here."},{"Start":"01:30.875 ","End":"01:33.560","Text":"Then we get a contradiction but,"},{"Start":"01:33.560 ","End":"01:35.555","Text":"of course, it could also be non 0."},{"Start":"01:35.555 ","End":"01:37.450","Text":"Really what we\u0027re going to do now,"},{"Start":"01:37.450 ","End":"01:39.990","Text":"is consider this element here,"},{"Start":"01:39.990 ","End":"01:42.630","Text":"everything depends really on this."},{"Start":"01:42.630 ","End":"01:44.730","Text":"Well, maybe also on this."},{"Start":"01:44.730 ","End":"01:45.930","Text":"Let\u0027s see."},{"Start":"01:45.930 ","End":"01:50.025","Text":"I\u0027m going to factorize this quadratic."},{"Start":"01:50.025 ","End":"01:51.830","Text":"This is the result of that."},{"Start":"01:51.830 ","End":"01:54.145","Text":"There\u0027s many ways to factorize."},{"Start":"01:54.145 ","End":"01:58.575","Text":"1 way to solve a quadratic equation this equals 0."},{"Start":"01:58.575 ","End":"02:02.430","Text":"If you do that, you get that 2 and 3 other solutions"},{"Start":"02:02.430 ","End":"02:08.300","Text":"and so k minus 2, k minus 3 is the factorization."},{"Start":"02:08.300 ","End":"02:13.350","Text":"Anyway, there are other ways."},{"Start":"02:13.350 ","End":"02:15.375","Text":"Really there\u0027s 3 cases here,"},{"Start":"02:15.375 ","End":"02:18.720","Text":"or apparently 3 cases could be 2,"},{"Start":"02:18.720 ","End":"02:22.295","Text":"for k could be 3 or it could be neither."},{"Start":"02:22.295 ","End":"02:28.840","Text":"Now, let\u0027s see what happens if k is equal to 3."},{"Start":"02:28.840 ","End":"02:33.380","Text":"If k is 3 obviously this is 0 because that\u0027s a root."},{"Start":"02:33.380 ","End":"02:38.270","Text":"Here we already have a 0 and this when k is 3,"},{"Start":"02:38.270 ","End":"02:41.190","Text":"3 minus 2 is 1."},{"Start":"02:41.600 ","End":"02:45.280","Text":"Immediately we can say that there are no solutions,"},{"Start":"02:45.280 ","End":"02:47.905","Text":"because here we have a contradiction row."},{"Start":"02:47.905 ","End":"02:51.325","Text":"When k is 3, then no solutions."},{"Start":"02:51.325 ","End":"02:54.305","Text":"Now note, if k is not 3,"},{"Start":"02:54.305 ","End":"02:56.295","Text":"I really have to look at 2 cases."},{"Start":"02:56.295 ","End":"02:58.410","Text":"If k is 2,"},{"Start":"02:58.410 ","End":"03:01.695","Text":"then this is 0 but so is this,"},{"Start":"03:01.695 ","End":"03:06.325","Text":"and all 0s means that we can just draw this row out."},{"Start":"03:06.325 ","End":"03:13.425","Text":"If k is 2, then we have 1 row and 3 columns."},{"Start":"03:13.425 ","End":"03:20.460","Text":"If k is not equal to 2."},{"Start":"03:20.460 ","End":"03:23.790","Text":"It should be down here, and it\u0027s still not equal to 3,"},{"Start":"03:23.790 ","End":"03:27.255","Text":"then this is not going to be 0."},{"Start":"03:27.255 ","End":"03:33.655","Text":"We\u0027re going to have 2 rows and still 3 columns."},{"Start":"03:33.655 ","End":"03:38.290","Text":"When I say columns, I\u0027m talking about the restricted matrix."},{"Start":"03:38.290 ","End":"03:41.710","Text":"Of course, we do this consideration on the restricted."},{"Start":"03:41.710 ","End":"03:46.245","Text":"Either way we have 1 or 2,"},{"Start":"03:46.245 ","End":"03:52.050","Text":"they\u0027re both less than 3 so it means infinite solutions."},{"Start":"03:52.050 ","End":"03:54.615","Text":"That\u0027s the result,"},{"Start":"03:54.615 ","End":"03:57.735","Text":"if k is 3 no solutions."},{"Start":"03:57.735 ","End":"04:00.490","Text":"K not equal to 3 infinitely many,"},{"Start":"04:00.490 ","End":"04:04.000","Text":"although we had to divide this up into 2 cases, 2 and not 2."},{"Start":"04:04.000 ","End":"04:05.425","Text":"Now I just want to show you."},{"Start":"04:05.425 ","End":"04:08.015","Text":"If we go back to the beginning,"},{"Start":"04:08.015 ","End":"04:15.375","Text":"then soon as you see that there\u0027s 2 equations in 3 unknowns,"},{"Start":"04:15.375 ","End":"04:20.349","Text":"it can\u0027t be a single solution or just 1 solution"},{"Start":"04:20.349 ","End":"04:25.070","Text":"because there\u0027s more variables than equations."},{"Start":"04:25.070 ","End":"04:26.350","Text":"There\u0027s no way we can get"},{"Start":"04:26.350 ","End":"04:31.450","Text":"an equal number of rows and columns in the restricted"},{"Start":"04:31.450 ","End":"04:35.390","Text":"because echelon form can only get rid of rows;"},{"Start":"04:35.390 ","End":"04:37.555","Text":"it can\u0027t increase the number of rows,"},{"Start":"04:37.555 ","End":"04:41.680","Text":"so 3 columns and at most 2 rows,"},{"Start":"04:41.680 ","End":"04:44.740","Text":"we know it\u0027s not going to be a unique solution."},{"Start":"04:44.740 ","End":"04:48.040","Text":"We knew that it\u0027s only either 0 solutions or infinity"},{"Start":"04:48.040 ","End":"04:51.890","Text":"and that confirms what we saw and we\u0027re done."}],"ID":9839},{"Watched":false,"Name":"Exercise 9","Duration":"5m 57s","ChapterTopicVideoID":9517,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9517.jpeg","UploadDate":"2017-07-26T08:25:19.8570000","DurationForVideoObject":"PT5M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.210","Text":"Here we have a system and it\u0027s 4 equations, 3 unknowns, x, y, z,"},{"Start":"00:06.210 ","End":"00:08.430","Text":"and there\u0027s 1 parameter k."},{"Start":"00:08.430 ","End":"00:12.360","Text":"We have to decide which values of k give how many solutions."},{"Start":"00:12.360 ","End":"00:17.310","Text":"We know there\u0027s only 3 cases: 1, 0, or infinity."},{"Start":"00:17.310 ","End":"00:20.670","Text":"Let\u0027s represent this as a matrix,"},{"Start":"00:20.670 ","End":"00:23.565","Text":"an augmented matrix like so."},{"Start":"00:23.565 ","End":"00:27.940","Text":"We want to try and bring it to row-echelon form."},{"Start":"00:30.050 ","End":"00:35.780","Text":"We have here a non-zero element and we want 0 out everything else."},{"Start":"00:35.780 ","End":"00:39.150","Text":"But it would be nicer if it was a 1 here."},{"Start":"00:39.150 ","End":"00:42.875","Text":"Why don\u0027t we just exchange row 1 with row 3."},{"Start":"00:42.875 ","End":"00:46.130","Text":"This is the notation and this is what we get."},{"Start":"00:46.130 ","End":"00:50.345","Text":"You see this row, it comes to the top here and vice versa."},{"Start":"00:50.345 ","End":"00:52.385","Text":"Let\u0027s look at this now."},{"Start":"00:52.385 ","End":"00:58.245","Text":"We want to 0 out everything below the 1."},{"Start":"00:58.245 ","End":"01:01.255","Text":"We\u0027ll do 3 operations in 1."},{"Start":"01:01.255 ","End":"01:04.970","Text":"We\u0027ll subtract multiples of the first row from each of the other rows."},{"Start":"01:04.970 ","End":"01:06.560","Text":"We\u0027ll subtract it twice from here,"},{"Start":"01:06.560 ","End":"01:07.910","Text":"3 times from here,"},{"Start":"01:07.910 ","End":"01:09.800","Text":"and k times from here."},{"Start":"01:09.800 ","End":"01:13.990","Text":"This is just the row notation form for what I just said."},{"Start":"01:13.990 ","End":"01:15.580","Text":"If we do all that,"},{"Start":"01:15.580 ","End":"01:17.425","Text":"then we end up with this."},{"Start":"01:17.425 ","End":"01:19.450","Text":"I\u0027m not going to go through it in detail."},{"Start":"01:19.450 ","End":"01:21.910","Text":"You can just pause the clip and check it."},{"Start":"01:21.910 ","End":"01:24.060","Text":"We\u0027ve done enough of these."},{"Start":"01:24.060 ","End":"01:26.725","Text":"We\u0027ve got 0s here."},{"Start":"01:26.725 ","End":"01:28.030","Text":"Now I want to continue."},{"Start":"01:28.030 ","End":"01:31.370","Text":"We want to have 0 here and here."},{"Start":"01:32.090 ","End":"01:34.545","Text":"I notice something."},{"Start":"01:34.545 ","End":"01:39.265","Text":"You should take use of opportunities if you see something, some shortcut."},{"Start":"01:39.265 ","End":"01:43.345","Text":"Look, this is half of this and this is half of this."},{"Start":"01:43.345 ","End":"01:47.350","Text":"I\u0027m thinking, let\u0027s subtract."},{"Start":"01:47.350 ","End":"01:50.109","Text":"We could do it either way, but I\u0027m going to subtract"},{"Start":"01:50.109 ","End":"01:56.570","Text":"half of this row from this row and get a 0 here."},{"Start":"01:56.570 ","End":"02:02.135","Text":"Here\u0027s the row notation of what I just said and the result is this."},{"Start":"02:02.135 ","End":"02:04.445","Text":"I\u0027ll leave you to check the calculations."},{"Start":"02:04.445 ","End":"02:06.935","Text":"But notice that we got a bonus."},{"Start":"02:06.935 ","End":"02:08.930","Text":"I mean, we got 0s."},{"Start":"02:08.930 ","End":"02:11.670","Text":"This minus half this is 0."},{"Start":"02:11.670 ","End":"02:14.885","Text":"This minus half this also came out to be 0."},{"Start":"02:14.885 ","End":"02:17.585","Text":"Because if you look at it, if you divide this by 2,"},{"Start":"02:17.585 ","End":"02:19.750","Text":"you get 1 minus 1 1/2 k,"},{"Start":"02:19.750 ","End":"02:22.360","Text":"which is just this in reverse order."},{"Start":"02:22.360 ","End":"02:30.750","Text":"We\u0027re now down to just 3 rows."},{"Start":"02:30.750 ","End":"02:31.745","Text":"What do we do now?"},{"Start":"02:31.745 ","End":"02:36.290","Text":"I don\u0027t really like to use an element with k"},{"Start":"02:36.290 ","End":"02:41.375","Text":"to start dividing or adding multiples of this."},{"Start":"02:41.375 ","End":"02:43.310","Text":"We want to try and do something"},{"Start":"02:43.310 ","End":"02:48.320","Text":"to get a non-zero element here, but not containing k."},{"Start":"02:48.320 ","End":"02:51.860","Text":"I\u0027m going to do some swapping around."},{"Start":"02:51.860 ","End":"02:55.520","Text":"But first of all, let\u0027s do some canceling, reducing."},{"Start":"02:55.520 ","End":"02:58.400","Text":"This 1, I\u0027m going to divide by minus 20."},{"Start":"02:58.400 ","End":"03:02.720","Text":"That\u0027ll give us a 1 here with the idea of then swapping it."},{"Start":"03:02.720 ","End":"03:09.180","Text":"Here also, I\u0027m going to divide by, let\u0027s see, 2 or minus,"},{"Start":"03:09.180 ","End":"03:13.305","Text":"let\u0027s divide this by minus 2 and make this positive."},{"Start":"03:13.305 ","End":"03:15.755","Text":"Here\u0027s the row notation,"},{"Start":"03:15.755 ","End":"03:17.795","Text":"and here\u0027s the result."},{"Start":"03:17.795 ","End":"03:23.930","Text":"Dividing in this row by minus 2 and dividing this row by minus 20."},{"Start":"03:23.930 ","End":"03:26.315","Text":"Now this 1 is useful, and like I said,"},{"Start":"03:26.315 ","End":"03:30.495","Text":"my aim is to put it here because then it\u0027s easier."},{"Start":"03:30.495 ","End":"03:39.720","Text":"What I\u0027m going to do is just swap row 2 with row 3 with this notation."},{"Start":"03:39.720 ","End":"03:43.500","Text":"We get this with the 1 here."},{"Start":"03:43.500 ","End":"03:47.360","Text":"Now we can zero out the rest of the column,"},{"Start":"03:47.360 ","End":"03:48.800","Text":"which is just this element."},{"Start":"03:48.800 ","End":"03:55.070","Text":"If I subtract 1 plus 4k of the second row from the third row,"},{"Start":"03:55.070 ","End":"03:58.190","Text":"let\u0027s see, give me some space here,"},{"Start":"03:58.190 ","End":"04:00.425","Text":"okay, this is what I\u0027m going to do,"},{"Start":"04:00.425 ","End":"04:02.645","Text":"and this is what we get."},{"Start":"04:02.645 ","End":"04:06.980","Text":"It is indeed in row-echelon form already and there\u0027s the 0."},{"Start":"04:06.980 ","End":"04:10.550","Text":"But look what a mess this element is."},{"Start":"04:10.550 ","End":"04:13.130","Text":"I don\u0027t so much care about its neighbor over here"},{"Start":"04:13.130 ","End":"04:17.465","Text":"because I really only more care about the reduced matrix."},{"Start":"04:17.465 ","End":"04:19.400","Text":"Let\u0027s simplify this element,"},{"Start":"04:19.400 ","End":"04:24.720","Text":"and its friend to the right, I\u0027ll just leave alone."},{"Start":"04:25.730 ","End":"04:27.979","Text":"If you do the algebra,"},{"Start":"04:27.979 ","End":"04:31.370","Text":"this boils down to just this,"},{"Start":"04:31.370 ","End":"04:34.225","Text":"which is a lot easier to deal with."},{"Start":"04:34.225 ","End":"04:36.995","Text":"I colored these 3 elements."},{"Start":"04:36.995 ","End":"04:39.740","Text":"I mean, the last row might as well just thrown it out."},{"Start":"04:39.740 ","End":"04:40.910","Text":"I\u0027m dragging it along,"},{"Start":"04:40.910 ","End":"04:43.425","Text":"but it\u0027s like it\u0027s only 3 rows here."},{"Start":"04:43.425 ","End":"04:48.395","Text":"The big question is if this is 0 or not 0."},{"Start":"04:48.395 ","End":"04:51.950","Text":"If this is not 0, I have non zeros on the diagonal."},{"Start":"04:51.950 ","End":"04:59.015","Text":"I have the same number of rows and columns in the reduced matrix."},{"Start":"04:59.015 ","End":"05:01.310","Text":"I\u0027ll have a single solution."},{"Start":"05:01.310 ","End":"05:06.200","Text":"Really, there\u0027s only 2 possibilities that k is 1 or not 1."},{"Start":"05:06.200 ","End":"05:08.705","Text":"Like I said, if k is not 1,"},{"Start":"05:08.705 ","End":"05:14.035","Text":"then this is going to be non-zero and then we get exactly 1 solution."},{"Start":"05:14.035 ","End":"05:20.140","Text":"The alternative is that k is equal to 1 and then I\u0027ll get a 0 here."},{"Start":"05:20.140 ","End":"05:24.490","Text":"In fact, this is what I end up with."},{"Start":"05:26.420 ","End":"05:30.470","Text":"We immediately spot a problem in this row."},{"Start":"05:30.470 ","End":"05:33.670","Text":"This is a contradiction row,"},{"Start":"05:33.670 ","End":"05:37.100","Text":"0 is equal to something non-zero."},{"Start":"05:37.100 ","End":"05:41.820","Text":"And right away we know that this means that the system has no solution."},{"Start":"05:41.820 ","End":"05:43.145","Text":"There\u0027s only 2 cases."},{"Start":"05:43.145 ","End":"05:46.940","Text":"If k is not 1, we have a single solution,"},{"Start":"05:46.940 ","End":"05:50.540","Text":"and if k is equal to 1, no solutions."},{"Start":"05:50.540 ","End":"05:52.670","Text":"The case of infinite solutions,"},{"Start":"05:52.670 ","End":"05:55.130","Text":"we don\u0027t achieve for any value of k."},{"Start":"05:55.130 ","End":"05:58.080","Text":"We\u0027re done."}],"ID":9840},{"Watched":false,"Name":"Exercise 10","Duration":"4m 17s","ChapterTopicVideoID":9518,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9518.jpeg","UploadDate":"2017-07-26T08:25:38.7570000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.385","Text":"Here we have another 1 of those exercises,"},{"Start":"00:02.385 ","End":"00:04.215","Text":"system of linear equations,"},{"Start":"00:04.215 ","End":"00:07.170","Text":"4 equations, 3 unknowns, x, y, and z,"},{"Start":"00:07.170 ","End":"00:09.200","Text":"and with 2 parameters, a and b."},{"Start":"00:09.200 ","End":"00:10.425","Text":"Let\u0027s see if we can find them."},{"Start":"00:10.425 ","End":"00:12.750","Text":"Here\u0027s a and here\u0027s b,"},{"Start":"00:12.750 ","End":"00:13.960","Text":"and they only occur there"},{"Start":"00:13.960 ","End":"00:19.810","Text":"and we have to classify number of solutions the usual."},{"Start":"00:19.970 ","End":"00:23.550","Text":"As usual, we use matrix representation"},{"Start":"00:23.550 ","End":"00:25.740","Text":"because it\u0027s much easier in matrix form"},{"Start":"00:25.740 ","End":"00:33.150","Text":"and we want to bring this into row-echelon form to be precise,"},{"Start":"00:33.150 ","End":"00:35.505","Text":"we want to get 0s here,"},{"Start":"00:35.505 ","End":"00:40.200","Text":"but I\u0027d rather if it was the 1 here, then a 2 here."},{"Start":"00:40.200 ","End":"00:41.850","Text":"Why don\u0027t we swap,"},{"Start":"00:41.850 ","End":"00:46.565","Text":"and let\u0027s say the first 2 rows like this and if we do that,"},{"Start":"00:46.565 ","End":"00:50.390","Text":"then the result will be this,"},{"Start":"00:50.390 ","End":"00:52.520","Text":"it\u0027s easy to swap the top 2 rows,"},{"Start":"00:52.520 ","End":"00:56.900","Text":"and now what we want to do is start getting it into echelon form"},{"Start":"00:56.900 ","End":"01:01.130","Text":"by zeroing out the rest of the column below the 1."},{"Start":"01:01.130 ","End":"01:05.420","Text":"I want all these 3 to be 0 we can do that with 3 row operations."},{"Start":"01:05.420 ","End":"01:08.240","Text":"This minus twice this, this minus this,"},{"Start":"01:08.240 ","End":"01:13.640","Text":"this minus this as written here and this is what we get."},{"Start":"01:13.640 ","End":"01:19.850","Text":"But I notice that we\u0027ve also got 0 in the rest of the second column."},{"Start":"01:19.850 ","End":"01:23.120","Text":"We now have to proceed to the third column"},{"Start":"01:23.120 ","End":"01:26.740","Text":"and we want to 0 out below the a minus 8"},{"Start":"01:26.740 ","End":"01:31.250","Text":"but we don\u0027t like a parameter in the leading term"},{"Start":"01:31.250 ","End":"01:35.405","Text":"because we don\u0027t know if a minus 8 is 0 or not."},{"Start":"01:35.405 ","End":"01:38.240","Text":"But this a minus 8 is not convenient"},{"Start":"01:38.240 ","End":"01:40.190","Text":"because they don\u0027t know if it\u0027s 0 or non-zero"},{"Start":"01:40.190 ","End":"01:45.500","Text":"and if I start dividing by 8 or subtracting or adding multiples of 1 over this,"},{"Start":"01:45.500 ","End":"01:47.030","Text":"that\u0027s going to be a problem."},{"Start":"01:47.030 ","End":"01:49.325","Text":"I\u0027ll swap 2 rows,"},{"Start":"01:49.325 ","End":"01:50.930","Text":"maybe swap this with this,"},{"Start":"01:50.930 ","End":"01:53.250","Text":"but I\u0027d rather have a 1,"},{"Start":"01:53.250 ","End":"01:56.730","Text":"so why don\u0027t I just first take that third row"},{"Start":"01:56.730 ","End":"02:03.220","Text":"and just multiply by minus 1 over 8."},{"Start":"02:03.220 ","End":"02:05.160","Text":"I get a 1 in this place"},{"Start":"02:05.160 ","End":"02:11.103","Text":"and now I\u0027ll do the swap and we\u0027ll end up with this"},{"Start":"02:11.103 ","End":"02:14.300","Text":"and I\u0027ll leave you to check the calculations."},{"Start":"02:14.300 ","End":"02:16.205","Text":"You can always pause,"},{"Start":"02:16.205 ","End":"02:21.170","Text":"so what we\u0027ll do is we\u0027ll subtract 32 this from the last row"},{"Start":"02:21.170 ","End":"02:25.250","Text":"and a minus 8 times this from this row as written here"},{"Start":"02:25.250 ","End":"02:26.270","Text":"and if we do that,"},{"Start":"02:26.270 ","End":"02:27.890","Text":"these 2 should come out 0"},{"Start":"02:27.890 ","End":"02:30.450","Text":"and this is what we get."},{"Start":"02:30.450 ","End":"02:35.365","Text":"Before I analyze, let me just simplify this term that\u0027s better."},{"Start":"02:35.365 ","End":"02:41.545","Text":"Now we are in echelon form the other step can jump more than 1,"},{"Start":"02:41.545 ","End":"02:47.715","Text":"I\u0027m talking about the restricted matrix now that\u0027s in echelon form."},{"Start":"02:47.715 ","End":"02:53.100","Text":"Notice that it has 2 rows and 3 columns"},{"Start":"02:53.100 ","End":"02:56.740","Text":"so it\u0027s never going to have a unique solution"},{"Start":"02:56.740 ","End":"03:01.015","Text":"we\u0027re going to expect either infinite solutions or a contradiction."},{"Start":"03:01.015 ","End":"03:05.815","Text":"Now, a contradiction can arise if I have something not 0 here,"},{"Start":"03:05.815 ","End":"03:10.270","Text":"or something not 0 here if either of these is not 0,"},{"Start":"03:10.270 ","End":"03:13.785","Text":"then a contradiction so let me just rephrase that."},{"Start":"03:13.785 ","End":"03:20.820","Text":"This 1, I just wrote here and this I wrote here and if this is not 0,"},{"Start":"03:20.820 ","End":"03:26.850","Text":"it means that b is not 2.5 and this not 0 means that a is not minus 6."},{"Start":"03:26.850 ","End":"03:30.520","Text":"If either of these occurs, there\u0027s no solution."},{"Start":"03:30.520 ","End":"03:33.535","Text":"Now the logical reverse of this not 0,"},{"Start":"03:33.535 ","End":"03:38.425","Text":"or this not 0 is that both of them are equal."},{"Start":"03:38.425 ","End":"03:42.280","Text":"Think about it the logical negative of this condition is that"},{"Start":"03:42.280 ","End":"03:46.945","Text":"this is equal to this and this is this because if either 1 of these doesn\u0027t hold,"},{"Start":"03:46.945 ","End":"03:49.405","Text":"then we\u0027re back to here and in this case there are"},{"Start":"03:49.405 ","End":"03:53.200","Text":"infinitely many solutions because like I said,"},{"Start":"03:53.200 ","End":"04:00.615","Text":"there\u0027s 2 rows and 3 columns and 2 is less than 3 so yeah."},{"Start":"04:00.615 ","End":"04:05.180","Text":"No value of a and b is going to give us a single solution,"},{"Start":"04:05.180 ","End":"04:07.580","Text":"exactly 1 solution we\u0027re always going to get"},{"Start":"04:07.580 ","End":"04:10.040","Text":"either no solutions or infinitely many solutions"},{"Start":"04:10.040 ","End":"04:13.520","Text":"and this only occurs in the very special case"},{"Start":"04:13.520 ","End":"04:18.660","Text":"that b is this and a is this, and we\u0027re done"}],"ID":9841},{"Watched":false,"Name":"Exercise 11","Duration":"4m 33s","ChapterTopicVideoID":9519,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9519.jpeg","UploadDate":"2017-07-26T08:25:50.7070000","DurationForVideoObject":"PT4M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have a system of linear equations,"},{"Start":"00:03.720 ","End":"00:07.485","Text":"3 equations, 3 unknowns, but 2 parameters, a and b."},{"Start":"00:07.485 ","End":"00:11.160","Text":"We can find b over here and a here, here and here"},{"Start":"00:11.160 ","End":"00:15.720","Text":"and we\u0027re going to determine the values of the parameters,"},{"Start":"00:15.720 ","End":"00:19.395","Text":"which gives 0 solutions, 1 solution, infinite solutions."},{"Start":"00:19.395 ","End":"00:23.385","Text":"As usual, we convert to a matrix first."},{"Start":"00:23.385 ","End":"00:26.880","Text":"Here\u0027s the augmented matrix that represents this"},{"Start":"00:26.880 ","End":"00:31.980","Text":"and we can\u0027t say anything until we bring it into row-echelon form."},{"Start":"00:31.980 ","End":"00:33.210","Text":"Let\u0027s start."},{"Start":"00:33.210 ","End":"00:34.980","Text":"It\u0027s good that we have a 1 here."},{"Start":"00:34.980 ","End":"00:39.055","Text":"We can now subtract multiples of the first row from the other 2 rows."},{"Start":"00:39.055 ","End":"00:47.030","Text":"A times from the second row and 3 times from the third row."},{"Start":"00:47.030 ","End":"00:51.920","Text":"If we do that, then we end up with this and I\u0027m going to remind you,"},{"Start":"00:51.920 ","End":"00:55.220","Text":"I\u0027m not going to do every calculation."},{"Start":"00:55.220 ","End":"00:59.780","Text":"You can pause and verify and we\u0027re ready at an advanced stage."},{"Start":"00:59.780 ","End":"01:04.640","Text":"This is what we get 0s here and that\u0027s still not echelon form"},{"Start":"01:04.640 ","End":"01:07.990","Text":"because we want a 0 here also."},{"Start":"01:07.990 ","End":"01:11.310","Text":"But this element has a parameter in it and I don\u0027t"},{"Start":"01:11.310 ","End":"01:15.075","Text":"like that because you don\u0027t know then if it\u0027s 0 or not 0."},{"Start":"01:15.075 ","End":"01:18.950","Text":"Let\u0027s swap these 2 rows."},{"Start":"01:18.950 ","End":"01:21.770","Text":"This is the notation and if we do that,"},{"Start":"01:21.770 ","End":"01:26.120","Text":"then we get this and that\u0027s much easier to deal with."},{"Start":"01:26.120 ","End":"01:27.800","Text":"Could multiply by minus 1."},{"Start":"01:27.800 ","End":"01:32.735","Text":"I don\u0027t see the point, if I just added this to this row here,"},{"Start":"01:32.735 ","End":"01:38.570","Text":"sorry, I meant 1 minus a times this row to this row."},{"Start":"01:38.570 ","End":"01:44.220","Text":"If I do that, that should certainly give me the 0 here."},{"Start":"01:44.220 ","End":"01:49.505","Text":"It looks a bit of a mess elsewhere."},{"Start":"01:49.505 ","End":"01:53.570","Text":"Well, tidied up a little bit,"},{"Start":"01:53.570 ","End":"01:56.990","Text":"especially on the left-hand side in the restricted matrix."},{"Start":"01:56.990 ","End":"02:00.510","Text":"I don\u0027t so much care about this column."},{"Start":"02:02.630 ","End":"02:05.360","Text":"I leave you to check the algebra."},{"Start":"02:05.360 ","End":"02:08.285","Text":"I\u0027m not going to go through every detail."},{"Start":"02:08.285 ","End":"02:13.080","Text":"Now, we are the echelon form."},{"Start":"02:14.060 ","End":"02:22.165","Text":"For the analysis, I\u0027m just going to first look at the restricted matrix and look,"},{"Start":"02:22.165 ","End":"02:27.720","Text":"there\u0027s 4 columns and there\u0027s at most 3 rows."},{"Start":"02:27.720 ","End":"02:29.130","Text":"Why do I say at most 3?"},{"Start":"02:29.130 ","End":"02:34.180","Text":"Because it\u0027s possible that these are both 0 and we only have 2 rows,"},{"Start":"02:34.180 ","End":"02:36.190","Text":"but either way 2 or 3 rows,"},{"Start":"02:36.190 ","End":"02:37.900","Text":"it\u0027s still less than 4."},{"Start":"02:37.900 ","End":"02:42.200","Text":"You can\u0027t expect to have just 1 solution."},{"Start":"02:42.200 ","End":"02:45.260","Text":"It\u0027s going to be 0 or infinite number of solutions"},{"Start":"02:45.260 ","End":"02:48.725","Text":"and that depends on whether we can find a contradiction or not,"},{"Start":"02:48.725 ","End":"02:50.660","Text":"which can only occur in the last row."},{"Start":"02:50.660 ","End":"02:51.665","Text":"Let\u0027s see."},{"Start":"02:51.665 ","End":"02:55.310","Text":"When would we get a contradiction in the last row?"},{"Start":"02:55.310 ","End":"02:59.440","Text":"It would have to be 0, 0, non-zero."},{"Start":"02:59.440 ","End":"03:04.940","Text":"For this 1 to be 0, we would have to have a equals 2."},{"Start":"03:04.940 ","End":"03:06.620","Text":"But if a equals 2,"},{"Start":"03:06.620 ","End":"03:12.110","Text":"then this 1 is 0 also automatically and then the question is,"},{"Start":"03:12.110 ","End":"03:14.535","Text":"how do I make this non-zero?"},{"Start":"03:14.535 ","End":"03:20.460","Text":"Well, a equals 2 means that this bit is 0 and so"},{"Start":"03:20.460 ","End":"03:26.390","Text":"all that\u0027s left for it to be non-zero is for b minus a to be non-zero,"},{"Start":"03:26.390 ","End":"03:29.885","Text":"which means that b is not equal to a."},{"Start":"03:29.885 ","End":"03:35.140","Text":"See b minus a non-zero is the same as b not equal to a."},{"Start":"03:35.140 ","End":"03:39.200","Text":"That gives us the case for the no solutions."},{"Start":"03:39.200 ","End":"03:42.590","Text":"Just noticed here I wrote b not equal to 2 instead of b not equal to a."},{"Start":"03:42.590 ","End":"03:47.390","Text":"But that\u0027s okay because if a equals 2 and b is not equal to a,"},{"Start":"03:47.390 ","End":"03:49.505","Text":"then b is not equal to 2."},{"Start":"03:49.505 ","End":"03:51.140","Text":"So same thing."},{"Start":"03:51.140 ","End":"03:54.320","Text":"Let\u0027s look at the logical opposite of this."},{"Start":"03:54.320 ","End":"03:56.175","Text":"We have either,"},{"Start":"03:56.175 ","End":"03:59.075","Text":"I mean if this and this is not true,"},{"Start":"03:59.075 ","End":"04:01.700","Text":"then either this is not true or this is not true."},{"Start":"04:01.700 ","End":"04:09.950","Text":"It must be that a is not equal to 2 or b equals 2."},{"Start":"04:09.950 ","End":"04:13.103","Text":"If b equals 2, it could be,"},{"Start":"04:13.103 ","End":"04:15.860","Text":"doesn\u0027t matter if a equals b equals 2,"},{"Start":"04:15.860 ","End":"04:21.420","Text":"it\u0027s logically equivalent because the converse of this is that also a is 2."},{"Start":"04:21.420 ","End":"04:26.480","Text":"In this case, we have infinitely many solutions and like I said,"},{"Start":"04:26.480 ","End":"04:28.730","Text":"we cannot expect that to be the case"},{"Start":"04:28.730 ","End":"04:30.800","Text":"where there\u0027s just 1 solution."},{"Start":"04:30.800 ","End":"04:33.510","Text":"Okay, we\u0027re done."}],"ID":9842},{"Watched":false,"Name":"Exercise 12","Duration":"5m 1s","ChapterTopicVideoID":9520,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9520.jpeg","UploadDate":"2017-07-26T08:26:12.6230000","DurationForVideoObject":"PT5M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"This exercise is a bit different."},{"Start":"00:03.105 ","End":"00:08.985","Text":"We\u0027re given as usual a system of linear equations with parameters,"},{"Start":"00:08.985 ","End":"00:12.630","Text":"but this time we have 4 parameters, a, b, c, and d."},{"Start":"00:12.630 ","End":"00:14.190","Text":"There\u0027s a for example,"},{"Start":"00:14.190 ","End":"00:16.590","Text":"and here\u0027s b, here\u0027s c, here\u0027s d."},{"Start":"00:16.590 ","End":"00:20.400","Text":"We have to determine, well,"},{"Start":"00:20.400 ","End":"00:27.030","Text":"there\u0027s 2 things; what relation or equation exists between a,"},{"Start":"00:27.030 ","End":"00:30.300","Text":"b, c, and d to get exactly 1 solution."},{"Start":"00:30.300 ","End":"00:35.040","Text":"In part 2, we have to decide what values of b, c,"},{"Start":"00:35.040 ","End":"00:41.500","Text":"and d make the system have infinitely many solutions for all values of a."},{"Start":"00:41.500 ","End":"00:47.570","Text":"We\u0027ll start with part 1 and as usual,"},{"Start":"00:47.570 ","End":"00:51.340","Text":"we represent this system as a matrix."},{"Start":"00:51.340 ","End":"00:54.890","Text":"So this is the augmented matrix that represents this."},{"Start":"00:54.890 ","End":"00:56.855","Text":"Note that we had to add, say,"},{"Start":"00:56.855 ","End":"01:02.310","Text":"0 here and 0 here because some terms we\u0027re missing."},{"Start":"01:05.720 ","End":"01:07.925","Text":"We have a 1 here,"},{"Start":"01:07.925 ","End":"01:11.795","Text":"which is good because we want to bring it to echelon form and a 1."},{"Start":"01:11.795 ","End":"01:14.330","Text":"We don\u0027t have to do anything to the second row,"},{"Start":"01:14.330 ","End":"01:20.395","Text":"but we want to subtract b times the first row from the last end row notation."},{"Start":"01:20.395 ","End":"01:22.955","Text":"The result which I leave you to check."},{"Start":"01:22.955 ","End":"01:25.145","Text":"But we do have 0,"},{"Start":"01:25.145 ","End":"01:26.600","Text":"0 as we wanted."},{"Start":"01:26.600 ","End":"01:30.290","Text":"Now we\u0027re still of course not done because we"},{"Start":"01:30.290 ","End":"01:34.740","Text":"wanted the echelon form and we need this to be 0 also."},{"Start":"01:34.740 ","End":"01:41.520","Text":"We subtract c times the second row from the last row in row notation."},{"Start":"01:41.520 ","End":"01:46.295","Text":"After that, we get 0 here and you can see that"},{"Start":"01:46.295 ","End":"01:51.485","Text":"this minus c times this minus 2c and so on here."},{"Start":"01:51.485 ","End":"01:57.750","Text":"Anyway, we\u0027re at this point and it is in echelon form."},{"Start":"01:58.010 ","End":"02:04.600","Text":"Well, the restricted the matrix is in echelon formulas."},{"Start":"02:04.600 ","End":"02:07.700","Text":"That is what we examine for a number of solutions."},{"Start":"02:07.700 ","End":"02:12.710","Text":"Now, we have 3 rows and we would have 3 columns also here."},{"Start":"02:12.710 ","End":"02:14.435","Text":"If this is not 0,"},{"Start":"02:14.435 ","End":"02:18.325","Text":"and then we have a single unique solution."},{"Start":"02:18.325 ","End":"02:21.990","Text":"The condition for 1 is that this should not be 0"},{"Start":"02:21.990 ","End":"02:25.670","Text":"and this is the relation between a, b, c, and d."},{"Start":"02:25.670 ","End":"02:28.115","Text":"I\u0027ll give you an example."},{"Start":"02:28.115 ","End":"02:30.680","Text":"Check if you put all the a,"},{"Start":"02:30.680 ","End":"02:33.005","Text":"b, c, d, e to the mu equal to 1,"},{"Start":"02:33.005 ","End":"02:39.605","Text":"we get 1 minus 2."},{"Start":"02:39.605 ","End":"02:45.255","Text":"Anyway, it\u0027s minus 2, it\u0027s not 0."},{"Start":"02:45.255 ","End":"02:48.180","Text":"There is exactly 1 solution."},{"Start":"02:48.180 ","End":"02:52.155","Text":"Unless this expression is 0, exactly 1 solution."},{"Start":"02:52.155 ","End":"02:55.075","Text":"Now let\u0027s go to part 2."},{"Start":"02:55.075 ","End":"02:59.105","Text":"To get an infinite number of solutions,"},{"Start":"02:59.105 ","End":"03:09.075","Text":"the condition is that there should be less rows than columns in the restricted."},{"Start":"03:09.075 ","End":"03:14.985","Text":"This has to be 0, but there mustn\u0027t be a contradiction,"},{"Start":"03:14.985 ","End":"03:19.335","Text":"so this has to be also 0."},{"Start":"03:19.335 ","End":"03:21.950","Text":"Both of these things that I\u0027ve highlighted have to be"},{"Start":"03:21.950 ","End":"03:25.460","Text":"0 and then we\u0027ll get an infinite number of solutions."},{"Start":"03:25.460 ","End":"03:29.970","Text":"Because then we have 2 rows and 3 columns"},{"Start":"03:29.970 ","End":"03:36.800","Text":"and infinite solutions for all a."},{"Start":"03:36.800 ","End":"03:41.450","Text":"Now, what I said about these being both 0 that\u0027s written here."},{"Start":"03:41.450 ","End":"03:47.200","Text":"But how do I express the third condition of this is for all a?"},{"Start":"03:47.200 ","End":"03:49.310","Text":"Well, this has nothing to do with a,"},{"Start":"03:49.310 ","End":"03:51.920","Text":"but the first equation, you might think, well,"},{"Start":"03:51.920 ","End":"03:58.750","Text":"different values of a would give you different values of this expression."},{"Start":"03:58.750 ","End":"04:02.625","Text":"It would, except if b is 0."},{"Start":"04:02.625 ","End":"04:08.165","Text":"If b is 0, then this condition is not dependent on a."},{"Start":"04:08.165 ","End":"04:10.355","Text":"You don\u0027t get different values for different a,"},{"Start":"04:10.355 ","End":"04:16.310","Text":"so we have to add the condition that b equals 0."},{"Start":"04:16.310 ","End":"04:18.290","Text":"But look, if b is 0,"},{"Start":"04:18.290 ","End":"04:24.500","Text":"we can find the values of the other 2 parameters, c and d."},{"Start":"04:24.500 ","End":"04:27.125","Text":"Because if you put b equals 0 here,"},{"Start":"04:27.125 ","End":"04:30.620","Text":"you get 3 minus 2c is 0."},{"Start":"04:30.620 ","End":"04:37.340","Text":"C is equal to 1.5, 2c equals 3."},{"Start":"04:37.340 ","End":"04:41.605","Text":"Once we have that b is 0 and c is 1.5,"},{"Start":"04:41.605 ","End":"04:45.360","Text":"c is 1.5 so minus 2c is minus 3."},{"Start":"04:45.360 ","End":"04:49.965","Text":"That gives us that d equals 3."},{"Start":"04:49.965 ","End":"04:52.710","Text":"These are the values of b, c,"},{"Start":"04:52.710 ","End":"05:01.580","Text":"and d that make this system have infinitely many solutions for all a and we\u0027re done."}],"ID":9843},{"Watched":false,"Name":"Exercise 13 Parts a-c","Duration":"4m 40s","ChapterTopicVideoID":9521,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9521.jpeg","UploadDate":"2017-07-26T08:26:52.7400000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we\u0027re given a system of linear equations,"},{"Start":"00:03.870 ","End":"00:08.100","Text":"the 3 equations in 3 unknowns, x, y, and z."},{"Start":"00:08.100 ","End":"00:11.130","Text":"Actually, there are 8 parts to this question,"},{"Start":"00:11.130 ","End":"00:14.340","Text":"there\u0027s a through h,"},{"Start":"00:14.340 ","End":"00:19.410","Text":"but we are going to start with just the first 3,"},{"Start":"00:19.410 ","End":"00:22.080","Text":"and from those, we\u0027ll start with the first 1."},{"Start":"00:22.080 ","End":"00:25.665","Text":"Write the matrix corresponding to the system."},{"Start":"00:25.665 ","End":"00:28.200","Text":"Lo and behold, there it is."},{"Start":"00:28.200 ","End":"00:32.700","Text":"Just copy the coefficients and place them in the grid in the right place, 1,"},{"Start":"00:32.700 ","End":"00:35.440","Text":"1 minus 1, and then 1,"},{"Start":"00:35.440 ","End":"00:39.290","Text":"3 minus 7 and so on."},{"Start":"00:39.290 ","End":"00:40.790","Text":"That\u0027s all there is to it."},{"Start":"00:40.790 ","End":"00:42.530","Text":"I wish they were all as easy as that."},{"Start":"00:42.530 ","End":"00:44.790","Text":"Let\u0027s go on to the next."},{"Start":"00:45.410 ","End":"00:47.510","Text":"Now, in part B,"},{"Start":"00:47.510 ","End":"00:50.840","Text":"we have to bring the matrix to row-echelon form."},{"Start":"00:50.840 ","End":"00:54.200","Text":"This is the matrix we found in part A."},{"Start":"00:54.200 ","End":"00:56.210","Text":"I guess we\u0027re going to"},{"Start":"00:56.210 ","End":"01:01.920","Text":"start out by zeroing out the first column below the 1,"},{"Start":"01:01.920 ","End":"01:05.450","Text":"so I\u0027ll subtract 3 times the first row from"},{"Start":"01:05.450 ","End":"01:09.500","Text":"the second and subtract it 4 times from the third."},{"Start":"01:09.500 ","End":"01:12.290","Text":"Here it is in row notation."},{"Start":"01:12.290 ","End":"01:15.065","Text":"Let\u0027s actually do that."},{"Start":"01:15.065 ","End":"01:17.915","Text":"This is what we get."},{"Start":"01:17.915 ","End":"01:23.080","Text":"As usual, you can always pause the clip and check the computations."},{"Start":"01:23.080 ","End":"01:26.610","Text":"We\u0027re not in echelon form yet,"},{"Start":"01:26.610 ","End":"01:30.110","Text":"we still have to get rid of this to put a 0 here."},{"Start":"01:30.110 ","End":"01:31.310","Text":"I think that\u0027s pretty clear."},{"Start":"01:31.310 ","End":"01:35.870","Text":"We just have to subtract the second row from the third row,"},{"Start":"01:35.870 ","End":"01:38.930","Text":"which is this in row notation,"},{"Start":"01:38.930 ","End":"01:42.005","Text":"and the result is this."},{"Start":"01:42.005 ","End":"01:50.050","Text":"The restricted matrix is definitely in row-echelon form,"},{"Start":"01:50.050 ","End":"01:53.820","Text":"and that\u0027s part B."},{"Start":"01:53.820 ","End":"01:55.410","Text":"Let\u0027s go on to the next."},{"Start":"01:55.410 ","End":"01:59.180","Text":"In part C, we have to find which values of k,"},{"Start":"01:59.180 ","End":"02:03.290","Text":"the parameter, give the system 0,"},{"Start":"02:03.290 ","End":"02:05.600","Text":"1, or infinitely many solutions."},{"Start":"02:05.600 ","End":"02:08.750","Text":"Those are the only 3 possibilities."},{"Start":"02:08.750 ","End":"02:11.735","Text":"This is what we found in part B,"},{"Start":"02:11.735 ","End":"02:19.005","Text":"and the most important is this element here."},{"Start":"02:19.005 ","End":"02:21.460","Text":"If it\u0027s not 0,"},{"Start":"02:21.460 ","End":"02:25.930","Text":"then we have 3 rows and 3 columns in the reduced,"},{"Start":"02:25.930 ","End":"02:30.540","Text":"and we know we have a unique single solution."},{"Start":"02:30.540 ","End":"02:35.465","Text":"If this is 0, then we have 2 subcases."},{"Start":"02:35.465 ","End":"02:37.770","Text":"Either this is also 0,"},{"Start":"02:37.770 ","End":"02:43.850","Text":"in which case we have infinite solutions because we just have then 2 rows but 3 columns."},{"Start":"02:43.850 ","End":"02:46.550","Text":"But if this is not 0 and this is 0,"},{"Start":"02:46.550 ","End":"02:49.355","Text":"then we get a contradiction, so no solutions."},{"Start":"02:49.355 ","End":"02:51.365","Text":"Those are the cases."},{"Start":"02:51.365 ","End":"02:54.875","Text":"First of all, hinges on what this is."},{"Start":"02:54.875 ","End":"02:57.770","Text":"If we want this not to be 0,"},{"Start":"02:57.770 ","End":"02:59.840","Text":"we can just solve the quadratic,"},{"Start":"02:59.840 ","End":"03:01.735","Text":"get the 2 roots,"},{"Start":"03:01.735 ","End":"03:04.760","Text":"which happen to be 2 and minus 1."},{"Start":"03:04.760 ","End":"03:06.530","Text":"I\u0027m not going to solve the quadratic."},{"Start":"03:06.530 ","End":"03:14.265","Text":"But to say not 0 means that k has to avoid being 2 or negative 1,"},{"Start":"03:14.265 ","End":"03:15.720","Text":"can\u0027t be either of these,"},{"Start":"03:15.720 ","End":"03:18.815","Text":"then we have a single solution,"},{"Start":"03:18.815 ","End":"03:21.275","Text":"sometimes a unique solution,"},{"Start":"03:21.275 ","End":"03:24.540","Text":"1 and only 1 solution."},{"Start":"03:24.650 ","End":"03:26.970","Text":"Let\u0027s continue."},{"Start":"03:26.970 ","End":"03:28.590","Text":"The 2 other cases,"},{"Start":"03:28.590 ","End":"03:30.385","Text":"we just have to try out,"},{"Start":"03:30.385 ","End":"03:34.010","Text":"all that\u0027s missing is k equals 2 and k equals 1."},{"Start":"03:34.010 ","End":"03:35.635","Text":"Let\u0027s start with 1 of them."},{"Start":"03:35.635 ","End":"03:38.295","Text":"Say, k equals 2 first."},{"Start":"03:38.295 ","End":"03:40.200","Text":"Then if k is 2,"},{"Start":"03:40.200 ","End":"03:41.220","Text":"this is going to be 0,"},{"Start":"03:41.220 ","End":"03:42.680","Text":"that\u0027s for sure, but look,"},{"Start":"03:42.680 ","End":"03:47.035","Text":"this is also going to be 0 because 4 minus 2 squared is 0."},{"Start":"03:47.035 ","End":"03:50.195","Text":"If this is 0 and this is 0,"},{"Start":"03:50.195 ","End":"03:53.600","Text":"it\u0027s like we cross this out and then"},{"Start":"03:53.600 ","End":"03:57.335","Text":"surely there\u0027s infinite solutions because now we definitely have"},{"Start":"03:57.335 ","End":"04:06.660","Text":"2 rows and 3 columns and there\u0027s no contradiction rows or anything."},{"Start":"04:07.840 ","End":"04:10.805","Text":"That\u0027s infinitely many."},{"Start":"04:10.805 ","End":"04:15.985","Text":"Let\u0027s try the only other case which is k equals minus 1."},{"Start":"04:15.985 ","End":"04:18.074","Text":"Well, in this case,"},{"Start":"04:18.074 ","End":"04:19.620","Text":"this is still 0,"},{"Start":"04:19.620 ","End":"04:22.560","Text":"the 4 minus 1 squared is 3,"},{"Start":"04:22.560 ","End":"04:24.570","Text":"in any event is not 0."},{"Start":"04:24.570 ","End":"04:28.220","Text":"Look, this last row is a contradiction row,"},{"Start":"04:28.220 ","End":"04:31.295","Text":"which means that the system has no solution."},{"Start":"04:31.295 ","End":"04:35.720","Text":"There we are. We\u0027ve classified all 3 cases, then that\u0027s part C."},{"Start":"04:35.720 ","End":"04:37.295","Text":"Let\u0027s move on."},{"Start":"04:37.295 ","End":"04:40.020","Text":"That will be in the next clip."}],"ID":9844},{"Watched":false,"Name":"Exercise 13 Parts d-f","Duration":"6m 12s","ChapterTopicVideoID":9522,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9522.jpeg","UploadDate":"2017-07-26T08:27:38.0700000","DurationForVideoObject":"PT6M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.335","Text":"In this exercise, we\u0027re continuing with the same system of linear equations as before."},{"Start":"00:07.335 ","End":"00:11.790","Text":"Well, it doesn\u0027t look the same because it\u0027s in echelon form."},{"Start":"00:11.790 ","End":"00:15.450","Text":"Back here in part b where we found the echelon form,"},{"Start":"00:15.450 ","End":"00:21.495","Text":"we just put this now back to equation form from matrix form,"},{"Start":"00:21.495 ","End":"00:25.560","Text":"then this is what you get and it\u0027s equivalent to the original system,"},{"Start":"00:25.560 ","End":"00:29.100","Text":"but because it\u0027s much easier to work with in echelon form."},{"Start":"00:29.100 ","End":"00:31.680","Text":"Now, we want to write the general solution"},{"Start":"00:31.680 ","End":"00:34.705","Text":"for the case where there are infinitely many solutions."},{"Start":"00:34.705 ","End":"00:37.165","Text":"Now in part c,"},{"Start":"00:37.165 ","End":"00:45.235","Text":"we showed that infinitely many solutions occurs when k equals 2."},{"Start":"00:45.235 ","End":"00:51.110","Text":"We just plug in k equals 2 everywhere and this is the system we get."},{"Start":"00:51.110 ","End":"00:53.060","Text":"Notice that the last 1,"},{"Start":"00:53.060 ","End":"00:55.790","Text":"when k is 2, this is 0 and this is 0."},{"Start":"00:55.790 ","End":"00:58.550","Text":"So this is a 0 equation."},{"Start":"00:58.550 ","End":"01:03.335","Text":"We just cross it out. We really only have 2 equations."},{"Start":"01:03.335 ","End":"01:13.130","Text":"Now I highlighted or boxed in the leading term in each."},{"Start":"01:13.130 ","End":"01:16.230","Text":"Actually, the leading term is the whole thing,"},{"Start":"01:17.360 ","End":"01:19.725","Text":"maybe the leading variable."},{"Start":"01:19.725 ","End":"01:25.850","Text":"Anyway, these are dependent and the variables that are not listed here, i.e,"},{"Start":"01:25.850 ","End":"01:32.210","Text":"the variable z becomes a free variable,"},{"Start":"01:32.210 ","End":"01:35.075","Text":"which means we can assign whatever we like to it."},{"Start":"01:35.075 ","End":"01:37.910","Text":"Then we can figure x and y from that."},{"Start":"01:37.910 ","End":"01:41.310","Text":"That\u0027s why there\u0027s an infinite number of solutions."},{"Start":"01:41.480 ","End":"01:44.790","Text":"Typically, I assign the letters t,"},{"Start":"01:44.790 ","End":"01:46.730","Text":"u, v for free variables,"},{"Start":"01:46.730 ","End":"01:49.070","Text":"we only have 1 here that z equals t."},{"Start":"01:49.070 ","End":"01:52.520","Text":"Now we go for the back substitution."},{"Start":"01:52.520 ","End":"01:56.315","Text":"If you plug in z equals t here,"},{"Start":"01:56.315 ","End":"02:02.660","Text":"then what you get if you isolate y is 0.8t."},{"Start":"02:02.660 ","End":"02:06.410","Text":"I mean, you get 10y equals 8t divide by 10."},{"Start":"02:06.410 ","End":"02:09.560","Text":"Now that we have y and z,"},{"Start":"02:09.560 ","End":"02:16.535","Text":"we can put in z equals t here and 0.8t here,"},{"Start":"02:16.535 ","End":"02:19.590","Text":"and this is what we get."},{"Start":"02:20.630 ","End":"02:24.275","Text":"Well, this is what you get at first and then you simplify it."},{"Start":"02:24.275 ","End":"02:31.120","Text":"Then we get that x is 1 plus 0.2t."},{"Start":"02:31.120 ","End":"02:35.975","Text":"This is the general solution involving a parameter t,"},{"Start":"02:35.975 ","End":"02:39.230","Text":"and that\u0027s why there are infinitely many solutions"},{"Start":"02:39.230 ","End":"02:41.930","Text":"because t can be any real number you want,"},{"Start":"02:41.930 ","End":"02:47.650","Text":"and you\u0027ll get a different solution for different t. That\u0027s it."},{"Start":"02:48.350 ","End":"02:51.840","Text":"Don\u0027t go yet, that\u0027s just part d."},{"Start":"02:51.840 ","End":"02:54.060","Text":"Yeah, we got to continue to part e."},{"Start":"02:54.060 ","End":"02:58.680","Text":"In part e, we have to find for which value of"},{"Start":"02:58.680 ","End":"03:06.645","Text":"k does the system have a solution with z equals 0."},{"Start":"03:06.645 ","End":"03:08.810","Text":"X equals something, y equals something,"},{"Start":"03:08.810 ","End":"03:10.670","Text":"and z equals 0."},{"Start":"03:10.670 ","End":"03:14.554","Text":"Let\u0027s just take the equation, the system,"},{"Start":"03:14.554 ","End":"03:18.820","Text":"and plug in z equals 0 in all of them,"},{"Start":"03:18.820 ","End":"03:21.725","Text":"and this is what we get."},{"Start":"03:21.725 ","End":"03:25.550","Text":"The most interesting is the third equation."},{"Start":"03:25.550 ","End":"03:28.070","Text":"If this is true,"},{"Start":"03:28.070 ","End":"03:30.755","Text":"I mean that\u0027s what we got when we let z equals 0,"},{"Start":"03:30.755 ","End":"03:32.315","Text":"0 times anything is 0,"},{"Start":"03:32.315 ","End":"03:36.115","Text":"then the only 2 possible values of k,"},{"Start":"03:36.115 ","End":"03:40.235","Text":"and those are k equals 2 and minus 2,"},{"Start":"03:40.235 ","End":"03:45.185","Text":"but we can\u0027t just stop here because we have to continue and show that"},{"Start":"03:45.185 ","End":"03:51.000","Text":"there really is a solution that we don\u0027t get some contradiction along the way."},{"Start":"03:51.200 ","End":"03:56.315","Text":"Well, the continuation is the same in both cases,"},{"Start":"03:56.315 ","End":"03:59.675","Text":"because if 4 minus k squared is 0,"},{"Start":"03:59.675 ","End":"04:02.105","Text":"then so is k squared minus 4,"},{"Start":"04:02.105 ","End":"04:05.910","Text":"because they\u0027re just negatives of each other."},{"Start":"04:05.960 ","End":"04:11.180","Text":"The continuation for 2 or minus 2 will be the same in both cases,"},{"Start":"04:11.180 ","End":"04:14.120","Text":"we\u0027ll put 0 here and here,"},{"Start":"04:14.120 ","End":"04:17.790","Text":"because we throw out the third equation, 0 equals 0,"},{"Start":"04:17.790 ","End":"04:19.340","Text":"so what we\u0027re left with is this,"},{"Start":"04:19.340 ","End":"04:22.765","Text":"x plus y is 1 is the same and minus 10y equals 0."},{"Start":"04:22.765 ","End":"04:26.900","Text":"From here, we first solve for y,"},{"Start":"04:26.900 ","End":"04:29.645","Text":"and clearly y is 0."},{"Start":"04:29.645 ","End":"04:32.945","Text":"If I put y equals 0,"},{"Start":"04:32.945 ","End":"04:35.195","Text":"that gives us x equals 1."},{"Start":"04:35.195 ","End":"04:40.940","Text":"It really does pan out to a solution, which is this."},{"Start":"04:40.940 ","End":"04:43.895","Text":"But as far as answering the question,"},{"Start":"04:43.895 ","End":"04:45.880","Text":"it asked which values of k,"},{"Start":"04:45.880 ","End":"04:53.480","Text":"then 2 and minus 2 are both valid possibilities for k."},{"Start":"04:53.480 ","End":"05:00.520","Text":"Now that we have the last part in this 3,"},{"Start":"05:00.520 ","End":"05:04.545","Text":"there\u0027ll still be a part g and an h. But Part f,"},{"Start":"05:04.545 ","End":"05:07.539","Text":"and what value of k"},{"Start":"05:11.690 ","End":"05:17.150","Text":"gives the system a unique solution with z equals 0?"},{"Start":"05:17.150 ","End":"05:20.000","Text":"Well, we can combine 2 things."},{"Start":"05:20.000 ","End":"05:24.530","Text":"First of all, unique solution and then z equals 0."},{"Start":"05:24.530 ","End":"05:32.450","Text":"Unique solution from part c gives the restriction that k should not be 2 or minus 1."},{"Start":"05:32.450 ","End":"05:34.834","Text":"On the other hand,"},{"Start":"05:34.834 ","End":"05:37.825","Text":"the condition z equals 0,"},{"Start":"05:37.825 ","End":"05:39.380","Text":"from the previous part,"},{"Start":"05:39.380 ","End":"05:44.750","Text":"we showed that this happens when k is 2 or negative 2."},{"Start":"05:44.750 ","End":"05:49.640","Text":"Now, we want both these conditions to hold that k has to be 2 or minus 2,"},{"Start":"05:49.640 ","End":"05:52.145","Text":"but cannot be 2 or minus 1."},{"Start":"05:52.145 ","End":"05:59.405","Text":"All you\u0027re left with is k equals minus 2 because we\u0027ve ruled this 1 out,"},{"Start":"05:59.405 ","End":"06:01.565","Text":"and so then only this 1 remains,"},{"Start":"06:01.565 ","End":"06:04.490","Text":"and that\u0027s all there is to it."},{"Start":"06:04.490 ","End":"06:09.855","Text":"That ends this clip and in the next clip we\u0027ll have couple of more,"},{"Start":"06:09.855 ","End":"06:12.370","Text":"parts g and h."}],"ID":9845},{"Watched":false,"Name":"Exercise 13 Parts g-h","Duration":"2m 3s","ChapterTopicVideoID":9523,"CourseChapterTopicPlaylistID":7281,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9523.jpeg","UploadDate":"2017-07-26T08:27:52.8370000","DurationForVideoObject":"PT2M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.440 ","End":"00:03.870","Text":"Continuing with the same system of linear equations"},{"Start":"00:03.870 ","End":"00:05.580","Text":"I went back to the original form,"},{"Start":"00:05.580 ","End":"00:08.400","Text":"not the row-echelon form."},{"Start":"00:08.860 ","End":"00:13.319","Text":"Now we\u0027re asked for which value of k"},{"Start":"00:13.319 ","End":"00:24.990","Text":"will the set of values 1, 2, 3 be a solution to the third equation."},{"Start":"00:24.990 ","End":"00:31.785","Text":"This means x equals 1, y equals 2, z equals 3."},{"Start":"00:31.785 ","End":"00:34.815","Text":"The third equation is this 1."},{"Start":"00:34.815 ","End":"00:42.165","Text":"All we have to do is substitute these values of x, y, and z in the third equation."},{"Start":"00:42.165 ","End":"00:45.100","Text":"This is what we get."},{"Start":"00:45.200 ","End":"00:49.620","Text":"As you can see, this is an equation in k."},{"Start":"00:49.620 ","End":"00:52.190","Text":"Leave this on this side,"},{"Start":"00:52.190 ","End":"00:53.900","Text":"bring the other stuff to the other side,"},{"Start":"00:53.900 ","End":"00:56.840","Text":"so it\u0027s 4 minus 4 plus 12."},{"Start":"00:56.840 ","End":"01:01.495","Text":"Then if we divide by 3k plus 2 is 4,"},{"Start":"01:01.495 ","End":"01:03.825","Text":"and we get k equals 2."},{"Start":"01:03.825 ","End":"01:06.000","Text":"That\u0027s the answer to this part."},{"Start":"01:06.000 ","End":"01:09.990","Text":"Now the last part in this series of 8."},{"Start":"01:09.990 ","End":"01:13.200","Text":"This time as opposed to g,"},{"Start":"01:13.200 ","End":"01:20.450","Text":"we want the 1, 2, 3 to be a solution of the whole system,"},{"Start":"01:20.450 ","End":"01:22.715","Text":"not just the third equation."},{"Start":"01:22.715 ","End":"01:26.840","Text":"I could choose, say, the first equation and plug it in there."},{"Start":"01:26.840 ","End":"01:28.715","Text":"It should work for all 3."},{"Start":"01:28.715 ","End":"01:30.740","Text":"Well, just substitute here,"},{"Start":"01:30.740 ","End":"01:34.285","Text":"x is 1, y equals 2, z equals 3."},{"Start":"01:34.285 ","End":"01:39.630","Text":"We ask ourselves, is 1 plus 2 minus 3 equals 1."},{"Start":"01:39.630 ","End":"01:41.270","Text":"Well, k doesn\u0027t appear here,"},{"Start":"01:41.270 ","End":"01:44.450","Text":"so that can\u0027t help us and the answer is no"},{"Start":"01:44.450 ","End":"01:49.585","Text":"because 1 plus 2 minus 3 is 0, and 0 is not equal to 1."},{"Start":"01:49.585 ","End":"01:53.015","Text":"It\u0027s never going to happen for any value of k."},{"Start":"01:53.015 ","End":"01:56.090","Text":"The answer is there is no value of k"},{"Start":"01:56.090 ","End":"02:00.410","Text":"for which 1, 2, 3 will be a solution to the whole system."},{"Start":"02:00.410 ","End":"02:01.550","Text":"That\u0027s it."},{"Start":"02:01.550 ","End":"02:04.020","Text":"We\u0027re done for the series."}],"ID":9846}],"Thumbnail":null,"ID":7281},{"Name":"SLE over Zp","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"8m 8s","ChapterTopicVideoID":9485,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9485.jpeg","UploadDate":"2017-07-26T08:22:09.4270000","DurationForVideoObject":"PT8M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"This exercise is 2 in 1."},{"Start":"00:03.705 ","End":"00:06.030","Text":"In both Parts A and B,"},{"Start":"00:06.030 ","End":"00:11.010","Text":"we have apparently the same system of linear equations,"},{"Start":"00:11.010 ","End":"00:13.065","Text":"2 equations and 2 unknowns."},{"Start":"00:13.065 ","End":"00:18.960","Text":"But 1 time we consider it over the field of real numbers R."},{"Start":"00:18.960 ","End":"00:22.065","Text":"In Part B,"},{"Start":"00:22.065 ","End":"00:28.065","Text":"we consider it over the integers Modulo 5,"},{"Start":"00:28.065 ","End":"00:31.075","Text":"what we call Z_5,"},{"Start":"00:31.075 ","End":"00:34.970","Text":"which is a finite field containing just 5 numbers."},{"Start":"00:34.970 ","End":"00:36.680","Text":"Now if you don\u0027t know what this means,"},{"Start":"00:36.680 ","End":"00:40.820","Text":"then you probably don\u0027t want to be in this exercise"},{"Start":"00:40.820 ","End":"00:45.050","Text":"and feel free to just go."},{"Start":"00:45.050 ","End":"00:50.170","Text":"But so I\u0027m assuming that you\u0027ve studied these fields."},{"Start":"00:50.170 ","End":"00:53.600","Text":"In general, it\u0027s not just with a 5,"},{"Start":"00:53.600 ","End":"00:56.090","Text":"it could be any prime number p."},{"Start":"00:56.090 ","End":"01:01.405","Text":"But Part A is just completely regular."},{"Start":"01:01.405 ","End":"01:03.980","Text":"We\u0027re going to do it with matrices"},{"Start":"01:03.980 ","End":"01:06.885","Text":"and with the Gauss elimination method."},{"Start":"01:06.885 ","End":"01:09.129","Text":"The matrix part is standard."},{"Start":"01:09.129 ","End":"01:11.305","Text":"I want to bring it to echelon form,"},{"Start":"01:11.305 ","End":"01:13.450","Text":"and normally I might do something like"},{"Start":"01:13.450 ","End":"01:16.885","Text":"subtract half the 1st row from the 2nd,"},{"Start":"01:16.885 ","End":"01:18.940","Text":"but I don\u0027t want to mess with fractions,"},{"Start":"01:18.940 ","End":"01:21.900","Text":"so the easier thing to do is just to swap"},{"Start":"01:21.900 ","End":"01:25.920","Text":"these 2 rows because a 1 is easier to deal with."},{"Start":"01:25.920 ","End":"01:29.025","Text":"That\u0027s just what we switched the 2 rows around,"},{"Start":"01:29.025 ","End":"01:30.600","Text":"this is what we get."},{"Start":"01:30.600 ","End":"01:36.990","Text":"Now we want do is subtract twice this row from this row."},{"Start":"01:36.990 ","End":"01:41.815","Text":"There\u0027s a notation, and this is how we write the notation for what I just said."},{"Start":"01:41.815 ","End":"01:45.340","Text":"The result, I\u0027ll leave you to check the computation"},{"Start":"01:45.340 ","End":"01:49.515","Text":"just like 2 minus twice 1 is 0, and so on."},{"Start":"01:49.515 ","End":"01:52.110","Text":"3 minus twice 4 is minus 5."},{"Start":"01:52.110 ","End":"01:54.530","Text":"This is now in echelon form,"},{"Start":"01:54.530 ","End":"01:57.080","Text":"at least the restricted matrix."},{"Start":"01:57.080 ","End":"01:59.180","Text":"This whole thing is the augmented,"},{"Start":"01:59.180 ","End":"02:02.335","Text":"remember, and just this part is the restricted."},{"Start":"02:02.335 ","End":"02:08.260","Text":"Now we do what I usually do when I see a common factor,"},{"Start":"02:08.260 ","End":"02:11.630","Text":"it\u0027s not mandatory, but it\u0027s nicer to work with smaller numbers."},{"Start":"02:11.630 ","End":"02:16.535","Text":"Divide by minus 5 on the bottom row."},{"Start":"02:16.535 ","End":"02:20.580","Text":"Doesn\u0027t this look much nicer?"},{"Start":"02:20.800 ","End":"02:26.345","Text":"It\u0027s time to go back from matrix to SLE."},{"Start":"02:26.345 ","End":"02:29.195","Text":"I don\u0027t think any explanation is necessary."},{"Start":"02:29.195 ","End":"02:32.135","Text":"Now we do what we call the backward substitution."},{"Start":"02:32.135 ","End":"02:33.650","Text":"The y we have already,"},{"Start":"02:33.650 ","End":"02:35.830","Text":"so we substitute it here."},{"Start":"02:35.830 ","End":"02:40.460","Text":"Y is 1, so we take this 1 and substitute it instead of y here,"},{"Start":"02:40.460 ","End":"02:43.680","Text":"and this top equation becomes this."},{"Start":"02:43.680 ","End":"02:45.475","Text":"From here, x is 2."},{"Start":"02:45.475 ","End":"02:47.825","Text":"Now we have x and y."},{"Start":"02:47.825 ","End":"02:51.050","Text":"This is a solution written nicely in a box."},{"Start":"02:51.050 ","End":"02:53.300","Text":"But that was Part A with the real numbers."},{"Start":"02:53.300 ","End":"02:59.495","Text":"Let\u0027s see how much is similar and how much is different over the field Z_5."},{"Start":"02:59.495 ","End":"03:05.990","Text":"As before, we\u0027re going to want to bring the system to matrix form."},{"Start":"03:05.990 ","End":"03:09.710","Text":"It\u0027s the same matrix that we had before."},{"Start":"03:09.710 ","End":"03:14.105","Text":"Now, I\u0027d like to bring things into the standard form."},{"Start":"03:14.105 ","End":"03:19.070","Text":"A minus 1 does exist in this field,"},{"Start":"03:19.070 ","End":"03:22.360","Text":"but it has the same name as 4."},{"Start":"03:22.360 ","End":"03:26.610","Text":"We say minus 1 is congruent to 4 Modulo 5."},{"Start":"03:26.610 ","End":"03:30.830","Text":"When 2 numbers differ by a multiple of 5 they are considered the same."},{"Start":"03:30.830 ","End":"03:36.385","Text":"But we prefer to take the representative from this set of numbers."},{"Start":"03:36.385 ","End":"03:40.455","Text":"Everything is in range except this minus 1,"},{"Start":"03:40.455 ","End":"03:44.290","Text":"which now becomes a 4."},{"Start":"03:45.500 ","End":"03:50.050","Text":"I\u0027m going to use the same initial trick"},{"Start":"03:50.050 ","End":"03:53.710","Text":"as in the real numbers of getting the 1 up here."},{"Start":"03:53.710 ","End":"03:56.510","Text":"Swapping the top and bottom rows."},{"Start":"03:56.510 ","End":"03:59.305","Text":"After swapping them, which we write like this,"},{"Start":"03:59.305 ","End":"04:01.390","Text":"then this is what we get to the 1, 2, 4"},{"Start":"04:01.390 ","End":"04:04.720","Text":"that\u0027s now up here 1, 2, 4,"},{"Start":"04:04.720 ","End":"04:07.690","Text":"and the, sorry, this 1, and the 2, 4, 3"},{"Start":"04:07.690 ","End":"04:10.520","Text":"is down here as 2, 4, 3."},{"Start":"04:10.880 ","End":"04:13.060","Text":"I\u0027m not sure that we\u0027ll need it,"},{"Start":"04:13.060 ","End":"04:19.310","Text":"but I brought in the multiplication table for Z_5,"},{"Start":"04:19.310 ","End":"04:22.580","Text":"just the multiplication and also without"},{"Start":"04:22.580 ","End":"04:26.420","Text":"the 0 row because we know that 0 times anything is 0."},{"Start":"04:26.420 ","End":"04:28.805","Text":"I don\u0027t need to put that in the table."},{"Start":"04:28.805 ","End":"04:35.374","Text":"In fact, I could have even eliminated the 1 row because 1 times anything is itself."},{"Start":"04:35.374 ","End":"04:38.465","Text":"But anyway, otherwise it wouldn\u0027t be left with hardly anything."},{"Start":"04:38.465 ","End":"04:41.420","Text":"This is the multiplication table in case we need it."},{"Start":"04:41.420 ","End":"04:47.085","Text":"Like 3 times 2 in this field is equal to 1."},{"Start":"04:47.085 ","End":"04:52.400","Text":"Anyway, continuing, I want to hear,"},{"Start":"04:52.400 ","End":"04:58.310","Text":"subtract twice the top row from the bottom row."},{"Start":"04:58.310 ","End":"05:02.660","Text":"We write that like this. This is what we get."},{"Start":"05:02.660 ","End":"05:04.430","Text":"2 minus twice 1 is 0,"},{"Start":"05:04.430 ","End":"05:07.040","Text":"4 minus twice 2 is 0,"},{"Start":"05:07.040 ","End":"05:10.594","Text":"3 minus twice 4 is minus 5."},{"Start":"05:10.594 ","End":"05:19.125","Text":"Except that minus 5 in Modulo 5 numbers is just the same as 0,"},{"Start":"05:19.125 ","End":"05:23.210","Text":"so I rewrite this as follows."},{"Start":"05:23.210 ","End":"05:24.470","Text":"Now a 0 row,"},{"Start":"05:24.470 ","End":"05:27.305","Text":"we just completely eliminate it."},{"Start":"05:27.305 ","End":"05:30.380","Text":"Now look in the restricted matrix,"},{"Start":"05:30.380 ","End":"05:31.825","Text":"just this part,"},{"Start":"05:31.825 ","End":"05:34.305","Text":"there\u0027s 1 row and 2 columns."},{"Start":"05:34.305 ","End":"05:36.230","Text":"When you have less rows than columns,"},{"Start":"05:36.230 ","End":"05:40.505","Text":"you\u0027re not going to get your unique single solution,"},{"Start":"05:40.505 ","End":"05:42.845","Text":"that\u0027s going to be more than 1 solution."},{"Start":"05:42.845 ","End":"05:47.260","Text":"Let\u0027s see though how it differs from the real numbers."},{"Start":"05:47.260 ","End":"05:52.470","Text":"Well, we get 1 equation in 2 unknowns."},{"Start":"05:52.470 ","End":"05:56.520","Text":"Remember the leading term in each row,"},{"Start":"05:56.520 ","End":"05:57.960","Text":"in this case it\u0027s the x,"},{"Start":"05:57.960 ","End":"06:01.340","Text":"is the dependent term and y is independent."},{"Start":"06:01.340 ","End":"06:04.490","Text":"It can be whatever we like it to be."},{"Start":"06:04.490 ","End":"06:07.220","Text":"We call it say,"},{"Start":"06:07.220 ","End":"06:11.180","Text":"a parameter t. If y is t,"},{"Start":"06:11.180 ","End":"06:12.995","Text":"where t is anything,"},{"Start":"06:12.995 ","End":"06:15.530","Text":"but in this field,"},{"Start":"06:15.530 ","End":"06:19.145","Text":"then x plus 2t is 4,"},{"Start":"06:19.145 ","End":"06:22.120","Text":"so x is 4 minus 2t."},{"Start":"06:22.120 ","End":"06:24.875","Text":"I prefer a plus than a minus."},{"Start":"06:24.875 ","End":"06:30.625","Text":"Minus 2 is the same as plus 3 in this field modulo 5."},{"Start":"06:30.625 ","End":"06:33.440","Text":"I write it as 4 plus 3t,"},{"Start":"06:33.440 ","End":"06:35.900","Text":"although I don\u0027t think it\u0027s wrong to write it this way,"},{"Start":"06:35.900 ","End":"06:39.150","Text":"just prefer to have everything but the plus."},{"Start":"06:39.920 ","End":"06:43.430","Text":"Now, if I package it nicely, x and y,"},{"Start":"06:43.430 ","End":"06:44.840","Text":"y is t from here,"},{"Start":"06:44.840 ","End":"06:46.550","Text":"x is 4 plus 3t from here,"},{"Start":"06:46.550 ","End":"06:48.275","Text":"I\u0027ve got the general solution."},{"Start":"06:48.275 ","End":"06:50.905","Text":"But what can t be?"},{"Start":"06:50.905 ","End":"06:56.030","Text":"T can only be from our field and there is only 5 possibilities."},{"Start":"06:56.030 ","End":"07:00.760","Text":"Sure some of them have different names like 2 and minus 3 are the same,"},{"Start":"07:00.760 ","End":"07:02.360","Text":"2 and 12 are the same,"},{"Start":"07:02.360 ","End":"07:06.530","Text":"but there\u0027s only 5 elements in this field."},{"Start":"07:06.530 ","End":"07:12.050","Text":"In contrast to the case of real numbers where we would say infinite solutions,"},{"Start":"07:12.050 ","End":"07:14.310","Text":"here we have just 5."},{"Start":"07:14.310 ","End":"07:16.890","Text":"In both cases, I guess for t,"},{"Start":"07:16.890 ","End":"07:23.600","Text":"you could say every t only the every in the real numbers is infinitely many."},{"Start":"07:23.600 ","End":"07:28.370","Text":"In the case of Modulo 5 is only 5 different ones."},{"Start":"07:28.370 ","End":"07:30.830","Text":"T could be any 1 of these 5."},{"Start":"07:30.830 ","End":"07:32.720","Text":"If I really want to spell it out,"},{"Start":"07:32.720 ","End":"07:34.780","Text":"which I can do here,"},{"Start":"07:34.780 ","End":"07:36.930","Text":"there\u0027s only 5 combinations."},{"Start":"07:36.930 ","End":"07:40.335","Text":"For each t, when t is y."},{"Start":"07:40.335 ","End":"07:42.570","Text":"You see 0, 1, 2, 3, and 4."},{"Start":"07:42.570 ","End":"07:47.640","Text":"Then we get the x by 4 plus 3t,"},{"Start":"07:47.640 ","End":"07:50.675","Text":"and we get these 5 numbers."},{"Start":"07:50.675 ","End":"07:52.520","Text":"This was not feasible in the case of"},{"Start":"07:52.520 ","End":"07:55.860","Text":"the real numbers because it would be infinitely many."},{"Start":"07:56.810 ","End":"07:59.640","Text":"That does it for this exercise,"},{"Start":"07:59.640 ","End":"08:02.510","Text":"and shows you some of the differences between the real numbers"},{"Start":"08:02.510 ","End":"08:07.860","Text":"and a modulo prime, Modulo 5 field."}],"ID":9847},{"Watched":false,"Name":"Exercise 2","Duration":"5m 39s","ChapterTopicVideoID":9486,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9486.jpeg","UploadDate":"2017-07-26T08:22:40.7770000","DurationForVideoObject":"PT5M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:09.780","Text":"This exercise assumes that you\u0027ve studied the field of integers modulo of prime number,"},{"Start":"00:09.780 ","End":"00:13.200","Text":"say modulo 5, and you know what this symbol means,"},{"Start":"00:13.200 ","End":"00:17.940","Text":"and that its elements we choose are 0,"},{"Start":"00:17.940 ","End":"00:19.840","Text":"1, 2, 3 and 4."},{"Start":"00:19.840 ","End":"00:21.650","Text":"If you don\u0027t know what this is,"},{"Start":"00:21.650 ","End":"00:25.330","Text":"you probably haven\u0027t studied it and you should skip this exercise."},{"Start":"00:25.330 ","End":"00:27.620","Text":"Just like with real numbers,"},{"Start":"00:27.620 ","End":"00:29.135","Text":"we\u0027re going to use matrices."},{"Start":"00:29.135 ","End":"00:31.160","Text":"It makes it easier, we don\u0027t have to drag the x,"},{"Start":"00:31.160 ","End":"00:32.945","Text":"y, z everywhere we go."},{"Start":"00:32.945 ","End":"00:34.250","Text":"This is the matrix form,"},{"Start":"00:34.250 ","End":"00:36.080","Text":"it\u0027s an augmented matrix."},{"Start":"00:36.080 ","End":"00:38.420","Text":"Now we want to bring it into"},{"Start":"00:38.420 ","End":"00:42.800","Text":"row echelon form we\u0027re using is called the Gauss elimination method."},{"Start":"00:42.800 ","End":"00:45.365","Text":"I\u0027m going to make this and this 0,"},{"Start":"00:45.365 ","End":"00:47.930","Text":"we\u0027re lucky we have a 1 here, makes it easier."},{"Start":"00:47.930 ","End":"00:51.040","Text":"I subtract twice this row from this row,"},{"Start":"00:51.040 ","End":"00:53.510","Text":"and 3 times this row from this row,"},{"Start":"00:53.510 ","End":"00:55.240","Text":"that should do it."},{"Start":"00:55.240 ","End":"00:58.310","Text":"Indeed I wrote the row operations here,"},{"Start":"00:58.310 ","End":"01:00.380","Text":"bit small, but you can read it,"},{"Start":"01:00.380 ","End":"01:02.700","Text":"and we get this."},{"Start":"01:02.800 ","End":"01:09.080","Text":"Now, what I mean is that we have numbers that are not 0, 1, 2, 3,"},{"Start":"01:09.080 ","End":"01:12.145","Text":"4 because really we only have 0, 1,"},{"Start":"01:12.145 ","End":"01:15.970","Text":"2, 3, and 4 in the number system."},{"Start":"01:15.970 ","End":"01:19.705","Text":"But the numbers have different names."},{"Start":"01:19.705 ","End":"01:22.509","Text":"In the modulo 5 system,"},{"Start":"01:22.509 ","End":"01:27.100","Text":"any number that\u0027s differs by 5 or a multiple is the same number."},{"Start":"01:27.100 ","End":"01:29.310","Text":"For example, 2 is the same as 7,"},{"Start":"01:29.310 ","End":"01:30.960","Text":"is the same as 12,"},{"Start":"01:30.960 ","End":"01:33.775","Text":"is the same as minus 3,"},{"Start":"01:33.775 ","End":"01:39.700","Text":"is the same as minus 8 and so on."},{"Start":"01:39.700 ","End":"01:41.950","Text":"If it differs by multiple of 5,"},{"Start":"01:41.950 ","End":"01:43.315","Text":"it\u0027s the same thing."},{"Start":"01:43.315 ","End":"01:46.610","Text":"Now we can write the standard form."},{"Start":"01:46.610 ","End":"01:48.690","Text":"This becomes this. Now,"},{"Start":"01:48.690 ","End":"01:49.895","Text":"how did I get to that?"},{"Start":"01:49.895 ","End":"01:53.705","Text":"Well, for example, the minus 8,"},{"Start":"01:53.705 ","End":"01:55.700","Text":"we looked at it, it\u0027s here,"},{"Start":"01:55.700 ","End":"02:02.160","Text":"it differs by multiple of 5 from the 2."},{"Start":"02:02.160 ","End":"02:07.890","Text":"Minus 8 gives me 2 and minus 6,"},{"Start":"02:07.890 ","End":"02:11.855","Text":"if I add or subtract multiples of 5 to bring it into range,"},{"Start":"02:11.855 ","End":"02:14.615","Text":"actually gives me 4."},{"Start":"02:14.615 ","End":"02:17.615","Text":"That\u0027s this 4 and that\u0027s here."},{"Start":"02:17.615 ","End":"02:24.060","Text":"Minus 2, I can add 5 to 8 is the same as 3,"},{"Start":"02:24.060 ","End":"02:26.565","Text":"and that gives me that."},{"Start":"02:26.565 ","End":"02:32.140","Text":"Minus 3 is the same as 2."},{"Start":"02:33.080 ","End":"02:35.265","Text":"Let\u0027s continue."},{"Start":"02:35.265 ","End":"02:43.530","Text":"Now, we want to try and bring the entry here to be 0,"},{"Start":"02:43.530 ","End":"02:45.810","Text":"because we\u0027re still not in echelon form."},{"Start":"02:45.810 ","End":"02:48.070","Text":"The 0 here is a bit of a problem,"},{"Start":"02:48.070 ","End":"02:49.905","Text":"it\u0027s only a bit."},{"Start":"02:49.905 ","End":"02:56.030","Text":"What I\u0027ll do is I\u0027ll swap the second and third rows and then the 0 will be down here."},{"Start":"02:56.030 ","End":"02:58.760","Text":"Yeah, that\u0027s better. Just need to swap the 2 rows,"},{"Start":"02:58.760 ","End":"03:01.100","Text":"that 4 goes down here and the 0 goes up here,"},{"Start":"03:01.100 ","End":"03:03.675","Text":"and similarly for the other entries."},{"Start":"03:03.675 ","End":"03:08.430","Text":"Now it is in row echelon,"},{"Start":"03:08.430 ","End":"03:12.150","Text":"just talking about the restricted matrix."},{"Start":"03:13.550 ","End":"03:15.800","Text":"Once it\u0027s in echelon form,"},{"Start":"03:15.800 ","End":"03:18.095","Text":"we can use back substitution."},{"Start":"03:18.095 ","End":"03:23.270","Text":"First thing we do is to go back from matrix form to x,"},{"Start":"03:23.270 ","End":"03:24.710","Text":"y, z form,"},{"Start":"03:24.710 ","End":"03:27.410","Text":"so that\u0027s these 3 equations."},{"Start":"03:27.410 ","End":"03:30.695","Text":"Then we start from the bottom and work our way up."},{"Start":"03:30.695 ","End":"03:33.335","Text":"From the last 1 if 3z is 0,"},{"Start":"03:33.335 ","End":"03:35.525","Text":"that means that z is 0."},{"Start":"03:35.525 ","End":"03:38.700","Text":"If z is 0, I substituted in here,"},{"Start":"03:38.700 ","End":"03:43.575","Text":"if it\u0027s 0, I get 4y equals 2."},{"Start":"03:43.575 ","End":"03:52.790","Text":"Let me at the side see what we do about 4y equals 2 in the field of integers modulo 5."},{"Start":"03:52.790 ","End":"03:58.055","Text":"Normally, I would be taking a multiplication table with me."},{"Start":"03:58.055 ","End":"04:02.525","Text":"We don\u0027t usually include the zeros in the table because clearly that\u0027s all zeros."},{"Start":"04:02.525 ","End":"04:04.850","Text":"But 1 times 1 is 1,"},{"Start":"04:04.850 ","End":"04:06.680","Text":"so this is 1, 2, 3, 4."},{"Start":"04:06.680 ","End":"04:08.165","Text":"This is also pretty easy."},{"Start":"04:08.165 ","End":"04:10.530","Text":"Anything times 1, 3, 4,"},{"Start":"04:10.530 ","End":"04:12.885","Text":"so 2 times 2 is 4,"},{"Start":"04:12.885 ","End":"04:14.460","Text":"2 times 3 is 6,"},{"Start":"04:14.460 ","End":"04:16.355","Text":"but 6 is 1."},{"Start":"04:16.355 ","End":"04:20.030","Text":"2 times 4 is 8, which is 3 modulo 5."},{"Start":"04:20.030 ","End":"04:22.310","Text":"3 times 2 is 6, which is 1."},{"Start":"04:22.310 ","End":"04:24.845","Text":"3 times 3 is 9, which is 4."},{"Start":"04:24.845 ","End":"04:28.230","Text":"3 times 4 is 12, which is 2."},{"Start":"04:28.250 ","End":"04:31.695","Text":"4 times 2 is 8, which is 3."},{"Start":"04:31.695 ","End":"04:35.280","Text":"4 times 3 is 12, which is 2."},{"Start":"04:35.280 ","End":"04:38.620","Text":"4 times 4 is16, which is 1."},{"Start":"04:38.620 ","End":"04:41.690","Text":"Should have had this ready and printed. Okay, never mind."},{"Start":"04:41.690 ","End":"04:43.550","Text":"4y equals 2."},{"Start":"04:43.550 ","End":"04:49.955","Text":"We look at the row for 4 and we look for where do we get the 2,"},{"Start":"04:49.955 ","End":"04:52.085","Text":"we see the 2 is here."},{"Start":"04:52.085 ","End":"04:57.005","Text":"Then we go up and we see that 4 times 3 is 2,"},{"Start":"04:57.005 ","End":"04:59.340","Text":"or the other way around."},{"Start":"04:59.780 ","End":"05:02.120","Text":"We look for where they get the 2."},{"Start":"05:02.120 ","End":"05:03.545","Text":"3 times 4 is 2,"},{"Start":"05:03.545 ","End":"05:07.110","Text":"so we get that y is 3."},{"Start":"05:07.600 ","End":"05:11.720","Text":"Now, we go back to the first 1,"},{"Start":"05:11.720 ","End":"05:12.890","Text":"we\u0027re doing back substitution."},{"Start":"05:12.890 ","End":"05:16.699","Text":"This time we have that y is 3, z is 0."},{"Start":"05:16.699 ","End":"05:19.090","Text":"We have to figure out x."},{"Start":"05:19.090 ","End":"05:25.664","Text":"What we get is that x plus 6 is 1."},{"Start":"05:25.664 ","End":"05:27.465","Text":"Well, 6 is 1,"},{"Start":"05:27.465 ","End":"05:31.440","Text":"so we get that x plus 1 is 1,"},{"Start":"05:31.440 ","End":"05:34.200","Text":"so x is 0,"},{"Start":"05:34.200 ","End":"05:39.910","Text":"and that\u0027s the answer and we\u0027re done."}],"ID":9848},{"Watched":false,"Name":"Exercise 3","Duration":"5m 17s","ChapterTopicVideoID":9487,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9487.jpeg","UploadDate":"2017-07-26T08:23:37.7870000","DurationForVideoObject":"PT5M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have a system of linear equations,"},{"Start":"00:04.650 ","End":"00:07.065","Text":"3 equations and 3 unknowns."},{"Start":"00:07.065 ","End":"00:10.110","Text":"But it\u0027s not over the field of real numbers."},{"Start":"00:10.110 ","End":"00:13.830","Text":"It\u0027s over the field of integers modulo 5,"},{"Start":"00:13.830 ","End":"00:16.425","Text":"written Z_5 like this,"},{"Start":"00:16.425 ","End":"00:21.615","Text":"a field with only 5 elements from 0 through 4."},{"Start":"00:21.615 ","End":"00:24.120","Text":"If all this is new to you,"},{"Start":"00:24.120 ","End":"00:31.215","Text":"you don\u0027t know what this means and you probably shouldn\u0027t be in this lecture clip,"},{"Start":"00:31.215 ","End":"00:33.615","Text":"so just skip it."},{"Start":"00:33.615 ","End":"00:36.720","Text":"I\u0027m assuming you have learned this."},{"Start":"00:36.720 ","End":"00:42.970","Text":"As usual, we like to solve SLEs with matrices."},{"Start":"00:43.730 ","End":"00:46.090","Text":"Just in case we need it,"},{"Start":"00:46.090 ","End":"00:52.430","Text":"I brought a multiplication table for Z_5."},{"Start":"00:54.380 ","End":"00:58.270","Text":"Back here, the idea is to bring it into"},{"Start":"00:58.270 ","End":"01:04.025","Text":"row echelon form and to make it so that there are zeros here and here."},{"Start":"01:04.025 ","End":"01:07.690","Text":"1 possible approach, and we\u0027re not going to use it,"},{"Start":"01:07.690 ","End":"01:11.005","Text":"which is why I wrote it in this faint color,"},{"Start":"01:11.005 ","End":"01:14.095","Text":"is we could leave the 3 here and say,"},{"Start":"01:14.095 ","End":"01:18.860","Text":"\"Okay, 3 times this row minus 4 times this row will give me 0."},{"Start":"01:18.860 ","End":"01:24.945","Text":"Also 3 times this row minus twice this row would give me 0 here.\""},{"Start":"01:24.945 ","End":"01:27.740","Text":"But that\u0027s not ideal."},{"Start":"01:27.740 ","End":"01:31.330","Text":"I would like there to be a 1 here."},{"Start":"01:31.330 ","End":"01:36.575","Text":"The thing with modulo fields is we don\u0027t need fractions,"},{"Start":"01:36.575 ","End":"01:39.260","Text":"we don\u0027t have to divide by 3."},{"Start":"01:39.260 ","End":"01:42.750","Text":"We can multiply by the inverse of 3,"},{"Start":"01:42.750 ","End":"01:47.045","Text":"and the way we find the inverse is by looking in the table."},{"Start":"01:47.045 ","End":"01:51.895","Text":"I start with the 3 and I go along until I come across a 1."},{"Start":"01:51.895 ","End":"01:56.470","Text":"I see that 3 times 2 is 1 in this field."},{"Start":"01:56.480 ","End":"01:59.390","Text":"The equivalent of dividing by 3,"},{"Start":"01:59.390 ","End":"02:01.685","Text":"which you would have done with reals may be,"},{"Start":"02:01.685 ","End":"02:04.990","Text":"is to multiply by the inverse of 3."},{"Start":"02:04.990 ","End":"02:07.650","Text":"As we saw, the inverse of 3 is 2,"},{"Start":"02:07.650 ","End":"02:14.015","Text":"so my row operation will be to double the 1st row."},{"Start":"02:14.015 ","End":"02:17.590","Text":"Now if we double it,"},{"Start":"02:17.590 ","End":"02:23.645","Text":"and here I just put 2 times in front of the 1st row in everything."},{"Start":"02:23.645 ","End":"02:25.150","Text":"Like we expected,"},{"Start":"02:25.150 ","End":"02:26.970","Text":"2 times 3 is 1."},{"Start":"02:26.970 ","End":"02:29.405","Text":"You have the multiplication table there."},{"Start":"02:29.405 ","End":"02:31.190","Text":"In fact, if we look at row 2,"},{"Start":"02:31.190 ","End":"02:33.860","Text":"here we\u0027ve got 2 times 1,"},{"Start":"02:33.860 ","End":"02:36.140","Text":"we\u0027ve got 2 times 4, 2 times 3."},{"Start":"02:36.140 ","End":"02:40.115","Text":"Anyway if you check the computations,"},{"Start":"02:40.115 ","End":"02:45.395","Text":"then you see that we have the top row becomes 1, 2, 3, 1."},{"Start":"02:45.395 ","End":"02:50.090","Text":"Anyway, we have our 1 here which was our main objective."},{"Start":"02:50.090 ","End":"02:58.820","Text":"Now we can subtract 4 times this row from here and twice the top row from the bottom row,"},{"Start":"02:58.820 ","End":"03:02.645","Text":"which in row notation looks like this."},{"Start":"03:02.645 ","End":"03:05.350","Text":"See if we do that,"},{"Start":"03:05.350 ","End":"03:10.130","Text":"we will get this."},{"Start":"03:10.130 ","End":"03:13.070","Text":"Well, this is an intermediate stage where I allow myself"},{"Start":"03:13.070 ","End":"03:16.540","Text":"to write numbers outside the range of 0-4."},{"Start":"03:16.540 ","End":"03:20.240","Text":"Then I find the equivalence, for example,"},{"Start":"03:20.240 ","End":"03:30.240","Text":"minus 5 in this field is 0 and minus 9 if I add 10 is 1 and so on."},{"Start":"03:30.240 ","End":"03:33.270","Text":"Then these are the numbers we get."},{"Start":"03:33.270 ","End":"03:40.320","Text":"We\u0027ve replaced everything by its modulo equivalent from 0 through 4."},{"Start":"03:41.210 ","End":"03:46.060","Text":"Let\u0027s see, this is not yet in echelon form."},{"Start":"03:46.060 ","End":"03:49.845","Text":"Now here I have a 0 here,"},{"Start":"03:49.845 ","End":"03:52.065","Text":"so what I\u0027m going to do,"},{"Start":"03:52.065 ","End":"03:54.510","Text":"I can\u0027t use this to 0 out the 1,"},{"Start":"03:54.510 ","End":"03:56.880","Text":"but I can just swap rows."},{"Start":"03:56.880 ","End":"04:03.585","Text":"Yeah, just switched around row 2 with row 3,"},{"Start":"04:03.585 ","End":"04:08.460","Text":"and this is what we get."},{"Start":"04:08.460 ","End":"04:10.365","Text":"You can see top rows unchanged."},{"Start":"04:10.365 ","End":"04:11.400","Text":"This 0, 0, 1,"},{"Start":"04:11.400 ","End":"04:13.350","Text":"0 is down here and the 0, 1, 3,"},{"Start":"04:13.350 ","End":"04:16.580","Text":"3 is here. Now everything\u0027s fine."},{"Start":"04:16.580 ","End":"04:19.475","Text":"Not only is it in echelon form,"},{"Start":"04:19.475 ","End":"04:24.875","Text":"but we\u0027ve even got ones on the diagonal, which is great."},{"Start":"04:24.875 ","End":"04:32.920","Text":"Back to the SLE system of linear equations from the matrix."},{"Start":"04:32.920 ","End":"04:35.065","Text":"Now we use the back substitution,"},{"Start":"04:35.065 ","End":"04:37.040","Text":"from the bottom, we work our way up."},{"Start":"04:37.040 ","End":"04:40.010","Text":"This last row tells us exactly what z is."},{"Start":"04:40.010 ","End":"04:41.780","Text":"We then go back up a step,"},{"Start":"04:41.780 ","End":"04:44.150","Text":"substitute z is 0."},{"Start":"04:44.150 ","End":"04:46.880","Text":"It just tells us that y equals 3."},{"Start":"04:46.880 ","End":"04:51.960","Text":"Now, put z equals 0 and y equals 3 here."},{"Start":"04:51.960 ","End":"04:57.885","Text":"We get that x equals 1 minus twice 3,"},{"Start":"04:57.885 ","End":"04:59.710","Text":"well, minus 3 times 0,"},{"Start":"04:59.710 ","End":"05:01.190","Text":"I didn\u0027t write that."},{"Start":"05:01.190 ","End":"05:05.020","Text":"This is minus 5, which is 0."},{"Start":"05:05.020 ","End":"05:11.765","Text":"Now we\u0027ve got the variable z, y, and x."},{"Start":"05:11.765 ","End":"05:14.120","Text":"Just arrange them nicely in a box,"},{"Start":"05:14.120 ","End":"05:17.010","Text":"and this is our answer."}],"ID":9849},{"Watched":false,"Name":"Exercise 4","Duration":"7m 6s","ChapterTopicVideoID":9488,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9488.jpeg","UploadDate":"2017-07-26T08:25:07.3200000","DurationForVideoObject":"PT7M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have a system of linear equations."},{"Start":"00:04.650 ","End":"00:09.030","Text":"It\u0027s 4 equations and 4 unknowns, x, y, z, t."},{"Start":"00:09.030 ","End":"00:12.585","Text":"The field is not the real numbers."},{"Start":"00:12.585 ","End":"00:15.480","Text":"It\u0027s the field Z_7,"},{"Start":"00:15.480 ","End":"00:18.270","Text":"which means integers modulo 7,"},{"Start":"00:18.270 ","End":"00:21.345","Text":"and contains the following 7 elements."},{"Start":"00:21.345 ","End":"00:24.510","Text":"If all this looks new to you and you haven\u0027t studied this,"},{"Start":"00:24.510 ","End":"00:31.000","Text":"then you probably shouldn\u0027t be in this clip and no harm done."},{"Start":"00:31.100 ","End":"00:33.900","Text":"For those of you who have studied this time,"},{"Start":"00:33.900 ","End":"00:35.295","Text":"I\u0027m going to continue."},{"Start":"00:35.295 ","End":"00:41.099","Text":"So we\u0027ll convert this SLE into matrix form."},{"Start":"00:41.099 ","End":"00:44.485","Text":"We get this augmented matrix,"},{"Start":"00:44.485 ","End":"00:47.480","Text":"the augmented because the extra bit,"},{"Start":"00:47.480 ","End":"00:54.650","Text":"it\u0027s the 4 by 4 co-efficient matrix on the free numbers after the separator."},{"Start":"00:54.650 ","End":"00:57.320","Text":"Now the ones are marked in red are the ones that are not in"},{"Start":"00:57.320 ","End":"00:59.940","Text":"the standard form for the field Z_7."},{"Start":"00:59.940 ","End":"01:05.875","Text":"We just add or subtract multiples of 7 to bring them in line."},{"Start":"01:05.875 ","End":"01:11.550","Text":"Minus 1 becomes 6 when we add 7 to it and minus 2 becomes 5,"},{"Start":"01:11.550 ","End":"01:14.385","Text":"and this is now our matrix."},{"Start":"01:14.385 ","End":"01:17.270","Text":"We probably need a multiplication table."},{"Start":"01:17.270 ","End":"01:20.520","Text":"This is the multiplication table for the field."},{"Start":"01:20.520 ","End":"01:30.275","Text":"Z_7 I omitted the zero from this probably even I omitted the 1 but anyway, here it is."},{"Start":"01:30.275 ","End":"01:35.130","Text":"Like 3 times 4 gives us 5 and so on."},{"Start":"01:35.210 ","End":"01:40.040","Text":"Now, we want to bring this into echelon form."},{"Start":"01:40.040 ","End":"01:44.300","Text":"So I want to use this 1 to 0 out the rest of the column."},{"Start":"01:44.300 ","End":"01:45.830","Text":"Well, this is already 0."},{"Start":"01:45.830 ","End":"01:49.265","Text":"If I subtract the top row from both the 2nd and the fourth,"},{"Start":"01:49.265 ","End":"01:53.490","Text":"that should start off nicely with the 1st column."},{"Start":"01:54.140 ","End":"02:00.420","Text":"This is what I just said in row notation for what I\u0027m going to do."},{"Start":"02:00.440 ","End":"02:02.720","Text":"This is what we end up with,"},{"Start":"02:02.720 ","End":"02:04.760","Text":"although I did 2 steps in 1,"},{"Start":"02:04.760 ","End":"02:08.600","Text":"I also converted the numbers, for example,"},{"Start":"02:08.600 ","End":"02:12.785","Text":"when I take 2 minus 4,"},{"Start":"02:12.785 ","End":"02:14.975","Text":"I want to put a minus 2 here,"},{"Start":"02:14.975 ","End":"02:20.885","Text":"but minus 2 is the same as 5 in modulo 7, that\u0027s a 5."},{"Start":"02:20.885 ","End":"02:23.495","Text":"Another example say here,"},{"Start":"02:23.495 ","End":"02:32.100","Text":"I did the 0 minus 1,"},{"Start":"02:32.100 ","End":"02:34.740","Text":"which is minus 1, which is 6;"},{"Start":"02:34.740 ","End":"02:38.265","Text":"3 minus 4 is minus 1 also 6 and so on and so on,"},{"Start":"02:38.265 ","End":"02:40.365","Text":"this is now the matrix."},{"Start":"02:40.365 ","End":"02:48.500","Text":"It\u0027s still not in echelon form because I have a 1 here and I want to zero here."},{"Start":"02:48.770 ","End":"02:51.930","Text":"There\u0027s more than 1 way to go about this."},{"Start":"02:51.930 ","End":"02:54.095","Text":"I\u0027d really like a 1 here,"},{"Start":"02:54.095 ","End":"02:57.985","Text":"cause I could multiply by the inverse of 5,"},{"Start":"02:57.985 ","End":"03:00.075","Text":"but this is a 1 here,"},{"Start":"03:00.075 ","End":"03:03.769","Text":"easiest thing to do is just to swap these 2 rows,"},{"Start":"03:03.769 ","End":"03:08.850","Text":"change the places of row 3 and row 2."},{"Start":"03:09.170 ","End":"03:11.760","Text":"Then we get to this,"},{"Start":"03:11.760 ","End":"03:15.435","Text":"and now we want to get rid of this 5 and 6, make them 0s."},{"Start":"03:15.435 ","End":"03:21.705","Text":"We\u0027ll subtract 5 times this row from this row and 6 times this row from this row,"},{"Start":"03:21.705 ","End":"03:28.600","Text":"and that such in row notation and here\u0027s our result."},{"Start":"03:28.600 ","End":"03:32.320","Text":"The ones in color are the ones who have converted."},{"Start":"03:32.320 ","End":"03:33.985","Text":"Well, I\u0027ll give you an example,"},{"Start":"03:33.985 ","End":"03:38.770","Text":"this minus 5 times this is 4 minus 5 is minus 1,"},{"Start":"03:38.770 ","End":"03:40.240","Text":"but minus 1 is 6."},{"Start":"03:40.240 ","End":"03:42.680","Text":"So I wrote a 6 here."},{"Start":"03:44.900 ","End":"03:48.870","Text":"Sorry, minus 6 ones is minus 2,"},{"Start":"03:48.870 ","End":"03:52.575","Text":"but minus 2 is 5, and so on."},{"Start":"03:52.575 ","End":"03:56.155","Text":"We\u0027ve got this we\u0027re getting very close."},{"Start":"03:56.155 ","End":"03:59.245","Text":"I need a 0 here,"},{"Start":"03:59.245 ","End":"04:02.620","Text":"so to get it into echelon form,"},{"Start":"04:02.620 ","End":"04:11.300","Text":"and so what we\u0027ll do is to divide this row by 6 to get a 1 here."},{"Start":"04:11.300 ","End":"04:20.060","Text":"When I say sort of we don\u0027t really divide by 6 or we can multiply by the inverse of 6."},{"Start":"04:20.060 ","End":"04:23.345","Text":"Back here at the multiplication table,"},{"Start":"04:23.345 ","End":"04:30.995","Text":"I can tell what the inverse of 6 is by going along the row where the 6 is until I get 1."},{"Start":"04:30.995 ","End":"04:34.520","Text":"Here\u0027s the 1, I go back up and see it\u0027s 6."},{"Start":"04:34.520 ","End":"04:37.445","Text":"In this field, 6 times 6 is 1."},{"Start":"04:37.445 ","End":"04:45.380","Text":"If you think about it, 36 is the same as 1 modulo 7 because 35 is nothing."},{"Start":"04:45.380 ","End":"04:47.910","Text":"So 36 is 1."},{"Start":"04:47.980 ","End":"04:52.640","Text":"Now back here what we\u0027re doing, as I said,"},{"Start":"04:52.640 ","End":"04:55.250","Text":"is like dividing this row by 6,"},{"Start":"04:55.250 ","End":"04:57.620","Text":"but instead we\u0027re multiplying by the inverse,"},{"Start":"04:57.620 ","End":"04:59.420","Text":"which happens to also be 6."},{"Start":"04:59.420 ","End":"05:05.570","Text":"So 6 times this row will give me this if you use the multiplication table,"},{"Start":"05:05.570 ","End":"05:07.220","Text":"6 times 6 is 1,"},{"Start":"05:07.220 ","End":"05:09.835","Text":"6 times 5 is 2."},{"Start":"05:09.835 ","End":"05:11.940","Text":"We think about 6 times 5 is 30,"},{"Start":"05:11.940 ","End":"05:14.915","Text":"which is 2 over 28 and so on,"},{"Start":"05:14.915 ","End":"05:17.210","Text":"6 times 1 is just 6."},{"Start":"05:17.210 ","End":"05:21.215","Text":"That\u0027s all we did, and now we\u0027ve got 1,1,1."},{"Start":"05:21.215 ","End":"05:25.880","Text":"Now it\u0027s easy, subtract 5 times this row from the last row."},{"Start":"05:25.880 ","End":"05:28.895","Text":"Same thing when I said in row notation."},{"Start":"05:28.895 ","End":"05:31.970","Text":"Let\u0027s see what the result comes out."},{"Start":"05:31.970 ","End":"05:35.150","Text":"There we are, and it already is in"},{"Start":"05:35.150 ","End":"05:41.135","Text":"echelon form but it\u0027s nicer if you have ones on the diagonal."},{"Start":"05:41.135 ","End":"05:43.715","Text":"The last 1 I\u0027m going to divide by 6."},{"Start":"05:43.715 ","End":"05:46.670","Text":"We already saw that the inverse of 6 is 6."},{"Start":"05:46.670 ","End":"05:49.945","Text":"So we can multiply the last row by 6."},{"Start":"05:49.945 ","End":"05:52.290","Text":"That\u0027s what I\u0027m doing in row notation."},{"Start":"05:52.290 ","End":"05:55.280","Text":"We remember that the inverse of 6 is 6."},{"Start":"05:55.280 ","End":"05:57.620","Text":"If we do that operation,"},{"Start":"05:57.620 ","End":"05:59.780","Text":"we get these 6 times 6 is 1,"},{"Start":"05:59.780 ","End":"06:03.420","Text":"6 times 5 is 2 in this field."},{"Start":"06:04.100 ","End":"06:11.420","Text":"Again, I think it\u0027s time from here to go back to the SLE,"},{"Start":"06:11.420 ","End":"06:14.310","Text":"system of linear equations."},{"Start":"06:14.900 ","End":"06:18.230","Text":"Sorry this matrix corresponds to the system."},{"Start":"06:18.230 ","End":"06:19.610","Text":"Just copy the coefficients,"},{"Start":"06:19.610 ","End":"06:21.800","Text":"don\u0027t bother with the 0s."},{"Start":"06:21.800 ","End":"06:24.170","Text":"We use backward substitution."},{"Start":"06:24.170 ","End":"06:25.760","Text":"We start from here."},{"Start":"06:25.760 ","End":"06:30.905","Text":"The last row we know tells us explicitly that t is 2."},{"Start":"06:30.905 ","End":"06:32.675","Text":"We plug in here,"},{"Start":"06:32.675 ","End":"06:36.350","Text":"we get z plus 4 is 6,"},{"Start":"06:36.350 ","End":"06:37.760","Text":"so z is 2,"},{"Start":"06:37.760 ","End":"06:42.450","Text":"then we plug in t equals 2 and z equals 2 here."},{"Start":"06:42.450 ","End":"06:45.990","Text":"We get y plus 4 is 1,"},{"Start":"06:45.990 ","End":"06:47.670","Text":"y is minus 3,"},{"Start":"06:47.670 ","End":"06:50.055","Text":"but minus 3 is 4,"},{"Start":"06:50.055 ","End":"06:52.470","Text":"and I\u0027ll leave you to check the last 1,"},{"Start":"06:52.470 ","End":"06:55.560","Text":"then we have all the values."},{"Start":"06:55.560 ","End":"06:58.850","Text":"Just arrange them neatly and declare that yes,"},{"Start":"06:58.850 ","End":"07:05.550","Text":"this is the result of the system of linear equations over Z_7."}],"ID":9850},{"Watched":false,"Name":"Exercise 5","Duration":"9m 27s","ChapterTopicVideoID":9489,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9489.jpeg","UploadDate":"2017-07-26T08:26:16.1330000","DurationForVideoObject":"PT9M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.885","Text":"In this exercise, we have a system of linear equations,"},{"Start":"00:04.885 ","End":"00:08.250","Text":"3 equations and 3 unknowns, x, y, z,"},{"Start":"00:08.250 ","End":"00:11.340","Text":"with a parameter that\u0027s that k."},{"Start":"00:11.340 ","End":"00:14.730","Text":"That\u0027s here, here, 3 places,"},{"Start":"00:14.730 ","End":"00:18.810","Text":"and the field is not the real numbers."},{"Start":"00:18.810 ","End":"00:25.305","Text":"It\u0027s the integer\u0027s modulo 3, Z_3, which contains 3 elements: 0, 1, or 2."},{"Start":"00:25.305 ","End":"00:28.560","Text":"If you don\u0027t know what this means and you haven\u0027t learned it,"},{"Start":"00:28.560 ","End":"00:35.740","Text":"and you probably shouldn\u0027t be in this lecture, clip, whatever. It\u0027s not for you."},{"Start":"00:37.280 ","End":"00:39.590","Text":"Assuming that you have, the question is as follows,"},{"Start":"00:39.590 ","End":"00:47.270","Text":"we have to decide which values of the parameter k for which will the system have"},{"Start":"00:47.270 ","End":"00:51.500","Text":"exactly 1 solution, or no solutions,"},{"Start":"00:51.500 ","End":"00:55.940","Text":"or which values of k would give infinite many solutions,"},{"Start":"00:55.940 ","End":"00:59.100","Text":"or maybe there\u0027s another possibility."},{"Start":"00:59.600 ","End":"01:05.150","Text":"As we usually do, we\u0027ll solve it with matrices."},{"Start":"01:05.150 ","End":"01:08.750","Text":"It\u0027s easier. We don\u0027t have to drag the x, y, z everywhere."},{"Start":"01:08.750 ","End":"01:18.905","Text":"Something that might come in useful about the addition and multiplication tables for Z_3."},{"Start":"01:18.905 ","End":"01:21.245","Text":"Both for this field,"},{"Start":"01:21.245 ","End":"01:23.465","Text":"Z_3 is the addition table."},{"Start":"01:23.465 ","End":"01:27.440","Text":"Only thing unusual, 2 plus 2 is 1"},{"Start":"01:27.440 ","End":"01:31.890","Text":"because modulo 3, 4 is the same as 1."},{"Start":"01:31.890 ","End":"01:35.135","Text":"Here\u0027s the multiplication table also."},{"Start":"01:35.135 ","End":"01:37.415","Text":"The only 1 unusual is this 1."},{"Start":"01:37.415 ","End":"01:41.250","Text":"2 times 2 is 1 in this field."},{"Start":"01:42.380 ","End":"01:44.450","Text":"You know the drill."},{"Start":"01:44.450 ","End":"01:49.740","Text":"We want to bring it into echelon form."},{"Start":"01:49.870 ","End":"01:52.475","Text":"I have a 1 here,"},{"Start":"01:52.475 ","End":"01:55.985","Text":"so I can use it to zero out the rest of the column."},{"Start":"01:55.985 ","End":"01:59.840","Text":"I\u0027m going to subtract the first row from the second row."},{"Start":"01:59.840 ","End":"02:03.260","Text":"I\u0027m going to subtract k times the first row from the last row,"},{"Start":"02:03.260 ","End":"02:05.305","Text":"and that\u0027s what I say here."},{"Start":"02:05.305 ","End":"02:08.060","Text":"I\u0027m not going to go into each computation."},{"Start":"02:08.060 ","End":"02:11.195","Text":"You can just pause the clip and verify."},{"Start":"02:11.195 ","End":"02:15.860","Text":"Perhaps I can give you 1 example, like this entry."},{"Start":"02:15.860 ","End":"02:19.485","Text":"I take this minus k times this,"},{"Start":"02:19.485 ","End":"02:21.599","Text":"so I\u0027ve got 1 minus k squared."},{"Start":"02:21.599 ","End":"02:25.470","Text":"That\u0027s from the row 3 minus k times row 1."},{"Start":"02:25.470 ","End":"02:32.720","Text":"That\u0027s already zeroed out the rest of the first column."},{"Start":"02:32.720 ","End":"02:37.745","Text":"Next, we want a 0 here."},{"Start":"02:37.745 ","End":"02:40.970","Text":"To help me with that, I\u0027ve used a bit of algebra."},{"Start":"02:40.970 ","End":"02:47.285","Text":"I\u0027ve rewritten this as this using the famous difference of squares formula for"},{"Start":"02:47.285 ","End":"02:50.990","Text":"a squared minus b squared is a minus b, a plus b."},{"Start":"02:50.990 ","End":"02:54.020","Text":"I don\u0027t want to even write it. This becomes this."},{"Start":"02:54.020 ","End":"03:00.920","Text":"Now it\u0027s easier to see that we have to multiply this row by"},{"Start":"03:00.920 ","End":"03:04.085","Text":"1 plus k and then subtract,"},{"Start":"03:04.085 ","End":"03:06.700","Text":"and that will zero out this entry."},{"Start":"03:06.700 ","End":"03:11.570","Text":"This is just what I said written in proper row notation."},{"Start":"03:11.570 ","End":"03:15.905","Text":"If we do it, the result comes out looking like this."},{"Start":"03:15.905 ","End":"03:18.440","Text":"This is before simplification."},{"Start":"03:18.440 ","End":"03:24.515","Text":"After we tidy up this element with a bit of algebra, then we get this."},{"Start":"03:24.515 ","End":"03:25.820","Text":"Perhaps I\u0027ll just show you."},{"Start":"03:25.820 ","End":"03:30.885","Text":"1 minus k minus,"},{"Start":"03:30.885 ","End":"03:33.810","Text":"now 1 plus k times k minus 1."},{"Start":"03:33.810 ","End":"03:34.500","Text":"Let\u0027s see."},{"Start":"03:34.500 ","End":"03:36.240","Text":"1 with k is k,"},{"Start":"03:36.240 ","End":"03:41.055","Text":"but also, k minus 1 is nothing."},{"Start":"03:41.055 ","End":"03:45.360","Text":"I just get minus 1 plus k squared,"},{"Start":"03:45.360 ","End":"03:56.145","Text":"and then 1 minus minus 1 is 2."},{"Start":"03:56.145 ","End":"04:01.776","Text":"This is 2 minus k,"},{"Start":"04:01.776 ","End":"04:07.950","Text":"and then minus k squared,"},{"Start":"04:07.950 ","End":"04:11.810","Text":"and that happens to factorize into"},{"Start":"04:11.810 ","End":"04:16.860","Text":"1 minus k, 2 plus k."},{"Start":"04:16.860 ","End":"04:18.330","Text":"Let\u0027s see if it does."},{"Start":"04:18.330 ","End":"04:27.080","Text":"1 times 2 is 2, minus 2k, plus k is minus k, and minus k squared."},{"Start":"04:27.080 ","End":"04:29.570","Text":"You know how to factorize if you had to."},{"Start":"04:29.570 ","End":"04:34.095","Text":"We could just solve the quadratic equation."},{"Start":"04:34.095 ","End":"04:38.630","Text":"You can multiply it by minus 1"},{"Start":"04:38.630 ","End":"04:42.830","Text":"and get k squared minus k plus 2 equals 0."},{"Start":"04:42.830 ","End":"04:45.120","Text":"You\u0027d get the roots."},{"Start":"04:45.500 ","End":"04:49.640","Text":"Anyway, I\u0027m not going to go into that."},{"Start":"04:49.640 ","End":"04:51.390","Text":"This factors into this,"},{"Start":"04:51.390 ","End":"04:54.380","Text":"and this is also good for us"},{"Start":"04:54.380 ","End":"04:59.720","Text":"because actually we could also make this a 0."},{"Start":"04:59.720 ","End":"05:03.750","Text":"If you multiply this row by 2 plus k,"},{"Start":"05:04.150 ","End":"05:07.805","Text":"changed my mind, we don\u0027t have to make it 0 here."},{"Start":"05:07.805 ","End":"05:14.915","Text":"I\u0027m going to already stop at this point in the echelon form."},{"Start":"05:14.915 ","End":"05:18.170","Text":"We can do all our analysis from here."},{"Start":"05:18.170 ","End":"05:21.365","Text":"But look at the restricted matrix,"},{"Start":"05:21.365 ","End":"05:26.780","Text":"we don\u0027t really know how many rows there are because as a parameter, really,"},{"Start":"05:26.780 ","End":"05:30.140","Text":"the important thing to know, at least at first,"},{"Start":"05:30.140 ","End":"05:33.880","Text":"is whether this is 0 or not,"},{"Start":"05:33.880 ","End":"05:37.550","Text":"and we\u0027re also interested in this element."},{"Start":"05:37.550 ","End":"05:41.765","Text":"Because really when all the diagonal is not 0,"},{"Start":"05:41.765 ","End":"05:44.390","Text":"then we know we\u0027re going to get a unique solution."},{"Start":"05:44.390 ","End":"05:47.950","Text":"Anyway, let\u0027s do the analysis."},{"Start":"05:47.950 ","End":"05:52.340","Text":"For exactly 1 solution to the system,"},{"Start":"05:52.340 ","End":"05:56.065","Text":"we need for this to be non-zero,"},{"Start":"05:56.065 ","End":"05:59.660","Text":"and we also need this to be non-zero,"},{"Start":"05:59.660 ","End":"06:05.450","Text":"which is in a way redundant because if 1 minus k is 0,"},{"Start":"06:05.450 ","End":"06:07.685","Text":"then this product is also 0."},{"Start":"06:07.685 ","End":"06:08.870","Text":"If this is non-zero,"},{"Start":"06:08.870 ","End":"06:11.349","Text":"then this is nonzero anyway."},{"Start":"06:11.349 ","End":"06:14.330","Text":"We get that k is not equal to 1,"},{"Start":"06:14.330 ","End":"06:18.080","Text":"and from here, k not equal to minus 2."},{"Start":"06:18.080 ","End":"06:19.940","Text":"But in Z_3,"},{"Start":"06:19.940 ","End":"06:22.370","Text":"minus 2 is the same as 1."},{"Start":"06:22.370 ","End":"06:25.720","Text":"You can really write mod 3."},{"Start":"06:25.720 ","End":"06:31.130","Text":"K not equal to 1 is the case that gives exactly 1 solution."},{"Start":"06:31.130 ","End":"06:34.700","Text":"But in a finite field like Z_3,"},{"Start":"06:34.700 ","End":"06:36.770","Text":"we can actually write the possibilities."},{"Start":"06:36.770 ","End":"06:40.620","Text":"If it\u0027s not 1, then it\u0027s 0 or 2."},{"Start":"06:40.820 ","End":"06:45.365","Text":"The only remaining case is when k is equal to 1."},{"Start":"06:45.365 ","End":"06:49.295","Text":"Let\u0027s see what happens if I plugin k equals 1 here and here."},{"Start":"06:49.295 ","End":"06:51.905","Text":"This becomes 0 and this becomes 0,"},{"Start":"06:51.905 ","End":"06:53.860","Text":"but so does this."},{"Start":"06:53.860 ","End":"06:57.980","Text":"In fact, what we get is just the top row."},{"Start":"06:57.980 ","End":"07:02.010","Text":"Both these rows have become zeros."},{"Start":"07:04.520 ","End":"07:08.045","Text":"You might be tempted to say, yes,"},{"Start":"07:08.045 ","End":"07:14.809","Text":"we have less rows than columns in the restricted matrix, and so infinite solutions."},{"Start":"07:14.809 ","End":"07:20.730","Text":"That would be true for the field of real numbers whip away in the field Z_3,"},{"Start":"07:21.080 ","End":"07:26.638","Text":"so maybe that\u0027s not infinite solutions."},{"Start":"07:26.638 ","End":"07:34.550","Text":"Things behave a little bit differently for a different field. What does happen?"},{"Start":"07:34.550 ","End":"07:40.250","Text":"Let\u0027s go back from the matrix to the system."},{"Start":"07:40.250 ","End":"07:45.410","Text":"We\u0027re left with only 1 equation in the system with 3 unknowns."},{"Start":"07:45.410 ","End":"07:50.790","Text":"Remember, the leading element is the dependent part,"},{"Start":"07:50.790 ","End":"07:52.879","Text":"and the rest are independent,"},{"Start":"07:52.879 ","End":"07:56.630","Text":"meaning they can be whatever they like, and we let them be a parameter."},{"Start":"07:56.630 ","End":"08:02.280","Text":"For y and z, I can substitute, lets say, t and s."},{"Start":"08:02.280 ","End":"08:03.240","Text":"Y is t."},{"Start":"08:03.240 ","End":"08:04.260","Text":"Z is s,"},{"Start":"08:04.260 ","End":"08:07.030","Text":"but x follows from these 2."},{"Start":"08:07.030 ","End":"08:11.960","Text":"Just bring these to the other side and you get x is 1 minus t minus s."},{"Start":"08:11.960 ","End":"08:16.220","Text":"We already have y and z."},{"Start":"08:16.220 ","End":"08:25.145","Text":"This is the general solution for the case when k equals 1."},{"Start":"08:25.145 ","End":"08:29.015","Text":"Now I\u0027d like to ask how many solutions are there?"},{"Start":"08:29.015 ","End":"08:32.075","Text":"It\u0027s not infinite in this case."},{"Start":"08:32.075 ","End":"08:34.340","Text":"Actually in the case of the reals,"},{"Start":"08:34.340 ","End":"08:36.680","Text":"it would even be infinity times infinity,"},{"Start":"08:36.680 ","End":"08:38.105","Text":"although that\u0027s still infinity."},{"Start":"08:38.105 ","End":"08:41.224","Text":"Because t could be anything and s could be anything."},{"Start":"08:41.224 ","End":"08:45.414","Text":"But here, we\u0027re on a finite field. There\u0027s only 3 elements,"},{"Start":"08:45.414 ","End":"08:47.845","Text":"the 3 possibilities for t,"},{"Start":"08:47.845 ","End":"08:50.210","Text":"3 possibilities for s."},{"Start":"08:50.210 ","End":"08:53.805","Text":"I mean each of them could be 0, 1, or 2."},{"Start":"08:53.805 ","End":"08:57.705","Text":"As a matter of fact, there are 9 combinations."},{"Start":"08:57.705 ","End":"08:59.470","Text":"If you put s and t,"},{"Start":"08:59.470 ","End":"09:01.520","Text":"each one can be anyone of the 3,"},{"Start":"09:01.520 ","End":"09:05.280","Text":"and we\u0027ll get 9 possible solutions."},{"Start":"09:05.390 ","End":"09:11.360","Text":"If I want to summarize k equals 1, I get 9 solutions."},{"Start":"09:11.360 ","End":"09:18.030","Text":"K not 1, then we get a unique solution."},{"Start":"09:18.030 ","End":"09:23.930","Text":"The case where there are no solutions just doesn\u0027t happen here in this problem."},{"Start":"09:23.930 ","End":"09:27.360","Text":"That\u0027s it."}],"ID":9851},{"Watched":false,"Name":"Exercise 6","Duration":"9m 12s","ChapterTopicVideoID":9490,"CourseChapterTopicPlaylistID":7282,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9490.jpeg","UploadDate":"2017-07-26T08:27:35.2570000","DurationForVideoObject":"PT9M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"Here we have a system of linear equations,"},{"Start":"00:03.450 ","End":"00:09.210","Text":"3 equations, and 3 unknowns with a parameter k."},{"Start":"00:09.210 ","End":"00:13.545","Text":"The field is not the real numbers."},{"Start":"00:13.545 ","End":"00:17.360","Text":"It\u0027s the finite field Z_5,"},{"Start":"00:17.360 ","End":"00:19.250","Text":"the integers modulo 5,"},{"Start":"00:19.250 ","End":"00:21.665","Text":"which contains just 5 elements."},{"Start":"00:21.665 ","End":"00:26.030","Text":"We want to sort out according to the value of k,"},{"Start":"00:26.030 ","End":"00:30.560","Text":"how many solutions the system has, the various possibilities,"},{"Start":"00:30.560 ","End":"00:36.480","Text":"no solutions, 1 solution, infinite solutions, may be something else."},{"Start":"00:36.830 ","End":"00:44.629","Text":"Let\u0027s start as usual by converting it to a matrix or to represent it by a matrix."},{"Start":"00:44.629 ","End":"00:47.450","Text":"This is what we get, the augmented matrix,"},{"Start":"00:47.450 ","End":"00:50.215","Text":"just copy the coefficients here."},{"Start":"00:50.215 ","End":"00:55.160","Text":"Make sure to put 0s where there is something missing and get them in the right order."},{"Start":"00:55.160 ","End":"00:58.440","Text":"The right-hand sides are here."},{"Start":"00:58.770 ","End":"01:02.995","Text":"As usual, we want to bring it into row echelon form."},{"Start":"01:02.995 ","End":"01:06.055","Text":"I have a 0 here already, that\u0027s lucky."},{"Start":"01:06.055 ","End":"01:10.900","Text":"I just need to subtract 3 times the top row from the bottom row."},{"Start":"01:10.900 ","End":"01:15.685","Text":"This is it in row notation, and I brought along a multiplication table."},{"Start":"01:15.685 ","End":"01:21.950","Text":"This is the multiplication for Z_5, might come in handy."},{"Start":"01:22.250 ","End":"01:24.120","Text":"Let\u0027s see."},{"Start":"01:24.120 ","End":"01:29.940","Text":"If we do the computation, this is what we get."},{"Start":"01:29.940 ","End":"01:33.430","Text":"Sorry, 3 minus 3 times 1 is 0,"},{"Start":"01:33.430 ","End":"01:36.955","Text":"minus 1 minus 3 times minus 1,"},{"Start":"01:36.955 ","End":"01:38.815","Text":"minus 1 plus 3 is 2."},{"Start":"01:38.815 ","End":"01:41.335","Text":"K plus 3 minus 3 is k."},{"Start":"01:41.335 ","End":"01:44.120","Text":"3 minus 3 times 1 is 0."},{"Start":"01:44.120 ","End":"01:51.355","Text":"This is what we have, still not in echelon form because I need to have a 0 here."},{"Start":"01:51.355 ","End":"01:54.090","Text":"Remember this is the field Z_5."},{"Start":"01:54.090 ","End":"01:57.755","Text":"If you think of addition,"},{"Start":"01:57.755 ","End":"02:03.390","Text":"notice that 3 plus 2 is 0 in this field."},{"Start":"02:03.390 ","End":"02:10.820","Text":"Actually, it\u0027s very easy to get a 0 here by just adding the second row to the third row,"},{"Start":"02:10.820 ","End":"02:13.805","Text":"which is this in row notation."},{"Start":"02:13.805 ","End":"02:17.904","Text":"If we do that, let\u0027s see what we get."},{"Start":"02:17.904 ","End":"02:20.985","Text":"3 plus 2 is 0."},{"Start":"02:20.985 ","End":"02:23.520","Text":"K squared plus 3 plus k is this,"},{"Start":"02:23.520 ","End":"02:25.830","Text":"k squared plus 1 plus 0 is this."},{"Start":"02:25.830 ","End":"02:29.170","Text":"This is the system we have now."},{"Start":"02:29.690 ","End":"02:37.440","Text":"By the way, yes, the minus 1 I changed into a 4"},{"Start":"02:37.440 ","End":"02:41.810","Text":"so we\u0027ll have everything in the standard form."},{"Start":"02:41.810 ","End":"02:47.270","Text":"0, 1, 2, 3, 4, the standard numbers we used in modulo 5,"},{"Start":"02:47.270 ","End":"02:52.170","Text":"and we\u0027re all ready to do the analysis, echelon."},{"Start":"02:52.760 ","End":"02:57.680","Text":"Diagonal containing a parameter leads to a constant."},{"Start":"02:57.680 ","End":"03:03.845","Text":"The fate of this whole thing is in the hands of this term here."},{"Start":"03:03.845 ","End":"03:06.170","Text":"This is the most important."},{"Start":"03:06.170 ","End":"03:10.595","Text":"We need to know when this thing is 0, when it\u0027s not."},{"Start":"03:10.595 ","End":"03:20.850","Text":"If it\u0027s not 0, then this restricted matrix has 3 rows, 3 columns, non-zero diagonal."},{"Start":"03:21.050 ","End":"03:24.225","Text":"We need to know when it\u0027s not 0."},{"Start":"03:24.225 ","End":"03:32.375","Text":"What we\u0027ll do is we\u0027ll find out when it is 0, and you don\u0027t have to solve quadratics here."},{"Start":"03:32.375 ","End":"03:35.375","Text":"This is not the real numbers."},{"Start":"03:35.375 ","End":"03:42.410","Text":"What you can do is just substitute values and use the trial and error."},{"Start":"03:42.410 ","End":"03:48.160","Text":"In other words, I only have 5 possibilities,"},{"Start":"03:48.160 ","End":"03:52.710","Text":"0, 1, 2, 3, 4, or all the numbers that there are and just try them."},{"Start":"03:52.710 ","End":"03:55.395","Text":"Plug in 0, no, we don\u0027t get 0."},{"Start":"03:55.395 ","End":"03:59.390","Text":"Plug in 1, 1 plus 1 plus 3 is 5, yes, 0."},{"Start":"03:59.390 ","End":"04:01.055","Text":"Plug in 2, it\u0027s not."},{"Start":"04:01.055 ","End":"04:07.590","Text":"Anyway, 3 and 1 are the solutions by trial and error."},{"Start":"04:07.850 ","End":"04:15.440","Text":"The case for exactly 1 solution is when k is not equal to 3 and not equal to 1."},{"Start":"04:15.440 ","End":"04:17.960","Text":"New page."},{"Start":"04:17.960 ","End":"04:26.720","Text":"Remember we said that when k is not equal to 1 or 3, then we have a unique solution."},{"Start":"04:26.720 ","End":"04:30.170","Text":"What remains to check is what happens when k equals 1"},{"Start":"04:30.170 ","End":"04:32.465","Text":"and what happens when k equals 3."},{"Start":"04:32.465 ","End":"04:35.150","Text":"Let\u0027s start with k equals 3."},{"Start":"04:35.150 ","End":"04:37.340","Text":"Everywhere we see k, we put 3."},{"Start":"04:37.340 ","End":"04:42.419","Text":"3 squared plus 3 is 12,"},{"Start":"04:42.419 ","End":"04:44.530","Text":"which is 2."},{"Start":"04:45.770 ","End":"04:48.870","Text":"K squared plus k plus 3"},{"Start":"04:48.870 ","End":"04:53.510","Text":"comes out to be that same 12 plus another 3 is 15."},{"Start":"04:53.510 ","End":"04:54.710","Text":"15 is 0."},{"Start":"04:54.710 ","End":"04:57.590","Text":"In short, check the calculations."},{"Start":"04:57.590 ","End":"04:59.320","Text":"When k is 3, we get this,"},{"Start":"04:59.320 ","End":"05:04.045","Text":"and the last row is a row of all zeros."},{"Start":"05:04.045 ","End":"05:12.510","Text":"If we look at the restricted, we have 3 columns and 2 rows,"},{"Start":"05:12.510 ","End":"05:15.830","Text":"so we know there\u0027s going to be more than 1 solution,"},{"Start":"05:15.830 ","End":"05:21.915","Text":"but it\u0027s not true to say infinite solutions in the case of finite fields."},{"Start":"05:21.915 ","End":"05:28.610","Text":"Like I said, you can\u0027t have infinite solutions over this field."},{"Start":"05:28.610 ","End":"05:33.230","Text":"That would be true only of the real numbers but not over."},{"Start":"05:33.230 ","End":"05:39.960","Text":"Sorry, this is Z_5."},{"Start":"05:39.960 ","End":"05:47.150","Text":"Turns out there\u0027s actually 5 solutions, and I\u0027ll show you how I reached that conclusion."},{"Start":"05:47.150 ","End":"05:50.609","Text":"I\u0027ll just briefly explain."},{"Start":"05:50.970 ","End":"05:58.600","Text":"We just take this matrix and go back to a system of linear equations,"},{"Start":"05:58.600 ","End":"06:07.200","Text":"then we see that the first non-zero entry in each row"},{"Start":"06:07.200 ","End":"06:13.030","Text":"comes a dependent variable that would be an x and y,"},{"Start":"06:13.030 ","End":"06:16.240","Text":"whereas z would be totally free,"},{"Start":"06:16.240 ","End":"06:20.340","Text":"so we let z be a parameter t."},{"Start":"06:20.340 ","End":"06:21.880","Text":"You know what, I\u0027m just going to solve it for you."},{"Start":"06:21.880 ","End":"06:29.920","Text":"Let\u0027s write it as x plus 4y plus z equals 1,"},{"Start":"06:29.920 ","End":"06:36.015","Text":"3y plus 2z equals 0."},{"Start":"06:36.015 ","End":"06:46.280","Text":"Then we see that the leading term in each, these make a dependent variable,"},{"Start":"06:46.280 ","End":"06:50.900","Text":"and z becomes a free variable,"},{"Start":"06:50.900 ","End":"06:55.085","Text":"and we call it, let\u0027s say, t."},{"Start":"06:55.085 ","End":"07:04.465","Text":"But there\u0027s only 5 possibilities because t could be 0, 1, 2, 3, or 4."},{"Start":"07:04.465 ","End":"07:12.530","Text":"Once we have that z is t, then we can solve for x and y."},{"Start":"07:12.530 ","End":"07:14.970","Text":"I\u0027ll skip that part."},{"Start":"07:15.670 ","End":"07:20.075","Text":"I mean, we have y in terms of z,"},{"Start":"07:20.075 ","End":"07:22.490","Text":"and then when we have y and z, we get x"},{"Start":"07:22.490 ","End":"07:27.210","Text":"also in terms of t actually."},{"Start":"07:27.210 ","End":"07:31.380","Text":"Then we can let t be any 1 of 4 possibilities."},{"Start":"07:31.380 ","End":"07:35.750","Text":"I\u0027ll leave the rest of it because we\u0027re not actually asked to solve it."},{"Start":"07:35.750 ","End":"07:38.425","Text":"Just how many solutions?"},{"Start":"07:38.425 ","End":"07:43.000","Text":"This gives the reasoning of why 5 solutions."},{"Start":"07:43.340 ","End":"07:46.600","Text":"Let\u0027s move on."},{"Start":"07:47.300 ","End":"07:50.660","Text":"In case you got lost as to where we are,"},{"Start":"07:50.660 ","End":"07:59.670","Text":"we had the case of k not equals 1 or 3 that gave a unique single solution,"},{"Start":"07:59.670 ","End":"08:03.080","Text":"then we tried the case k equals 3 here."},{"Start":"08:03.080 ","End":"08:09.484","Text":"What remains is the case k equals 1 and see what happens there."},{"Start":"08:09.484 ","End":"08:13.400","Text":"I just jump back so we can see what our matrix was anyway,"},{"Start":"08:13.400 ","End":"08:19.230","Text":"when k is 1, here we have 1 plus 1 plus 3 is 5, which is 0."},{"Start":"08:19.230 ","End":"08:22.425","Text":"Here, we have 1 plus 1 is 2."},{"Start":"08:22.425 ","End":"08:24.200","Text":"That\u0027s how we get this."},{"Start":"08:24.200 ","End":"08:29.235","Text":"Now look at that last row, 0s in the restricted part,"},{"Start":"08:29.235 ","End":"08:31.545","Text":"but when we look at the augmented, we have a 2."},{"Start":"08:31.545 ","End":"08:37.940","Text":"When we have all 0s on the not zero, that\u0027s what we call a contradiction row."},{"Start":"08:37.940 ","End":"08:42.150","Text":"When this happens, then there are no solutions."},{"Start":"08:42.150 ","End":"08:44.900","Text":"Really, we should make a summary table."},{"Start":"08:44.900 ","End":"08:46.040","Text":"Just do it informally."},{"Start":"08:46.040 ","End":"08:51.800","Text":"K not equals to 1 or 3, then we have 1 solution."},{"Start":"08:51.800 ","End":"08:55.980","Text":"When k equals 3, we don\u0027t have infinitely many,"},{"Start":"08:55.980 ","End":"08:58.920","Text":"we have 5 solutions."},{"Start":"08:58.920 ","End":"09:01.655","Text":"5 is the number of elements in the field."},{"Start":"09:01.655 ","End":"09:08.760","Text":"When k is 1, then we have 0, no solutions."},{"Start":"09:09.970 ","End":"09:12.990","Text":"That concludes this."}],"ID":9852}],"Thumbnail":null,"ID":7282}]

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1.1

1

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