Reminder - Coordinate Vectors and Change of Basis
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Matrix of Linear Transformation
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[{"Name":"Reminder - Coordinate Vectors and Change of Basis","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Coordinate Vectors","Duration":"9m 26s","ChapterTopicVideoID":9988,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"Starting a new topic in vector spaces,"},{"Start":"00:03.705 ","End":"00:06.765","Text":"something called coordinate vectors."},{"Start":"00:06.765 ","End":"00:09.090","Text":"I\u0027ll explain through an example,"},{"Start":"00:09.090 ","End":"00:10.890","Text":"and then also explain what I mean by"},{"Start":"00:10.890 ","End":"00:15.285","Text":"an ordered basis.. We\u0027ll consider the vector space R^3,"},{"Start":"00:15.285 ","End":"00:17.790","Text":"3-dimensional real space,"},{"Start":"00:17.790 ","End":"00:21.960","Text":"and we\u0027ll take the basis B,"},{"Start":"00:21.960 ","End":"00:26.570","Text":"and I\u0027ll just ask you to take my word for it that this is indeed a basis,"},{"Start":"00:26.570 ","End":"00:29.905","Text":"so I don\u0027t want to spend time verifying that."},{"Start":"00:29.905 ","End":"00:32.780","Text":"But all you would have to do would be to show that these"},{"Start":"00:32.780 ","End":"00:35.150","Text":"3 are linearly independent because there\u0027s 3 of them,"},{"Start":"00:35.150 ","End":"00:37.425","Text":"anyway, take my word for it,"},{"Start":"00:37.425 ","End":"00:39.270","Text":"this is a basis."},{"Start":"00:39.270 ","End":"00:43.245","Text":"Now, in this section,"},{"Start":"00:43.245 ","End":"00:45.165","Text":"makes a difference the order."},{"Start":"00:45.165 ","End":"00:46.950","Text":"The bases is in a specific order,"},{"Start":"00:46.950 ","End":"00:47.970","Text":"so this is the 1st,"},{"Start":"00:47.970 ","End":"00:50.710","Text":"this is the 2nd, this is the 3rd."},{"Start":"00:51.260 ","End":"00:55.820","Text":"It\u0027s not quite a set because in a set the order doesn\u0027t make a difference,"},{"Start":"00:55.820 ","End":"00:57.155","Text":"but here it does."},{"Start":"00:57.155 ","End":"00:59.705","Text":"That\u0027s far as audit goes."},{"Start":"00:59.705 ","End":"01:05.195","Text":"Now, because it\u0027s a basis of R^3, of course,"},{"Start":"01:05.195 ","End":"01:12.970","Text":"then any vector in R^3 can be written as a linear combination of these 3."},{"Start":"01:12.970 ","End":"01:15.365","Text":"In only 1 way,"},{"Start":"01:15.365 ","End":"01:16.520","Text":"it\u0027ll be something times this,"},{"Start":"01:16.520 ","End":"01:17.570","Text":"plus something times this,"},{"Start":"01:17.570 ","End":"01:19.670","Text":"plus something times this."},{"Start":"01:19.670 ","End":"01:22.640","Text":"I\u0027ll take just some random example."},{"Start":"01:22.640 ","End":"01:24.320","Text":"Let\u0027s take the vector 2,"},{"Start":"01:24.320 ","End":"01:27.005","Text":"8, 12 in R^3."},{"Start":"01:27.005 ","End":"01:35.580","Text":"You could check that this is equal to minus 2 times the 1st element of the basis,"},{"Start":"01:35.580 ","End":"01:37.740","Text":"and then plus 4 times the 2nd,"},{"Start":"01:37.740 ","End":"01:40.075","Text":"plus 10 times the 3rd."},{"Start":"01:40.075 ","End":"01:42.740","Text":"How I got to this minus 2, 4,"},{"Start":"01:42.740 ","End":"01:45.320","Text":"10 is not important at this moment."},{"Start":"01:45.320 ","End":"01:47.600","Text":"In fact, I might have even cooked it up backwards."},{"Start":"01:47.600 ","End":"01:50.915","Text":"I might have started with this and given you this."},{"Start":"01:50.915 ","End":"01:55.770","Text":"Just verify that this is so with computation."},{"Start":"01:57.080 ","End":"02:00.235","Text":"Now that we have these numbers,"},{"Start":"02:00.235 ","End":"02:05.125","Text":"these coefficients of the basis,"},{"Start":"02:05.125 ","End":"02:11.290","Text":"we say that the coordinate vector of the vector u,"},{"Start":"02:11.290 ","End":"02:13.210","Text":"which is 2, 8, 12,"},{"Start":"02:13.210 ","End":"02:16.000","Text":"with respect to the basis B,"},{"Start":"02:16.000 ","End":"02:18.335","Text":"it\u0027s going to be an ordered basis,"},{"Start":"02:18.335 ","End":"02:25.865","Text":"we write it as u in square brackets with the name of the basis here,"},{"Start":"02:25.865 ","End":"02:30.810","Text":"and these 3 components are these 3 scalars from here,"},{"Start":"02:30.810 ","End":"02:34.320","Text":"and that\u0027s what gives us the coordinate vector."},{"Start":"02:34.320 ","End":"02:41.019","Text":"Remember the basis, you have to keep the order of the 3 basis elements."},{"Start":"02:41.019 ","End":"02:46.185","Text":"If we didn\u0027t give the name u,"},{"Start":"02:46.185 ","End":"02:49.670","Text":"we could just write it like this to put the vector itself"},{"Start":"02:49.670 ","End":"02:52.970","Text":"in the square brackets to indicate the name of the basis,"},{"Start":"02:52.970 ","End":"02:55.745","Text":"and this is the coordinate vector."},{"Start":"02:55.745 ","End":"02:59.275","Text":"Here\u0027s the 2nd example."},{"Start":"02:59.275 ","End":"03:04.080","Text":"Another random vector, 7, 2, 7."},{"Start":"03:04.080 ","End":"03:07.245","Text":"Check that this holds."},{"Start":"03:07.245 ","End":"03:10.050","Text":"It doesn\u0027t matter where I got the 2,"},{"Start":"03:10.050 ","End":"03:11.980","Text":"5, 0 from."},{"Start":"03:11.980 ","End":"03:14.180","Text":"Later, we\u0027ll learn how to find them,"},{"Start":"03:14.180 ","End":"03:15.920","Text":"but I might have just cooked it up by"},{"Start":"03:15.920 ","End":"03:19.115","Text":"starting from the right-hand side and getting to the left."},{"Start":"03:19.115 ","End":"03:21.365","Text":"Anyway, just verify this,"},{"Start":"03:21.365 ","End":"03:24.125","Text":"and when you\u0027ve checked that,"},{"Start":"03:24.125 ","End":"03:33.290","Text":"then we\u0027ll be able to say that the coordinate vector of this vector 7,"},{"Start":"03:33.290 ","End":"03:39.750","Text":"2, 7 v is the vector 2, 5, 0,"},{"Start":"03:39.750 ","End":"03:41.850","Text":"which I get from these 3 here."},{"Start":"03:41.850 ","End":"03:48.845","Text":"Again, we put it in square brackets with the name of the audit basis here,"},{"Start":"03:48.845 ","End":"03:53.070","Text":"and if we haven\u0027t given this a name v,"},{"Start":"03:53.070 ","End":"03:57.660","Text":"we can just write like so."},{"Start":"03:57.660 ","End":"04:02.285","Text":"Now, it\u0027s time to discuss how we can actually compute"},{"Start":"04:02.285 ","End":"04:09.230","Text":"a coordinate vector of a given vector relative to a given basis."},{"Start":"04:09.230 ","End":"04:12.560","Text":"As usual, we do it with examples."},{"Start":"04:12.560 ","End":"04:19.505","Text":"I\u0027ll use the example we had earlier for this ordered basis of R^3."},{"Start":"04:19.505 ","End":"04:25.430","Text":"I\u0027ll go back and check and see if this was the example we had."},{"Start":"04:25.430 ","End":"04:28.040","Text":"We actually had 2 examples."},{"Start":"04:28.040 ","End":"04:30.635","Text":"Remember, we had u, which was this,"},{"Start":"04:30.635 ","End":"04:33.890","Text":"and we had v, which was this, and in each case,"},{"Start":"04:33.890 ","End":"04:36.860","Text":"I pulled these numbers out of a hat,"},{"Start":"04:36.860 ","End":"04:38.410","Text":"so to speak, the minus 2,"},{"Start":"04:38.410 ","End":"04:40.900","Text":"4, 10 I just asked you to verify."},{"Start":"04:40.900 ","End":"04:42.290","Text":"Once we verified,"},{"Start":"04:42.290 ","End":"04:44.135","Text":"then we said, okay,"},{"Start":"04:44.135 ","End":"04:53.750","Text":"the coordinate vector of u with respect to the basis B is this from these numbers,"},{"Start":"04:53.750 ","End":"04:59.290","Text":"and we said that the coordinate vector of this"},{"Start":"04:59.290 ","End":"05:06.275","Text":"v with respect to this same ordered basis was 2, 5, 0."},{"Start":"05:06.275 ","End":"05:07.775","Text":"All these we had,"},{"Start":"05:07.775 ","End":"05:09.110","Text":"but the question is,"},{"Start":"05:09.110 ","End":"05:14.660","Text":"how would we compute these numbers if I didn\u0027t just produce them?"},{"Start":"05:14.660 ","End":"05:18.320","Text":"The idea is instead of taking say,"},{"Start":"05:18.320 ","End":"05:21.125","Text":"2, 8, 12, or 7, 2, 7,"},{"Start":"05:21.125 ","End":"05:27.855","Text":"is to take a general vector like x, y, z."},{"Start":"05:27.855 ","End":"05:32.210","Text":"If I do the computation for a general x, y, z,"},{"Start":"05:32.210 ","End":"05:36.410","Text":"then I can afterwards plug in whatever I want,"},{"Start":"05:36.410 ","End":"05:38.000","Text":"2, 8, 12, 7,"},{"Start":"05:38.000 ","End":"05:40.355","Text":"2, 7, or anything else."},{"Start":"05:40.355 ","End":"05:42.420","Text":"Here is our vector x, y,"},{"Start":"05:42.420 ","End":"05:44.595","Text":"z, give it a name w,"},{"Start":"05:44.595 ","End":"05:50.165","Text":"you want to find it as the linear combination of the basis vectors,"},{"Start":"05:50.165 ","End":"05:52.850","Text":"and these scalars a, b,"},{"Start":"05:52.850 ","End":"05:55.880","Text":"c are the unknowns."},{"Start":"05:55.880 ","End":"05:58.160","Text":"In other words, we want to find a, b,"},{"Start":"05:58.160 ","End":"06:00.170","Text":"and c in terms of x,"},{"Start":"06:00.170 ","End":"06:01.310","Text":"y, z, as if x,"},{"Start":"06:01.310 ","End":"06:03.490","Text":"y, z were known."},{"Start":"06:03.490 ","End":"06:06.100","Text":"I multiply the scalars,"},{"Start":"06:06.100 ","End":"06:07.910","Text":"and then I do the addition,"},{"Start":"06:07.910 ","End":"06:10.135","Text":"then we get this,"},{"Start":"06:10.135 ","End":"06:17.670","Text":"and this gives us a system of 3 linear equations in 3 unknowns,"},{"Start":"06:17.670 ","End":"06:21.985","Text":"a, b, c. We\u0027ll do it using matrices."},{"Start":"06:21.985 ","End":"06:27.080","Text":"This is the corresponding augmented matrix."},{"Start":"06:27.080 ","End":"06:31.850","Text":"What we\u0027re going to do is not just bringing it into row echelon form,"},{"Start":"06:31.850 ","End":"06:36.200","Text":"let\u0027s make this part the identity matrix,"},{"Start":"06:36.200 ","End":"06:38.780","Text":"and then it will be easier,"},{"Start":"06:38.780 ","End":"06:40.760","Text":"at least that\u0027s 1 way of doing it."},{"Start":"06:40.760 ","End":"06:47.750","Text":"I\u0027ll start out by subtracting this top row from the other 2."},{"Start":"06:47.750 ","End":"06:49.790","Text":"In row notation,"},{"Start":"06:49.790 ","End":"06:51.380","Text":"this is what I mean,"},{"Start":"06:51.380 ","End":"06:55.430","Text":"and if we do that, we get this."},{"Start":"06:55.430 ","End":"06:59.435","Text":"This is already in row echelon form, but we\u0027re continuing."},{"Start":"06:59.435 ","End":"07:03.755","Text":"Let\u0027s subtract this row from this row,"},{"Start":"07:03.755 ","End":"07:05.570","Text":"this is the notation,"},{"Start":"07:05.570 ","End":"07:09.140","Text":"and then we get this."},{"Start":"07:09.140 ","End":"07:14.340","Text":"The next thing is to add this row to this row,"},{"Start":"07:14.340 ","End":"07:16.575","Text":"and that will make this 0,"},{"Start":"07:16.575 ","End":"07:19.090","Text":"and then finally we get this,"},{"Start":"07:19.090 ","End":"07:25.405","Text":"and all we have to do is multiply the middle row by minus 1."},{"Start":"07:25.405 ","End":"07:28.805","Text":"We have the identity matrix here,"},{"Start":"07:28.805 ","End":"07:34.130","Text":"which means that we have exactly what a,"},{"Start":"07:34.130 ","End":"07:36.290","Text":"b, and c are in terms of x,"},{"Start":"07:36.290 ","End":"07:40.985","Text":"y, z, and if we go back to where we came from,"},{"Start":"07:40.985 ","End":"07:45.170","Text":"we had that W was a times this vector,"},{"Start":"07:45.170 ","End":"07:46.880","Text":"plus b times this,"},{"Start":"07:46.880 ","End":"07:48.155","Text":"plus c times this,"},{"Start":"07:48.155 ","End":"07:50.540","Text":"and I just replaced a, b,"},{"Start":"07:50.540 ","End":"07:54.300","Text":"and c with what they are from here."},{"Start":"07:54.350 ","End":"07:56.900","Text":"We have this formula,"},{"Start":"07:56.900 ","End":"07:58.459","Text":"and now we can convert"},{"Start":"07:58.459 ","End":"08:08.090","Text":"any vector into its coordinate vector."},{"Start":"08:08.090 ","End":"08:11.810","Text":"For instance, let\u0027s take 1, 2, 3."},{"Start":"08:11.810 ","End":"08:14.410","Text":"I just follow this recipe."},{"Start":"08:14.410 ","End":"08:17.070","Text":"These 3 basis vectors stay,"},{"Start":"08:17.070 ","End":"08:18.945","Text":"and all I\u0027ll do is these computation,"},{"Start":"08:18.945 ","End":"08:21.135","Text":"x plus y minus z,"},{"Start":"08:21.135 ","End":"08:25.330","Text":"1 plus 2 minus 3, and so on."},{"Start":"08:26.090 ","End":"08:28.410","Text":"Here we get 0,"},{"Start":"08:28.410 ","End":"08:30.545","Text":"here 1, and here 2,"},{"Start":"08:30.545 ","End":"08:35.260","Text":"which means that the coordinate vector of this 1, 2,"},{"Start":"08:35.260 ","End":"08:38.920","Text":"3 with respect to that basis is 0,"},{"Start":"08:38.920 ","End":"08:41.010","Text":"1, 2, these are the 0,"},{"Start":"08:41.010 ","End":"08:42.885","Text":"1, 2 from here."},{"Start":"08:42.885 ","End":"08:45.540","Text":"1 more example, take 4,"},{"Start":"08:45.540 ","End":"08:47.490","Text":"5, 6 this time,"},{"Start":"08:47.490 ","End":"08:50.280","Text":"same setup, we put in 4,"},{"Start":"08:50.280 ","End":"08:53.160","Text":"5, 6 for x, y, z,"},{"Start":"08:53.160 ","End":"08:58.680","Text":"compute these 3 scalar coefficients,"},{"Start":"08:58.680 ","End":"09:01.390","Text":"make these into a vector,"},{"Start":"09:01.390 ","End":"09:03.730","Text":"and that\u0027s the coordinate vector of 4,"},{"Start":"09:03.730 ","End":"09:06.475","Text":"5, 6 with respect to the basis B."},{"Start":"09:06.475 ","End":"09:10.450","Text":"In general, we can say that our w,"},{"Start":"09:10.450 ","End":"09:13.460","Text":"which is the arbitrary vector x, y,"},{"Start":"09:13.460 ","End":"09:18.425","Text":"z, has this expression here as"},{"Start":"09:18.425 ","End":"09:24.380","Text":"the coordinate vector with respect to the basis."},{"Start":"09:24.380 ","End":"09:27.030","Text":"That\u0027s all for this clip."}],"Thumbnail":null,"ID":10244},{"Watched":false,"Name":"Change of Basis Matrix Part 1","Duration":"3m 47s","ChapterTopicVideoID":9989,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"Now, we\u0027re beginning a new topic in vector spaces,"},{"Start":"00:03.945 ","End":"00:07.075","Text":"the concept of change of basis matrix."},{"Start":"00:07.075 ","End":"00:11.880","Text":"It\u0027s going to be in 5 parts and it\u0027s directly"},{"Start":"00:11.880 ","End":"00:17.820","Text":"related to the concept of coordinate vectors so I suggest you review this."},{"Start":"00:17.820 ","End":"00:25.095","Text":"What I\u0027m going to do is explain it through a series of examples, solved exercises."},{"Start":"00:25.095 ","End":"00:27.005","Text":"Just to keep things simple,"},{"Start":"00:27.005 ","End":"00:32.525","Text":"I\u0027m going to have all the examples from the vector space R^3,"},{"Start":"00:32.525 ","End":"00:35.510","Text":"although you can see how it will be clearly"},{"Start":"00:35.510 ","End":"00:40.010","Text":"generalizable to any dimension or any vector space."},{"Start":"00:40.010 ","End":"00:46.580","Text":"Let\u0027s begin. Here\u0027s the first exercise that we\u0027re going to solve."},{"Start":"00:46.580 ","End":"00:52.445","Text":"We\u0027re given 2 basis, B_1 and B_2."},{"Start":"00:52.445 ","End":"00:59.330","Text":"I expect you to just take it on trust that these are bases or verify it yourself."},{"Start":"00:59.330 ","End":"01:01.820","Text":"I don\u0027t want to waste time doing this."},{"Start":"01:01.820 ","End":"01:06.095","Text":"In fact, B_2 will not be used in this particular exercise,"},{"Start":"01:06.095 ","End":"01:11.465","Text":"but this variants, this will appear in the following clips so it\u0027s included."},{"Start":"01:11.465 ","End":"01:13.400","Text":"Let\u0027s take a general vector, x,"},{"Start":"01:13.400 ","End":"01:16.250","Text":"y, z in R^3."},{"Start":"01:16.250 ","End":"01:19.730","Text":"The first part of the exercise is to compute"},{"Start":"01:19.730 ","End":"01:25.980","Text":"the coordinate vector of v relative to the basis B_1."},{"Start":"01:26.030 ","End":"01:30.410","Text":"Essentially, we\u0027re just repeating some of the material from"},{"Start":"01:30.410 ","End":"01:35.255","Text":"coordinate vectors but that\u0027s okay."},{"Start":"01:35.255 ","End":"01:37.460","Text":"B_1 is this."},{"Start":"01:37.460 ","End":"01:43.700","Text":"We create this augmented matrix by taking the rows or the vectors here"},{"Start":"01:43.700 ","End":"01:50.265","Text":"as columns of this augmented matrix and here we put x, y, z."},{"Start":"01:50.265 ","End":"01:58.265","Text":"Now, we go through a series of row operations until we get the identity matrix here."},{"Start":"01:58.265 ","End":"02:00.260","Text":"I\u0027ll go through this quickly,"},{"Start":"02:00.260 ","End":"02:08.215","Text":"subtract the top row from the second row to get a 0 here and here we get y minus x."},{"Start":"02:08.215 ","End":"02:14.520","Text":"Then we subtract the third row from the second row to get a 0 here."},{"Start":"02:14.740 ","End":"02:19.625","Text":"What we get is we take this x here,"},{"Start":"02:19.625 ","End":"02:24.450","Text":"and this comes from this column here,"},{"Start":"02:24.560 ","End":"02:33.430","Text":"this we put here and this is this column here."},{"Start":"02:33.670 ","End":"02:41.795","Text":"Then what\u0027s here goes here and it\u0027s the third element of the basis,"},{"Start":"02:41.795 ","End":"02:44.225","Text":"really, that\u0027s what these 3 are."},{"Start":"02:44.225 ","End":"02:52.290","Text":"That\u0027s the long-hand form of the coordinate vector."},{"Start":"02:52.290 ","End":"02:59.060","Text":"You could just write it for short that the coordinate vector of"},{"Start":"02:59.060 ","End":"03:08.940","Text":"v with respect to the basis B_1 is x,"},{"Start":"03:08.940 ","End":"03:13.485","Text":"y minus x minus z and z."},{"Start":"03:13.485 ","End":"03:18.635","Text":"To give you an example of how we use this if we have,"},{"Start":"03:18.635 ","End":"03:20.150","Text":"let\u0027s say 4, 5,"},{"Start":"03:20.150 ","End":"03:24.785","Text":"6 and we want to write it in terms of the basis B_1."},{"Start":"03:24.785 ","End":"03:27.700","Text":"We compute these 3,"},{"Start":"03:27.700 ","End":"03:30.480","Text":"these would be my x, y, z,"},{"Start":"03:30.480 ","End":"03:38.600","Text":"x is 4, y minus x minus z is minus 5 and z is 6."},{"Start":"03:38.600 ","End":"03:44.690","Text":"This is how we write this vector in terms of the 3 basis vectors."},{"Start":"03:44.690 ","End":"03:47.550","Text":"We\u0027re done for Part 1."}],"Thumbnail":null,"ID":10245},{"Watched":false,"Name":"Change of Basis Matrix Part 2","Duration":"3m 29s","ChapterTopicVideoID":9990,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.655","Text":"We\u0027re continuing with the concept of change of basis matrix,"},{"Start":"00:05.655 ","End":"00:11.475","Text":"again, with an exercise which is in fact a continuation of the previous exercise."},{"Start":"00:11.475 ","End":"00:15.570","Text":"It\u0027s almost identical to the previous one."},{"Start":"00:15.570 ","End":"00:22.230","Text":"In both cases, we have B1 and B2 are bases of 3."},{"Start":"00:22.230 ","End":"00:30.420","Text":"But this time we have to compute the general vector"},{"Start":"00:30.420 ","End":"00:35.715","Text":"as a coordinate vector relative to B2 this time,"},{"Start":"00:35.715 ","End":"00:39.075","Text":"we had B1 in the previous exercise."},{"Start":"00:39.075 ","End":"00:42.880","Text":"Here it\u0027s B2, that\u0027s the only difference."},{"Start":"00:42.980 ","End":"00:45.930","Text":"The solution, it\u0027s pretty much routine"},{"Start":"00:45.930 ","End":"00:48.380","Text":"because we\u0027re just repeating what we did before,"},{"Start":"00:48.380 ","End":"00:53.040","Text":"only this time with B2 instead of B1,"},{"Start":"00:53.040 ","End":"01:01.415","Text":"where we take the vectors in the basis and put them as columns in this augmented matrix."},{"Start":"01:01.415 ","End":"01:04.735","Text":"Here it\u0027s always the x, y, z."},{"Start":"01:04.735 ","End":"01:12.830","Text":"The idea is to get this matrix, the restricted part,"},{"Start":"01:12.830 ","End":"01:17.225","Text":"to be the identity matrix through a series of row operations."},{"Start":"01:17.225 ","End":"01:25.230","Text":"The first row operation will be to subtract the 2nd row from the 3rd row"},{"Start":"01:25.230 ","End":"01:27.515","Text":"and that will give us the 0 here."},{"Start":"01:27.515 ","End":"01:29.720","Text":"Here we get z minus x."},{"Start":"01:29.720 ","End":"01:39.305","Text":"Next we\u0027re going to subtract the 2nd row from the 3rd row to get 0 here."},{"Start":"01:39.305 ","End":"01:41.990","Text":"After we\u0027ve done that, this is what we get."},{"Start":"01:41.990 ","End":"01:45.785","Text":"Notice we already have the identity matrix here,"},{"Start":"01:45.785 ","End":"01:48.325","Text":"which is what we want."},{"Start":"01:48.325 ","End":"01:55.100","Text":"This, in fact, tells us how to write x, y, z in terms of the basis B2"},{"Start":"01:55.100 ","End":"01:58.025","Text":"and let me just go back a little bit."},{"Start":"01:58.025 ","End":"02:05.910","Text":"This is the first element of the basis, that\u0027s this."},{"Start":"02:05.910 ","End":"02:11.300","Text":"We take the entry here and put it here."},{"Start":"02:11.300 ","End":"02:14.570","Text":"Then we have this entry which goes here"},{"Start":"02:14.570 ","End":"02:19.820","Text":"and this is the second element of B1 as you can see from here"},{"Start":"02:19.820 ","End":"02:24.750","Text":"and this is what we put here,"},{"Start":"02:30.650 ","End":"02:35.505","Text":"the 3rd member of the basis."},{"Start":"02:35.505 ","End":"02:37.310","Text":"That\u0027s in long hand."},{"Start":"02:37.310 ","End":"02:42.230","Text":"We can also write this in shorthand form like so,"},{"Start":"02:42.230 ","End":"02:48.630","Text":"where we just abbreviate and we just say the coordinate vector of v,"},{"Start":"02:48.630 ","End":"02:51.130","Text":"with respect to the basis B2,"},{"Start":"02:51.130 ","End":"02:52.910","Text":"consists of these 3,"},{"Start":"02:52.910 ","End":"02:56.870","Text":"and that\u0027s the 3 I got from here or, if you like, from here."},{"Start":"02:56.870 ","End":"03:03.360","Text":"As an example, I\u0027ll take the same 4, 5, 6 I used in the previous."},{"Start":"03:03.360 ","End":"03:08.135","Text":"We can write this in terms of the basis B2."},{"Start":"03:08.135 ","End":"03:13.910","Text":"The coefficients are just computed from here or from here."},{"Start":"03:13.910 ","End":"03:18.960","Text":"X would be 4, y is 5,"},{"Start":"03:18.960 ","End":"03:23.790","Text":"and z minus x minus y is 6 minus 4 minus 5,"},{"Start":"03:23.790 ","End":"03:26.235","Text":"which is the minus 3."},{"Start":"03:26.235 ","End":"03:29.860","Text":"This completes part 2."}],"Thumbnail":null,"ID":10246},{"Watched":false,"Name":"Change of Basis Matrix Part 3","Duration":"3m 52s","ChapterTopicVideoID":9991,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"Continuing with the concept of change of basis matrix through a series of exercises."},{"Start":"00:06.450 ","End":"00:10.290","Text":"It\u0027s all 1 evolving exercise."},{"Start":"00:10.290 ","End":"00:13.470","Text":"We have the same 2 bases,"},{"Start":"00:13.470 ","End":"00:17.190","Text":"B_1 and B_2 as in the previous exercises."},{"Start":"00:17.190 ","End":"00:23.055","Text":"This time we\u0027re finally coming to the concept of change of basis matrix,"},{"Start":"00:23.055 ","End":"00:26.775","Text":"which I\u0027ll define through the example."},{"Start":"00:26.775 ","End":"00:30.390","Text":"The solution is broken up into 3 steps."},{"Start":"00:30.390 ","End":"00:38.070","Text":"In step 1, we compute the coordinate vector relative to the old basis."},{"Start":"00:38.070 ","End":"00:42.420","Text":"I guess I should say, when we say from B_1 to B_2,"},{"Start":"00:42.420 ","End":"00:48.144","Text":"this 1 is the old and this 1 is the new."},{"Start":"00:48.144 ","End":"00:52.940","Text":"But we don\u0027t have to do this computation because we\u0027ve already done it"},{"Start":"00:52.940 ","End":"00:58.925","Text":"in the previous part 1, I believe."},{"Start":"00:58.925 ","End":"01:01.610","Text":"We have the solution,"},{"Start":"01:01.610 ","End":"01:04.160","Text":"I just copied it."},{"Start":"01:04.160 ","End":"01:07.250","Text":"The next step is to write each vector in"},{"Start":"01:07.250 ","End":"01:10.835","Text":"the new basis as the linear combination of the old basis."},{"Start":"01:10.835 ","End":"01:15.620","Text":"What this means is that we\u0027re going to apply this formula here to each"},{"Start":"01:15.620 ","End":"01:21.660","Text":"of the 3 vectors in the new basis."},{"Start":"01:21.660 ","End":"01:29.690","Text":"Let me just emphasize again what\u0027s old and what is new that we have."},{"Start":"01:29.690 ","End":"01:31.650","Text":"The old is B_1,"},{"Start":"01:31.650 ","End":"01:41.570","Text":"the new is B_2 and we want to write each of these in terms of the old from this formula."},{"Start":"01:41.570 ","End":"01:44.505","Text":"Take this and put it in a box,"},{"Start":"01:44.505 ","End":"01:46.560","Text":"we have 3 computations."},{"Start":"01:46.560 ","End":"01:50.340","Text":"The first vector, plugging in x,"},{"Start":"01:50.340 ","End":"01:53.745","Text":"y, z is 1, 0, 1 in here."},{"Start":"01:53.745 ","End":"01:56.815","Text":"This is really the only the computation,"},{"Start":"01:56.815 ","End":"02:01.120","Text":"y minus x minus z comes out 0 minus 1 minus 1,"},{"Start":"02:01.120 ","End":"02:04.570","Text":"which is minus 2 and this is what we get."},{"Start":"02:04.570 ","End":"02:07.655","Text":"Then the next 1."},{"Start":"02:07.655 ","End":"02:13.100","Text":"I\u0027ll leave you to check that these numbers are what we get."},{"Start":"02:13.100 ","End":"02:17.450","Text":"The third 1 here is plugging in x,"},{"Start":"02:17.450 ","End":"02:19.100","Text":"y, z is 0, 0,"},{"Start":"02:19.100 ","End":"02:22.225","Text":"1, and this is what we get."},{"Start":"02:22.225 ","End":"02:29.260","Text":"That\u0027s step 2. Now, the last step, step 3,"},{"Start":"02:29.260 ","End":"02:32.330","Text":"is to take these coefficients,"},{"Start":"02:32.330 ","End":"02:35.090","Text":"the ones that are in color,"},{"Start":"02:35.090 ","End":"02:38.150","Text":"and put them into a matrix."},{"Start":"02:38.150 ","End":"02:45.529","Text":"But the transpose of that matrix, the transpose,"},{"Start":"02:45.529 ","End":"02:49.460","Text":"which means that the rows become columns like this 1 minus 2,"},{"Start":"02:49.460 ","End":"02:52.490","Text":"1 becomes the first column,"},{"Start":"02:52.490 ","End":"02:54.590","Text":"and this 0, 0,"},{"Start":"02:54.590 ","End":"02:57.775","Text":"1 becomes the second column,"},{"Start":"02:57.775 ","End":"03:00.180","Text":"and 0 minus 1,"},{"Start":"03:00.180 ","End":"03:02.940","Text":"1 becomes the third column."},{"Start":"03:02.940 ","End":"03:11.160","Text":"The notation is we write M matrix,"},{"Start":"03:11.160 ","End":"03:14.085","Text":"the new 1 on the top,"},{"Start":"03:14.085 ","End":"03:15.495","Text":"which is B_2,"},{"Start":"03:15.495 ","End":"03:16.740","Text":"and the old 1,"},{"Start":"03:16.740 ","End":"03:20.050","Text":"the old basis at the bottom here."},{"Start":"03:20.350 ","End":"03:25.159","Text":"I just have to point out and that\u0027s the asterisk"},{"Start":"03:25.159 ","End":"03:29.390","Text":"here but this is not universally agreed on."},{"Start":"03:29.390 ","End":"03:37.790","Text":"Some people write the inverse of this matrix as the change of basis matrix."},{"Start":"03:37.790 ","End":"03:41.735","Text":"If you see that, don\u0027t be alarmed."},{"Start":"03:41.735 ","End":"03:45.305","Text":"Many things in mathematics are not universally agreed on."},{"Start":"03:45.305 ","End":"03:47.375","Text":"Some define things other ways."},{"Start":"03:47.375 ","End":"03:52.680","Text":"But other than that, we\u0027re done with this third clip."}],"Thumbnail":null,"ID":10247},{"Watched":false,"Name":"Change of Basis Matrix Part 4","Duration":"2m 45s","ChapterTopicVideoID":9992,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.075","Text":"Continuing with change of basis matrix and this ongoing exercise."},{"Start":"00:06.075 ","End":"00:10.665","Text":"This time we have the reverse of the previous exercise."},{"Start":"00:10.665 ","End":"00:16.785","Text":"There we had to compute the change of basis matrix from B_1 to B_2."},{"Start":"00:16.785 ","End":"00:21.270","Text":"This time we\u0027re going to do it from B_2 to B_1."},{"Start":"00:21.270 ","End":"00:25.275","Text":"It\u0027s going to be the same 3 steps."},{"Start":"00:25.275 ","End":"00:27.300","Text":"The first step is to compute"},{"Start":"00:27.300 ","End":"00:36.764","Text":"the coordinate vector relative to the old basis at this time."},{"Start":"00:36.764 ","End":"00:45.600","Text":"This 1 is going to be considered the old and this 1 is the new."},{"Start":"00:45.600 ","End":"00:47.660","Text":"We already did that."},{"Start":"00:47.660 ","End":"00:51.460","Text":"I believe it was in step 2."},{"Start":"00:51.460 ","End":"00:55.640","Text":"I meant to say Part 2 of the 5-part series."},{"Start":"00:55.640 ","End":"00:57.289","Text":"Yeah, that\u0027s what we computed."},{"Start":"00:57.289 ","End":"01:00.650","Text":"Step 2 is still to write each vector"},{"Start":"01:00.650 ","End":"01:03.890","Text":"in the new basis as a linear combination of the old basis,"},{"Start":"01:03.890 ","End":"01:07.130","Text":"except that the roles of old and new have switched."},{"Start":"01:07.130 ","End":"01:10.250","Text":"Old is B_2 and new is B_1."},{"Start":"01:10.250 ","End":"01:19.955","Text":"What it means is that we take this formula and I put it in a nice box."},{"Start":"01:19.955 ","End":"01:29.230","Text":"We have to apply it 3 times at once to each of the members of the new basis,"},{"Start":"01:29.230 ","End":"01:34.195","Text":"beginning with 1, 1, 0."},{"Start":"01:34.195 ","End":"01:36.560","Text":"We just plugin,"},{"Start":"01:36.560 ","End":"01:38.130","Text":"these are x, y, and z."},{"Start":"01:38.130 ","End":"01:39.485","Text":"We plugin for x, y,"},{"Start":"01:39.485 ","End":"01:41.320","Text":"and z minus x minus y."},{"Start":"01:41.320 ","End":"01:42.849","Text":"This 1, for example,"},{"Start":"01:42.849 ","End":"01:49.890","Text":"would be 0 minus 1 minus 1,"},{"Start":"01:49.890 ","End":"01:52.185","Text":"which gives us minus 2."},{"Start":"01:52.185 ","End":"01:55.079","Text":"The next 1 is this."},{"Start":"01:55.079 ","End":"01:58.755","Text":"I\u0027ll leave you to verify the computation,"},{"Start":"01:58.755 ","End":"02:02.310","Text":"and similarly with the third 1."},{"Start":"02:02.310 ","End":"02:06.755","Text":"We\u0027re going to do what we did before with the coefficients,"},{"Start":"02:06.755 ","End":"02:10.475","Text":"the 1s that I\u0027ve colored in blue."},{"Start":"02:10.475 ","End":"02:16.080","Text":"We just make a matrix from these and then transpose it."},{"Start":"02:16.080 ","End":"02:18.400","Text":"So that for example,"},{"Start":"02:18.400 ","End":"02:23.135","Text":"these 3 coefficients become the first column,"},{"Start":"02:23.135 ","End":"02:29.600","Text":"and these 3 coefficients become the second column."},{"Start":"02:29.600 ","End":"02:33.455","Text":"This, this, and this goes in the third column,"},{"Start":"02:33.455 ","End":"02:36.470","Text":"rows become columns that is transpose."},{"Start":"02:36.470 ","End":"02:45.119","Text":"This concludes Step 3 and Part 2 of the series of clips."}],"Thumbnail":null,"ID":10248},{"Watched":false,"Name":"Change of Basis Matrix Part 5","Duration":"4m 15s","ChapterTopicVideoID":9993,"CourseChapterTopicPlaylistID":7318,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:04.605","Text":"This is the last in a 5-part series."},{"Start":"00:04.605 ","End":"00:10.425","Text":"This exercise puts together all the concepts in the previous parts."},{"Start":"00:10.425 ","End":"00:17.130","Text":"Once again, we have B_1 and B_2 the same as in the previous parts and a general vector v,"},{"Start":"00:17.130 ","End":"00:19.560","Text":"which is x, y, z."},{"Start":"00:19.560 ","End":"00:23.835","Text":"Here we have to show 3 things."},{"Start":"00:23.835 ","End":"00:29.515","Text":"The first part shows us that if we take the right change of basis matrix,"},{"Start":"00:29.515 ","End":"00:33.350","Text":"we can convert the coordinate vector relative to"},{"Start":"00:33.350 ","End":"00:38.020","Text":"one base into the coordinate vector relative to the other base."},{"Start":"00:38.020 ","End":"00:40.355","Text":"It\u0027s a little bit confusing."},{"Start":"00:40.355 ","End":"00:43.985","Text":"We call this the change of basis from B_2 to"},{"Start":"00:43.985 ","End":"00:49.685","Text":"B_1 even though really it should be the other way,"},{"Start":"00:49.685 ","End":"00:56.915","Text":"because we multiply by the B_1 coordinate vector and get the B_2 coordinate vector."},{"Start":"00:56.915 ","End":"01:00.150","Text":"We just write it in symbols."},{"Start":"01:01.010 ","End":"01:05.265","Text":"It\u0027s not even standard, the names."},{"Start":"01:05.265 ","End":"01:11.055","Text":"Like this is from B_1 to B_2 and B_2 to B_1 although we called it B_2 to B_1."},{"Start":"01:11.055 ","End":"01:15.390","Text":"Anyway, let\u0027s just stick to the way it was written."},{"Start":"01:15.430 ","End":"01:18.335","Text":"The third part, either way you look at it,"},{"Start":"01:18.335 ","End":"01:21.395","Text":"it says that 1 is the inverse of the other."},{"Start":"01:21.395 ","End":"01:28.250","Text":"Let\u0027s start with Part 1 or better still,"},{"Start":"01:28.250 ","End":"01:37.025","Text":"let\u0027s just review the values of all these matrices and vectors."},{"Start":"01:37.025 ","End":"01:42.990","Text":"Here they all are just copied from the previous parts."},{"Start":"01:43.820 ","End":"01:49.740","Text":"Number 1 says that we have to multiply,"},{"Start":"01:49.740 ","End":"01:51.570","Text":"let\u0027s see, B_1 on the top,"},{"Start":"01:51.570 ","End":"01:52.995","Text":"B_2 at the bottom,"},{"Start":"01:52.995 ","End":"01:59.030","Text":"this 1, and if we multiply it by this vector,"},{"Start":"01:59.030 ","End":"02:01.595","Text":"then we should get this vector,"},{"Start":"02:01.595 ","End":"02:03.755","Text":"this times this is this."},{"Start":"02:03.755 ","End":"02:07.110","Text":"Let\u0027s see, yeah."},{"Start":"02:07.110 ","End":"02:08.930","Text":"I want this times this to be this."},{"Start":"02:08.930 ","End":"02:10.385","Text":"That\u0027s what it says here."},{"Start":"02:10.385 ","End":"02:13.820","Text":"Should have really explained also that when we take"},{"Start":"02:13.820 ","End":"02:18.560","Text":"a vector and represent it as a matrix,"},{"Start":"02:18.560 ","End":"02:24.170","Text":"we make it a column matrix so that this x,"},{"Start":"02:24.170 ","End":"02:25.850","Text":"y minus x minus z,"},{"Start":"02:25.850 ","End":"02:27.170","Text":"z is this, x,"},{"Start":"02:27.170 ","End":"02:28.520","Text":"y minus x minus z,"},{"Start":"02:28.520 ","End":"02:32.510","Text":"z. I\u0027ll leave you to verify it."},{"Start":"02:32.510 ","End":"02:35.810","Text":"Perhaps I\u0027ll do 1 of them,"},{"Start":"02:35.810 ","End":"02:43.925","Text":"maybe the last 1 if I take minus 2 times x and minus 1 times this and 0 times this,"},{"Start":"02:43.925 ","End":"02:47.135","Text":"all I have to do is minus 1 times this reverses it."},{"Start":"02:47.135 ","End":"02:50.380","Text":"So it\u0027s x plus z minus y,"},{"Start":"02:50.380 ","End":"02:54.510","Text":"and then I subtract 2x and anyway,"},{"Start":"02:54.510 ","End":"02:57.950","Text":"you get this, I suggest just verify this."},{"Start":"02:57.950 ","End":"03:01.535","Text":"The second part, where was it?"},{"Start":"03:01.535 ","End":"03:07.950","Text":"Says that we want the B_2 on the top,"},{"Start":"03:07.950 ","End":"03:09.170","Text":"B_1 at the bottom."},{"Start":"03:09.170 ","End":"03:14.090","Text":"This 1 times this 1 has to be this 1."},{"Start":"03:14.090 ","End":"03:16.840","Text":"Let\u0027s check that."},{"Start":"03:16.840 ","End":"03:19.490","Text":"These are the right ones."},{"Start":"03:19.490 ","End":"03:25.680","Text":"What we have is we wanted this times this to equal this."},{"Start":"03:25.680 ","End":"03:27.710","Text":"If you do the computation,"},{"Start":"03:27.710 ","End":"03:29.675","Text":"you\u0027ll see that it\u0027s true."},{"Start":"03:29.675 ","End":"03:36.000","Text":"The last part is to show that these 2 are inverses of each other."},{"Start":"03:36.020 ","End":"03:38.300","Text":"To show that they\u0027re inverses,"},{"Start":"03:38.300 ","End":"03:43.504","Text":"easiest thing is to multiply them together and see that we get the identity matrix."},{"Start":"03:43.504 ","End":"03:51.300","Text":"Again, I\u0027ll leave it up to you to check as you know how to multiply matrices."},{"Start":"03:51.440 ","End":"03:58.490","Text":"The main thing here that we\u0027ve shown is that we can get a coordinate vector with"},{"Start":"03:58.490 ","End":"04:05.170","Text":"respect to 1 basis in terms of a coordinate vector relative to another basis,"},{"Start":"04:05.170 ","End":"04:08.510","Text":"by multiplication by the appropriate change"},{"Start":"04:08.510 ","End":"04:12.320","Text":"of basis matrix or change of coordinates matrix."},{"Start":"04:12.320 ","End":"04:15.960","Text":"We are finally done."}],"Thumbnail":null,"ID":10249}],"ID":7318},{"Name":"Matrix of Linear Transformation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"4m 51s","ChapterTopicVideoID":13595,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.325","Text":"This exercise is a continuation of a multi-part exercise."},{"Start":"00:05.325 ","End":"00:08.505","Text":"Here, we\u0027re given a transformation."},{"Start":"00:08.505 ","End":"00:11.505","Text":"I say otherwise it\u0027s a linear transformation."},{"Start":"00:11.505 ","End":"00:16.815","Text":"T from R^3 to R^3 given by the following formula."},{"Start":"00:16.815 ","End":"00:22.170","Text":"We have 2 bases of R^3, this and this."},{"Start":"00:22.170 ","End":"00:26.520","Text":"These appeared in the previous exercise."},{"Start":"00:26.520 ","End":"00:30.060","Text":"We have to find the matrix that represents"},{"Start":"00:30.060 ","End":"00:37.605","Text":"the transformation relative to the basis B_1 then the same thing,"},{"Start":"00:37.605 ","End":"00:40.750","Text":"but for basis B_2."},{"Start":"00:40.870 ","End":"00:46.060","Text":"I want to remind you that as far as notation goes,"},{"Start":"00:46.060 ","End":"00:50.290","Text":"that we write our vectors as row vectors for convenience,"},{"Start":"00:50.290 ","End":"00:53.530","Text":"but really, if you had the space for it,"},{"Start":"00:53.530 ","End":"01:00.830","Text":"this would be the correct way to write this transformation vertical column vectors."},{"Start":"01:02.150 ","End":"01:06.015","Text":"We\u0027ll start with part 1."},{"Start":"01:06.015 ","End":"01:11.775","Text":"Here, I just copied. Here\u0027s the transformation and here\u0027s the basis that we need."},{"Start":"01:11.775 ","End":"01:16.270","Text":"From a previous exercise in this series of exercises,"},{"Start":"01:16.270 ","End":"01:18.700","Text":"they\u0027re all part of a suite or a series."},{"Start":"01:18.700 ","End":"01:25.390","Text":"We got this result for the coordinate vector relative to basis B_1."},{"Start":"01:25.460 ","End":"01:31.790","Text":"Now we have to do a computation for each of these 3 basis vectors."},{"Start":"01:31.790 ","End":"01:34.475","Text":"Let\u0027s start with the first 1,1,0."},{"Start":"01:34.475 ","End":"01:36.740","Text":"We apply T to it."},{"Start":"01:36.740 ","End":"01:39.820","Text":"Using the formula x plus y,"},{"Start":"01:39.820 ","End":"01:42.870","Text":"it would be 2."},{"Start":"01:42.870 ","End":"01:45.390","Text":"Y plus z is 1,"},{"Start":"01:45.390 ","End":"01:48.930","Text":"z minus x is 0 minus 1."},{"Start":"01:48.930 ","End":"01:53.775","Text":"Then we use this formula but this is x, y, z."},{"Start":"01:53.775 ","End":"01:56.810","Text":"We plug them into here, here and here."},{"Start":"01:56.810 ","End":"01:59.975","Text":"What we get is this."},{"Start":"01:59.975 ","End":"02:02.360","Text":"Similarly, for the second,"},{"Start":"02:02.360 ","End":"02:07.955","Text":"we take the image of it under T or T does to it,"},{"Start":"02:07.955 ","End":"02:12.320","Text":"send it to 1,1,0 using this formula,"},{"Start":"02:12.320 ","End":"02:20.210","Text":"then we break this up using this into coordinates relative to the basis B_1."},{"Start":"02:20.210 ","End":"02:29.200","Text":"The same thing for the third vector, 0,1,1."},{"Start":"02:29.200 ","End":"02:30.575","Text":"I\u0027ll leave you to check."},{"Start":"02:30.575 ","End":"02:34.460","Text":"Now that we have all these calculations,"},{"Start":"02:34.460 ","End":"02:38.150","Text":"we get the result which is"},{"Start":"02:38.150 ","End":"02:45.415","Text":"a matrix for the transformation T relative to B_1."},{"Start":"02:45.415 ","End":"02:51.095","Text":"We get it from these numbers here but notice that there is a transpose."},{"Start":"02:51.095 ","End":"02:54.230","Text":"Here, I have 2,0 minus 1 is the row."},{"Start":"02:54.230 ","End":"02:55.490","Text":"Here, it\u0027s a column."},{"Start":"02:55.490 ","End":"02:58.250","Text":"1,0,0, that\u0027s the second column."},{"Start":"02:58.250 ","End":"03:00.380","Text":"Don\u0027t forget to transpose."},{"Start":"03:00.380 ","End":"03:03.055","Text":"Don\u0027t just copy them as they are."},{"Start":"03:03.055 ","End":"03:06.660","Text":"That\u0027s part 1."},{"Start":"03:06.660 ","End":"03:08.670","Text":"Now on to part 2,"},{"Start":"03:08.670 ","End":"03:10.455","Text":"which is going to be very similar."},{"Start":"03:10.455 ","End":"03:16.155","Text":"This transformation is the same and just copied it again."},{"Start":"03:16.155 ","End":"03:19.165","Text":"Here is the basis B_2."},{"Start":"03:19.165 ","End":"03:22.280","Text":"The same procedure. Again,"},{"Start":"03:22.280 ","End":"03:27.965","Text":"I\u0027m building on earlier work that we did in a previous exercise in this series."},{"Start":"03:27.965 ","End":"03:37.935","Text":"We got the coordinate vector relative to B_2 as this formula,"},{"Start":"03:37.935 ","End":"03:40.620","Text":"then we do 3 calculations,"},{"Start":"03:40.620 ","End":"03:43.010","Text":"1 for each basis vector."},{"Start":"03:43.010 ","End":"03:46.100","Text":"I first apply T to it."},{"Start":"03:46.100 ","End":"03:47.750","Text":"Let\u0027s say this 1."},{"Start":"03:47.750 ","End":"03:57.830","Text":"I get the result and then I break it up back in terms of the basis into its coordinate,"},{"Start":"03:57.830 ","End":"04:01.015","Text":"make it a coordinate vector."},{"Start":"04:01.015 ","End":"04:03.990","Text":"Here, x is 1,"},{"Start":"04:03.990 ","End":"04:06.475","Text":"y is also 1,"},{"Start":"04:06.475 ","End":"04:09.410","Text":"and z minus x minus y is 0,"},{"Start":"04:09.410 ","End":"04:12.905","Text":"minus 1, minus 1 is minus 2."},{"Start":"04:12.905 ","End":"04:15.700","Text":"Similarly for the others."},{"Start":"04:15.700 ","End":"04:17.790","Text":"You know what to do now,"},{"Start":"04:17.790 ","End":"04:21.980","Text":"we take these coefficients,"},{"Start":"04:21.980 ","End":"04:25.445","Text":"let\u0027s call them these numbers in blue."},{"Start":"04:25.445 ","End":"04:27.080","Text":"They form a matrix,"},{"Start":"04:27.080 ","End":"04:29.825","Text":"but we must transpose,"},{"Start":"04:29.825 ","End":"04:34.070","Text":"don\u0027t forget to transpose the first column,"},{"Start":"04:34.070 ","End":"04:36.140","Text":"1,1,0 becomes the first row."},{"Start":"04:36.140 ","End":"04:38.150","Text":"Or if we want to do the other way round the first row,"},{"Start":"04:38.150 ","End":"04:41.585","Text":"1,1 minus 2 is the first column, and so on."},{"Start":"04:41.585 ","End":"04:49.270","Text":"That gives us the transformation T as a matrix relative to basis B_2."},{"Start":"04:49.270 ","End":"04:52.260","Text":"That\u0027s part 2 and we\u0027re done."}],"Thumbnail":null,"ID":14313},{"Watched":false,"Name":"Exercise 1 - Part b","Duration":"8m 30s","ChapterTopicVideoID":13596,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.910","Text":"This exercise is part of a whole series of connected exercises,"},{"Start":"00:05.910 ","End":"00:11.565","Text":"so we\u0027ll be using some of the results from the earlier exercises."},{"Start":"00:11.565 ","End":"00:14.685","Text":"We have here a transformation,"},{"Start":"00:14.685 ","End":"00:17.980","Text":"it\u0027s linear unless I say otherwise."},{"Start":"00:18.140 ","End":"00:21.825","Text":"From R3 to R3 given as follows,"},{"Start":"00:21.825 ","End":"00:27.700","Text":"and we have the 2 bases we\u0027ve used before, B_1 and B_2."},{"Start":"00:27.800 ","End":"00:31.350","Text":"We have 3 claims we have to verify,"},{"Start":"00:31.350 ","End":"00:34.305","Text":"and I\u0027ll just read each 1 as we come to it."},{"Start":"00:34.305 ","End":"00:37.225","Text":"Note that v,"},{"Start":"00:37.225 ","End":"00:42.050","Text":"we use to represent a general vector x, y,"},{"Start":"00:42.050 ","End":"00:46.760","Text":"z so that v with respect to B_1 is"},{"Start":"00:46.760 ","End":"00:53.080","Text":"the coordinate vector for basis B_1 and so on."},{"Start":"00:53.180 ","End":"00:56.460","Text":"We\u0027ll start with Part 1,"},{"Start":"00:56.460 ","End":"00:57.630","Text":"I just copied it,"},{"Start":"00:57.630 ","End":"01:00.010","Text":"so there we are."},{"Start":"01:00.140 ","End":"01:02.505","Text":"Here\u0027s what it says."},{"Start":"01:02.505 ","End":"01:10.235","Text":"What it actually means is that if I take"},{"Start":"01:10.235 ","End":"01:19.250","Text":"the transformation matrix for T and multiply it by the coordinate vector,"},{"Start":"01:19.250 ","End":"01:21.845","Text":"but all these relative to B_1,"},{"Start":"01:21.845 ","End":"01:28.940","Text":"it\u0027s the same as if first I applied the transformation T to the vector v,"},{"Start":"01:28.940 ","End":"01:31.265","Text":"and then took the result of that,"},{"Start":"01:31.265 ","End":"01:34.950","Text":"it\u0027s coordinate vector with respect to B_1."},{"Start":"01:35.330 ","End":"01:42.680","Text":"Now I brought in a couple of results from earlier exercises and oops,"},{"Start":"01:42.680 ","End":"01:46.680","Text":"I got the wrong kind of brackets here."},{"Start":"01:47.480 ","End":"01:52.745","Text":"We have the coordinate vector with respect to B_1,"},{"Start":"01:52.745 ","End":"02:00.475","Text":"and we have the matrix of T with respect to B_1."},{"Start":"02:00.475 ","End":"02:03.090","Text":"Let\u0027s check if this is so."},{"Start":"02:03.090 ","End":"02:08.390","Text":"Let\u0027s check that the left-hand side is equal to the right-hand side."},{"Start":"02:08.390 ","End":"02:11.465","Text":"Left-hand side is this,"},{"Start":"02:11.465 ","End":"02:14.840","Text":"and I\u0027m just copying this from here,"},{"Start":"02:14.840 ","End":"02:20.064","Text":"this from here, but it\u0027s a column vector."},{"Start":"02:20.064 ","End":"02:23.630","Text":"Remember I said that we sometimes, just for convenience,"},{"Start":"02:23.630 ","End":"02:27.890","Text":"flatten them out, make them horizontal row instead of a column."},{"Start":"02:27.890 ","End":"02:30.425","Text":"If you do the multiplication,"},{"Start":"02:30.425 ","End":"02:31.940","Text":"let\u0027s see, 2, 1,"},{"Start":"02:31.940 ","End":"02:35.150","Text":"1 times this,"},{"Start":"02:35.150 ","End":"02:39.490","Text":"this times this should equal this entry."},{"Start":"02:39.490 ","End":"02:44.470","Text":"Let\u0027s see, twice x plus y minus x minus z."},{"Start":"02:44.470 ","End":"02:47.615","Text":"You know what? I\u0027ll leave you to check it."},{"Start":"02:47.615 ","End":"02:49.305","Text":"Similarly, for the other 2,"},{"Start":"02:49.305 ","End":"02:51.120","Text":"this with this gives 0."},{"Start":"02:51.120 ","End":"02:52.995","Text":"Of course, that\u0027s all zeros,"},{"Start":"02:52.995 ","End":"02:55.035","Text":"similarly for the last 1."},{"Start":"02:55.035 ","End":"03:00.590","Text":"That was the left-hand side and now the right-hand side, this is what we have,"},{"Start":"03:02.810 ","End":"03:05.010","Text":"T of v,"},{"Start":"03:05.010 ","End":"03:09.540","Text":"using the formula for T is in general,"},{"Start":"03:09.540 ","End":"03:13.120","Text":"x plus y, y plus z, z minus x."},{"Start":"03:13.120 ","End":"03:15.070","Text":"That was the formula for T,"},{"Start":"03:15.070 ","End":"03:18.650","Text":"and I have to figure this relative to B_1."},{"Start":"03:19.940 ","End":"03:23.040","Text":"I kept this formula here,"},{"Start":"03:23.040 ","End":"03:25.860","Text":"what we do is use this formula,"},{"Start":"03:25.860 ","End":"03:27.000","Text":"instead of on v,"},{"Start":"03:27.000 ","End":"03:31.840","Text":"on T of v. This is like"},{"Start":"03:32.330 ","End":"03:40.395","Text":"x and this is like y in the formula,"},{"Start":"03:40.395 ","End":"03:43.365","Text":"and this is z."},{"Start":"03:43.365 ","End":"03:47.085","Text":"When it says x, I just copy the x part,"},{"Start":"03:47.085 ","End":"03:52.670","Text":"y minus x is this bit minus this bit as here."},{"Start":"03:52.670 ","End":"03:54.920","Text":"Then the third component,"},{"Start":"03:54.920 ","End":"03:58.765","Text":"z, just z minus x as is."},{"Start":"03:58.765 ","End":"04:00.650","Text":"If we simplify it,"},{"Start":"04:00.650 ","End":"04:07.520","Text":"it comes out to be this and this is this."},{"Start":"04:07.520 ","End":"04:14.945","Text":"Like I said, we just sometimes write column vectors for convenience as row vectors."},{"Start":"04:14.945 ","End":"04:22.595","Text":"We\u0027ve proven the equality that indeed this times this is the same as this,"},{"Start":"04:22.595 ","End":"04:24.275","Text":"and that\u0027s just what I wrote here."},{"Start":"04:24.275 ","End":"04:26.645","Text":"That\u0027s Part 1."},{"Start":"04:26.645 ","End":"04:30.950","Text":"Now Part 2 is pretty much the same as Part 1,"},{"Start":"04:30.950 ","End":"04:35.315","Text":"except that we\u0027re using B_2 everywhere instead of B_1."},{"Start":"04:35.315 ","End":"04:37.885","Text":"This is what we have to prove,"},{"Start":"04:37.885 ","End":"04:43.550","Text":"and here I\u0027ve imported a couple of results from the earlier exercises in the series."},{"Start":"04:43.550 ","End":"04:50.600","Text":"The coordinate vector with respect to B_2 and the transformation,"},{"Start":"04:50.600 ","End":"04:56.475","Text":"its matrix representation relative to B_2."},{"Start":"04:56.475 ","End":"04:59.300","Text":"As before, we\u0027ll compute the left-hand side"},{"Start":"04:59.300 ","End":"05:02.570","Text":"than the right-hand side and see that they are equal."},{"Start":"05:02.570 ","End":"05:04.835","Text":"Left-hand side, this,"},{"Start":"05:04.835 ","End":"05:08.555","Text":"now do it properly with the column vector."},{"Start":"05:08.555 ","End":"05:10.850","Text":"It\u0027s this matrix here,"},{"Start":"05:10.850 ","End":"05:14.435","Text":"times this vector here."},{"Start":"05:14.435 ","End":"05:17.125","Text":"If you do the computation,"},{"Start":"05:17.125 ","End":"05:19.230","Text":"well, just check 1 of them."},{"Start":"05:19.230 ","End":"05:24.355","Text":"Let\u0027s see that, this 1 times this is got to equal this."},{"Start":"05:24.355 ","End":"05:31.490","Text":"Minus 2 times x plus minus 2 times y plus 0 times this is equal to this."},{"Start":"05:31.490 ","End":"05:34.465","Text":"Similarly, for the other 2, I\u0027ll leave you to check."},{"Start":"05:34.465 ","End":"05:36.950","Text":"Next, we\u0027ll go for the right-hand side."},{"Start":"05:36.950 ","End":"05:39.540","Text":"First of all, I just want to say what T of v is."},{"Start":"05:39.540 ","End":"05:43.520","Text":"If you look at the original definition of the transformation T,"},{"Start":"05:43.520 ","End":"05:45.030","Text":"what it does to x, y,"},{"Start":"05:45.030 ","End":"05:47.030","Text":"z is it sends it to x plus y,"},{"Start":"05:47.030 ","End":"05:49.205","Text":"y plus z, z minus x."},{"Start":"05:49.205 ","End":"05:55.460","Text":"Now we have to find the coordinates of this relative to B_2."},{"Start":"05:55.460 ","End":"06:05.795","Text":"We\u0027ll use this formula with v being replaced by T of v. This is like the x,"},{"Start":"06:05.795 ","End":"06:08.450","Text":"this is like the y,"},{"Start":"06:08.450 ","End":"06:12.310","Text":"and this is like the z."},{"Start":"06:12.310 ","End":"06:17.055","Text":"What we want is x, y,"},{"Start":"06:17.055 ","End":"06:20.530","Text":"z minus x minus y, and here it is,"},{"Start":"06:20.530 ","End":"06:23.255","Text":"the x part, the y part,"},{"Start":"06:23.255 ","End":"06:27.120","Text":"and the z minus x minus y part."},{"Start":"06:27.560 ","End":"06:30.920","Text":"Just a bit of simplification, tidying up,"},{"Start":"06:30.920 ","End":"06:32.195","Text":"and we get this,"},{"Start":"06:32.195 ","End":"06:35.130","Text":"that\u0027s the left-hand side."},{"Start":"06:35.480 ","End":"06:38.985","Text":"Sorry, I meant right-hand side."},{"Start":"06:38.985 ","End":"06:46.665","Text":"The left-hand side is equal to the right-hand side."},{"Start":"06:46.665 ","End":"06:48.570","Text":"This is a column vector,"},{"Start":"06:48.570 ","End":"06:51.045","Text":"this we\u0027ve written as a row the same,"},{"Start":"06:51.045 ","End":"06:55.520","Text":"so we\u0027ve proved indeed that this times this equals this,"},{"Start":"06:55.520 ","End":"06:57.230","Text":"which is what we had to show,"},{"Start":"06:57.230 ","End":"06:59.365","Text":"so that\u0027s Part 2."},{"Start":"06:59.365 ","End":"07:04.240","Text":"Now, we have still Part 3 to do."},{"Start":"07:04.400 ","End":"07:08.695","Text":"We have to show this formula."},{"Start":"07:08.695 ","End":"07:13.910","Text":"This is the transformation matrix of T relative to B_1,"},{"Start":"07:13.910 ","End":"07:15.905","Text":"and here it is relative to B_2."},{"Start":"07:15.905 ","End":"07:19.400","Text":"The way to get from here to here is to multiply on the left and"},{"Start":"07:19.400 ","End":"07:22.890","Text":"on the right by change of basis matrices,"},{"Start":"07:22.890 ","End":"07:26.000","Text":"once from B_2 to B_1 and once from B_1 to B_2."},{"Start":"07:26.000 ","End":"07:28.655","Text":"These are inverses of each other."},{"Start":"07:28.655 ","End":"07:33.255","Text":"It would look nicer if we wrote this,"},{"Start":"07:33.255 ","End":"07:37.430","Text":"for example as, if we called this p,"},{"Start":"07:37.430 ","End":"07:40.025","Text":"this would be p minus 1,"},{"Start":"07:40.025 ","End":"07:44.200","Text":"and so p minus 1 times something times p,"},{"Start":"07:44.200 ","End":"07:46.685","Text":"it would show that these 2 are similar."},{"Start":"07:46.685 ","End":"07:51.140","Text":"Whenever we sandwich something in between an inverse matrix and a matrix,"},{"Start":"07:51.140 ","End":"07:53.885","Text":"then it\u0027s a similar matrix."},{"Start":"07:53.885 ","End":"07:57.420","Text":"Anyway, it\u0027s just another way of looking at it."},{"Start":"07:58.490 ","End":"08:00.620","Text":"We have all these,"},{"Start":"08:00.620 ","End":"08:02.465","Text":"we\u0027ve computed them already."},{"Start":"08:02.465 ","End":"08:05.150","Text":"It\u0027s just a matter of verifying"},{"Start":"08:05.150 ","End":"08:10.950","Text":"a matrix multiplication that this times this times this equals this."},{"Start":"08:11.930 ","End":"08:15.210","Text":"It\u0027s just purely technical,"},{"Start":"08:15.210 ","End":"08:17.300","Text":"so I\u0027m not going to do that."},{"Start":"08:17.300 ","End":"08:19.580","Text":"I\u0027ll leave that for you to check."},{"Start":"08:19.580 ","End":"08:21.755","Text":"But that\u0027s the idea, we take the left-hand side,"},{"Start":"08:21.755 ","End":"08:23.390","Text":"compute this times, this, times this,"},{"Start":"08:23.390 ","End":"08:31.110","Text":"and see if at the end we get the right-hand side, hopefully. We\u0027re done."}],"Thumbnail":null,"ID":14314},{"Watched":false,"Name":"Exercise 1 - Part c","Duration":"4m 22s","ChapterTopicVideoID":13597,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.785","Text":"This exercise is part of a continuing series of related exercises."},{"Start":"00:07.785 ","End":"00:10.980","Text":"We\u0027ll use some of the results of the previous ones."},{"Start":"00:10.980 ","End":"00:17.020","Text":"In this one, we\u0027re given a basis for R^3, this one."},{"Start":"00:17.650 ","End":"00:21.965","Text":"We given that there\u0027s a linear transformation"},{"Start":"00:21.965 ","End":"00:29.240","Text":"whose representative matrix with respect to this basis is the following matrix."},{"Start":"00:29.240 ","End":"00:34.790","Text":"We have to figure out what the formula is for T to describe it."},{"Start":"00:34.790 ","End":"00:36.290","Text":"In other words, T of x, y,"},{"Start":"00:36.290 ","End":"00:38.450","Text":"z is what, what, what."},{"Start":"00:38.450 ","End":"00:41.095","Text":"This is like a reverse exercise."},{"Start":"00:41.095 ","End":"00:45.540","Text":"Previously, we would be given T and compute its matrix."},{"Start":"00:45.540 ","End":"00:47.700","Text":"This time, we\u0027re given the matrix,"},{"Start":"00:47.700 ","End":"00:50.690","Text":"and we have to figure out what T is."},{"Start":"00:50.690 ","End":"00:54.120","Text":"Let me just clear some space."},{"Start":"00:54.890 ","End":"01:03.640","Text":"I copied again what the matrix of T is with respect to basis B,"},{"Start":"01:03.640 ","End":"01:05.090","Text":"and the basis B."},{"Start":"01:05.090 ","End":"01:08.375","Text":"Now if you look at previous exercises,"},{"Start":"01:08.375 ","End":"01:11.485","Text":"this is what we called B_2."},{"Start":"01:11.485 ","End":"01:15.255","Text":"The coordinate vector we computed,"},{"Start":"01:15.255 ","End":"01:17.180","Text":"I forget which number exercise,"},{"Start":"01:17.180 ","End":"01:23.425","Text":"anyway, we computed the coordinate vector with respect to B_2 as this."},{"Start":"01:23.425 ","End":"01:30.595","Text":"Where our v, as usual is just a general vector x, y, z."},{"Start":"01:30.595 ","End":"01:32.990","Text":"Now we bring in this formula."},{"Start":"01:32.990 ","End":"01:38.915","Text":"This is the one that we need that relates the matrix for B,"},{"Start":"01:38.915 ","End":"01:40.925","Text":"and the coordinate vector,"},{"Start":"01:40.925 ","End":"01:45.840","Text":"and it tells us the coordinate vector"},{"Start":"01:45.840 ","End":"01:51.510","Text":"for T of v. From that we\u0027ll be able to find what T of v is."},{"Start":"01:51.510 ","End":"01:55.880","Text":"We\u0027re going to compute the left-hand side first because we have everything."},{"Start":"01:55.880 ","End":"02:00.170","Text":"This T with respect to B is this matrix,"},{"Start":"02:00.170 ","End":"02:02.520","Text":"and that one is here."},{"Start":"02:02.740 ","End":"02:06.770","Text":"The coordinate vector of v with respect to B is this,"},{"Start":"02:06.770 ","End":"02:09.395","Text":"but we want it as a column vector, x,"},{"Start":"02:09.395 ","End":"02:11.465","Text":"y, z minus x minus y."},{"Start":"02:11.465 ","End":"02:13.145","Text":"Same thing here."},{"Start":"02:13.145 ","End":"02:15.920","Text":"If you multiply this out,"},{"Start":"02:15.920 ","End":"02:19.400","Text":"and we get this, for example,"},{"Start":"02:19.400 ","End":"02:25.910","Text":"this we would get by this row with this column and 1 times x,"},{"Start":"02:25.910 ","End":"02:27.815","Text":"1 times y, 0 times this,"},{"Start":"02:27.815 ","End":"02:30.570","Text":"and then similarly for the other 2."},{"Start":"02:32.320 ","End":"02:37.234","Text":"What we\u0027ve just computed according to this formula is"},{"Start":"02:37.234 ","End":"02:47.610","Text":"the coordinate vector for T of v,"},{"Start":"02:47.610 ","End":"02:49.725","Text":"with respect to B."},{"Start":"02:49.725 ","End":"02:58.470","Text":"Here I just essentially copied it except that we got to remember that v is x, y, z."},{"Start":"02:59.900 ","End":"03:06.725","Text":"Now if we know that the coordinates of this thing with respect to B is this,"},{"Start":"03:06.725 ","End":"03:08.104","Text":"what does it mean?"},{"Start":"03:08.104 ","End":"03:15.260","Text":"It means that we can get this vector by taking the first component,"},{"Start":"03:15.260 ","End":"03:19.505","Text":"x plus y times the first vector in"},{"Start":"03:19.505 ","End":"03:25.655","Text":"the basis B plus this second component times the second vector,"},{"Start":"03:25.655 ","End":"03:29.410","Text":"the third component with the third vector."},{"Start":"03:29.410 ","End":"03:34.715","Text":"Here we just did the scalar multiplication 3 times,"},{"Start":"03:34.715 ","End":"03:36.950","Text":"like here, x plus y times 1,"},{"Start":"03:36.950 ","End":"03:39.320","Text":"x plus y times 0, x plus y times 1."},{"Start":"03:39.320 ","End":"03:41.190","Text":"Similarly, for the rest."},{"Start":"03:41.190 ","End":"03:45.970","Text":"Now all we have to do is do a vector addition of 3 vectors."},{"Start":"03:45.970 ","End":"03:49.975","Text":"The first component, x plus y plus 0 plus 0 is that."},{"Start":"03:49.975 ","End":"03:57.675","Text":"Second component 0 plus y plus z plus 0 is this,"},{"Start":"03:57.675 ","End":"04:04.450","Text":"and x plus y plus y plus z minus 2y is,"},{"Start":"04:05.270 ","End":"04:09.000","Text":"anyway, it comes out z minus x."},{"Start":"04:09.000 ","End":"04:10.665","Text":"That\u0027s the answer. Now,"},{"Start":"04:10.665 ","End":"04:16.335","Text":"we have the transformation T explicitly what it does to x, y, z."},{"Start":"04:16.335 ","End":"04:19.650","Text":"These were the 3 question marks that we had to find,"},{"Start":"04:19.650 ","End":"04:22.750","Text":"so we are done."}],"Thumbnail":null,"ID":14315},{"Watched":false,"Name":"Exercise 1 - Part d","Duration":"4m 47s","ChapterTopicVideoID":13598,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"We\u0027re continuing with this long series of"},{"Start":"00:03.030 ","End":"00:07.395","Text":"exercises all related and building on each other."},{"Start":"00:07.395 ","End":"00:14.370","Text":"This time, we have this transformation from R^3 to R^3 linear,"},{"Start":"00:14.370 ","End":"00:17.550","Text":"of course, given by this formula."},{"Start":"00:17.550 ","End":"00:20.415","Text":"We have the basis B_1."},{"Start":"00:20.415 ","End":"00:23.640","Text":"We\u0027ve seen this before."},{"Start":"00:23.640 ","End":"00:26.160","Text":"There are 2 questions."},{"Start":"00:26.160 ","End":"00:31.110","Text":"First of all, is t as a transformation invertible?"},{"Start":"00:31.110 ","End":"00:38.160","Text":"Secondly, to compute both the determinant and the trace of the transformation"},{"Start":"00:38.160 ","End":"00:47.305","Text":"T. This exercise is yet another in a long series of related exercises."},{"Start":"00:47.305 ","End":"00:54.460","Text":"Here\u0027s the transformation from R^3 to R^3 that we\u0027ve been using."},{"Start":"00:54.890 ","End":"00:57.730","Text":"We have the basis B_1,"},{"Start":"00:57.730 ","End":"01:01.600","Text":"which is this just like previous exercises."},{"Start":"01:01.600 ","End":"01:03.450","Text":"There are 2 questions."},{"Start":"01:03.450 ","End":"01:08.890","Text":"First is T invertible transformation?"},{"Start":"01:08.900 ","End":"01:16.500","Text":"Secondly to compute the determinant and the trace of the transformation T."},{"Start":"01:16.650 ","End":"01:27.080","Text":"Our strategy will be to represent T as a matrix with this particular basis,"},{"Start":"01:27.080 ","End":"01:29.105","Text":"it could be with any basis."},{"Start":"01:29.105 ","End":"01:34.900","Text":"Then answer questions about a matrix rather than a transformation."},{"Start":"01:34.900 ","End":"01:39.810","Text":"For the first part, what this says,"},{"Start":"01:39.810 ","End":"01:45.380","Text":"this is like a theorem that linear transformation is"},{"Start":"01:45.380 ","End":"01:52.310","Text":"invertible if and only if its matrix representation as a matrix is invertible."},{"Start":"01:52.310 ","End":"01:53.720","Text":"It\u0027s going to be in any,"},{"Start":"01:53.720 ","End":"01:55.460","Text":"in the sense of every basis,"},{"Start":"01:55.460 ","End":"01:59.340","Text":"whatever the basis, it\u0027s get to be invertible."},{"Start":"01:59.340 ","End":"02:03.660","Text":"Let\u0027s take this basis."},{"Start":"02:03.660 ","End":"02:08.195","Text":"I copy this result from a previous exercise that"},{"Start":"02:08.195 ","End":"02:14.345","Text":"the matrix of T with respect to B_1 came out to be this."},{"Start":"02:14.345 ","End":"02:18.140","Text":"Notice that there\u0027s a row of zeros in here."},{"Start":"02:18.140 ","End":"02:22.970","Text":"It\u0027s really not invertible for many reasons."},{"Start":"02:22.970 ","End":"02:28.050","Text":"For example, you could compute the determinant and see that it\u0027s 0."},{"Start":"02:28.090 ","End":"02:35.375","Text":"There are other ways of seeing how many when we put it in row echelon form,"},{"Start":"02:35.375 ","End":"02:38.750","Text":"how many rows you get and if it\u0027s less than 3,"},{"Start":"02:38.750 ","End":"02:41.970","Text":"it\u0027s not invertible, but you will be here."},{"Start":"02:42.790 ","End":"02:46.335","Text":"I just wrote that to what I just said."},{"Start":"02:46.335 ","End":"02:50.235","Text":"Now, let\u0027s go on to part 2."},{"Start":"02:50.235 ","End":"02:52.840","Text":"Let\u0027s see if I can keep it on screen."},{"Start":"02:52.840 ","End":"02:59.060","Text":"We\u0027ve got to compute the determinant and the trace. Here\u0027s the thing."},{"Start":"02:59.060 ","End":"03:04.850","Text":"We define the determinant of a transformation"},{"Start":"03:04.850 ","End":"03:11.085","Text":"to be the determinant of a matrix representing that transformation."},{"Start":"03:11.085 ","End":"03:15.170","Text":"It can be shown that it doesn\u0027t matter what basis you choose,"},{"Start":"03:15.170 ","End":"03:16.910","Text":"you\u0027ll get the same result."},{"Start":"03:16.910 ","End":"03:22.190","Text":"That\u0027s why it makes sense to say that the determinant of the transformation is"},{"Start":"03:22.190 ","End":"03:28.130","Text":"the determinant of its matrix because it doesn\u0027t depend on the basis that can be shown."},{"Start":"03:28.130 ","End":"03:29.780","Text":"Similarly, for the trace,"},{"Start":"03:29.780 ","End":"03:37.849","Text":"if I take the matrix that represents T in any basis and take the trace of that matrix,"},{"Start":"03:37.849 ","End":"03:40.550","Text":"I\u0027ll get the same result."},{"Start":"03:40.550 ","End":"03:48.840","Text":"We now converted the problem from transformations to matrices."},{"Start":"03:51.350 ","End":"03:56.270","Text":"In this case. This is in general for any b and we can take"},{"Start":"03:56.270 ","End":"04:04.030","Text":"b to be our particular B_1 that\u0027s just off-screen."},{"Start":"04:05.360 ","End":"04:09.090","Text":"We have this matrix here."},{"Start":"04:09.090 ","End":"04:11.905","Text":"Since we took b equals B_1,"},{"Start":"04:11.905 ","End":"04:18.530","Text":"we want to figure out its determinant of this matrix and the trace of this matrix."},{"Start":"04:18.530 ","End":"04:22.460","Text":"Now, the determined that this matrix we said is 0."},{"Start":"04:22.460 ","End":"04:24.230","Text":"When it has a 0 row,"},{"Start":"04:24.230 ","End":"04:27.145","Text":"the determinant is going to be 0."},{"Start":"04:27.145 ","End":"04:33.770","Text":"The trace, we add up the elements along the diagonal."},{"Start":"04:33.770 ","End":"04:40.345","Text":"This plus this plus this is the trace 2 plus 0 plus 1 is 3."},{"Start":"04:40.345 ","End":"04:43.190","Text":"That\u0027s the answer. The determinant of the transformation is"},{"Start":"04:43.190 ","End":"04:48.330","Text":"0 and the trace of the transformation is 3. We\u0027re done."}],"Thumbnail":null,"ID":14316},{"Watched":false,"Name":"Exercise 1 - Part e","Duration":"10m 13s","ChapterTopicVideoID":13599,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.580","Text":"This exercise, I think it\u0027s the last 1 in a long series of related exercises,"},{"Start":"00:08.580 ","End":"00:14.640","Text":"and we have the familiar transformation,"},{"Start":"00:14.640 ","End":"00:18.795","Text":"linear transformation from R^3 to R^3 with this formula."},{"Start":"00:18.795 ","End":"00:21.630","Text":"This time, we have 2 questions."},{"Start":"00:21.630 ","End":"00:26.340","Text":"Find the eigenvalues of T and"},{"Start":"00:26.340 ","End":"00:35.380","Text":"the eigenvectors relative to the standard basis E. Question 2,"},{"Start":"00:35.380 ","End":"00:39.280","Text":"is the transformation diagonalizable?"},{"Start":"00:39.740 ","End":"00:42.020","Text":"Let\u0027s start with part 1."},{"Start":"00:42.020 ","End":"00:48.800","Text":"Let me just make a comment that the eigenvalues don\u0027t depend on whatever basis we choose,"},{"Start":"00:48.800 ","End":"00:50.945","Text":"but the eigenvectors do,"},{"Start":"00:50.945 ","End":"00:54.450","Text":"so it makes a difference which basis."},{"Start":"00:56.150 ","End":"01:04.160","Text":"The first step is to find the matrix that represents T in the standard basis,"},{"Start":"01:04.160 ","End":"01:09.750","Text":"and I\u0027ll just copy this here so when I scroll, it doesn\u0027t disappear."},{"Start":"01:10.400 ","End":"01:15.800","Text":"You know what? Just in case you\u0027ve forgotten what the standard basis is,"},{"Start":"01:15.800 ","End":"01:18.410","Text":"I\u0027ll write it out for you."},{"Start":"01:18.410 ","End":"01:21.710","Text":"It has 3 members, 1,"},{"Start":"01:21.710 ","End":"01:34.850","Text":"0, 0, 0, 1, 0 and you can guess it, 0, 0, 1, Now,"},{"Start":"01:34.850 ","End":"01:38.900","Text":"what we do is we do something to each 1 of these."},{"Start":"01:38.900 ","End":"01:41.360","Text":"First, let\u0027s take 1, 0, 0."},{"Start":"01:41.360 ","End":"01:46.475","Text":"I compute T of this vector according to this formula,"},{"Start":"01:46.475 ","End":"01:50.260","Text":"x plus y is 1 plus 0 is 1 and so on."},{"Start":"01:50.260 ","End":"01:52.070","Text":"When I\u0027ve got this result,"},{"Start":"01:52.070 ","End":"01:56.650","Text":"I express it in terms of the standard basis,"},{"Start":"01:56.650 ","End":"02:01.910","Text":"and the standard basis is easy because all you have to do is copy these numbers."},{"Start":"02:01.910 ","End":"02:04.925","Text":"1 times the first vector,"},{"Start":"02:04.925 ","End":"02:07.279","Text":"0 times the second vector,"},{"Start":"02:07.279 ","End":"02:11.000","Text":"and minus 1 times the last vector."},{"Start":"02:11.000 ","End":"02:13.470","Text":"Am I missing a comma here?"},{"Start":"02:14.320 ","End":"02:17.860","Text":"We do the same thing for the other 2."},{"Start":"02:17.860 ","End":"02:20.490","Text":"The middle 1, 0, 1, 0."},{"Start":"02:20.490 ","End":"02:23.300","Text":"We figure out T of it from x plus y,"},{"Start":"02:23.300 ","End":"02:24.950","Text":"y plus z,"},{"Start":"02:24.950 ","End":"02:26.405","Text":"and we get this."},{"Start":"02:26.405 ","End":"02:28.730","Text":"Then we just expressed the 1, 1,"},{"Start":"02:28.730 ","End":"02:31.880","Text":"0 as the components of"},{"Start":"02:31.880 ","End":"02:38.160","Text":"the coordinate vector relative to standard basis which is 1, 1, 0."},{"Start":"02:38.600 ","End":"02:45.165","Text":"The last one, 0, 0, 1, x plus y is 0,"},{"Start":"02:45.165 ","End":"02:48.645","Text":"y plus z is 1, so on,"},{"Start":"02:48.645 ","End":"02:50.340","Text":"and then we take this 0, 1,"},{"Start":"02:50.340 ","End":"02:55.060","Text":"1 and place them in front of these 3 vectors."},{"Start":"02:55.190 ","End":"02:58.910","Text":"Now that we\u0027ve got this computed,"},{"Start":"02:58.910 ","End":"03:01.385","Text":"we take the numbers that I colored,"},{"Start":"03:01.385 ","End":"03:06.785","Text":"these coordinates, there\u0027s 9 of them and just put them in a matrix."},{"Start":"03:06.785 ","End":"03:11.390","Text":"But don\u0027t forget, you have to take the transpose,"},{"Start":"03:11.390 ","End":"03:15.770","Text":"rows become columns and columns become row. This is what we get."},{"Start":"03:15.770 ","End":"03:20.270","Text":"Notice 1, 0 minus 1 is the first column, 1,"},{"Start":"03:20.270 ","End":"03:22.070","Text":"1, 0 is the second column,"},{"Start":"03:22.070 ","End":"03:25.685","Text":"rows become columns, the transpose. Not to forget."},{"Start":"03:25.685 ","End":"03:29.585","Text":"That\u0027s the matrix for T with respect to standard basis,"},{"Start":"03:29.585 ","End":"03:33.300","Text":"E. That\u0027s step 1."},{"Start":"03:33.300 ","End":"03:35.860","Text":"Now, let\u0027s go on to the next step."},{"Start":"03:35.860 ","End":"03:40.610","Text":"We want to find the eigenvalues and eigenvectors but before that,"},{"Start":"03:40.610 ","End":"03:43.975","Text":"we have to find the characteristic polynomial."},{"Start":"03:43.975 ","End":"03:46.475","Text":"Sorry. Before the characteristic polynomial,"},{"Start":"03:46.475 ","End":"03:49.445","Text":"we first have to compute the characteristic matrix."},{"Start":"03:49.445 ","End":"03:50.930","Text":"That\u0027s the first step."},{"Start":"03:50.930 ","End":"03:56.680","Text":"That\u0027s x times the identity matrix minus A,"},{"Start":"03:56.680 ","End":"04:01.530","Text":"and the identity for a 3 by 3 is 1s on the diagonal, so multiplied by x,"},{"Start":"04:01.530 ","End":"04:04.050","Text":"we get x\u0027s on the diagonal minus A,"},{"Start":"04:04.050 ","End":"04:06.855","Text":"I just copied from here."},{"Start":"04:06.855 ","End":"04:10.355","Text":"I should have said that we give it a name."},{"Start":"04:10.355 ","End":"04:12.365","Text":"I wrote it here,"},{"Start":"04:12.365 ","End":"04:15.670","Text":"but let\u0027s say this is A."},{"Start":"04:15.830 ","End":"04:19.260","Text":"Now, it\u0027s just a straightforward subtraction,"},{"Start":"04:19.260 ","End":"04:21.645","Text":"and this is the characteristic matrix."},{"Start":"04:21.645 ","End":"04:24.725","Text":"Now, we\u0027ll get to the characteristic polynomial."},{"Start":"04:24.725 ","End":"04:29.180","Text":"The characteristic polynomial is just the determinant of"},{"Start":"04:29.180 ","End":"04:31.760","Text":"the characteristic matrix because what\u0027s written here"},{"Start":"04:31.760 ","End":"04:35.415","Text":"is xI minus A is the characteristic matrix."},{"Start":"04:35.415 ","End":"04:39.095","Text":"We get that, we just copy basically,"},{"Start":"04:39.095 ","End":"04:42.080","Text":"the characteristic matrix only instead of matrix,"},{"Start":"04:42.080 ","End":"04:43.730","Text":"we take off the square brackets,"},{"Start":"04:43.730 ","End":"04:48.575","Text":"put vertical lines, and that makes it the determinant."},{"Start":"04:48.575 ","End":"04:55.440","Text":"Let\u0027s see, let\u0027s expand along the top row."},{"Start":"04:55.580 ","End":"05:00.115","Text":"This won\u0027t contribute anything but we\u0027ll get something from here,"},{"Start":"05:00.115 ","End":"05:02.635","Text":"and this has a plus attached to it,"},{"Start":"05:02.635 ","End":"05:04.050","Text":"and we\u0027ll get something from here,"},{"Start":"05:04.050 ","End":"05:06.320","Text":"this has a minus attached to it."},{"Start":"05:06.320 ","End":"05:08.195","Text":"Remember the alternating signs."},{"Start":"05:08.195 ","End":"05:12.670","Text":"Now, for this entry,"},{"Start":"05:13.330 ","End":"05:18.710","Text":"we would eliminate the row and column for it and"},{"Start":"05:18.710 ","End":"05:23.830","Text":"take the 2 by 2 determinant that\u0027s left and multiply it by this,"},{"Start":"05:23.830 ","End":"05:26.745","Text":"and that\u0027s this first term here."},{"Start":"05:26.745 ","End":"05:32.090","Text":"This 1, we get from this element,"},{"Start":"05:32.090 ","End":"05:34.830","Text":"subtracting its row and column,"},{"Start":"05:34.830 ","End":"05:37.015","Text":"and what\u0027s left is this 0,"},{"Start":"05:37.015 ","End":"05:39.410","Text":"1, minus 1, x minus 1."},{"Start":"05:39.410 ","End":"05:44.935","Text":"Notice that the minus became a plus because there\u0027s a minus here and it was a minus here."},{"Start":"05:44.935 ","End":"05:46.910","Text":"This is just technical."},{"Start":"05:46.910 ","End":"05:49.970","Text":"This diagonal product minus this product,"},{"Start":"05:49.970 ","End":"05:52.010","Text":"this is just x minus 1 squared."},{"Start":"05:52.010 ","End":"05:54.965","Text":"This is 0. We get x minus 1 cubed."},{"Start":"05:54.965 ","End":"05:57.840","Text":"From here, we get this product minus this."},{"Start":"05:57.840 ","End":"05:59.955","Text":"This is 0 minus minus 1,"},{"Start":"05:59.955 ","End":"06:02.310","Text":"so we get this."},{"Start":"06:02.310 ","End":"06:06.305","Text":"Let\u0027s call it p of x is our characteristic polynomial."},{"Start":"06:06.305 ","End":"06:09.730","Text":"Now, in the past, we normally factorize it,"},{"Start":"06:09.730 ","End":"06:10.910","Text":"but in this case,"},{"Start":"06:10.910 ","End":"06:16.085","Text":"it\u0027s actually more convenient not to factorize it and to leave it as is."},{"Start":"06:16.085 ","End":"06:23.660","Text":"I\u0027m going to use the characteristic polynomial to find the eigenvalues."},{"Start":"06:23.660 ","End":"06:29.920","Text":"The eigenvalues are the roots of this polynomial so we set it equal to 0 and solve."},{"Start":"06:29.920 ","End":"06:33.245","Text":"Just by the way, this equation,"},{"Start":"06:33.245 ","End":"06:39.140","Text":"polynomial equals 0 characteristic is called the characteristic equation."},{"Start":"06:39.140 ","End":"06:42.540","Text":"If this is equal to 0,"},{"Start":"06:42.540 ","End":"06:46.025","Text":"we get x minus 1 cubed is minus 1."},{"Start":"06:46.025 ","End":"06:52.090","Text":"But remember, we are working over the real numbers,"},{"Start":"06:52.090 ","End":"06:55.635","Text":"and so there\u0027s only 1 solution to this,"},{"Start":"06:55.635 ","End":"06:59.065","Text":"the cube root of minus 1 is just minus 1,"},{"Start":"06:59.065 ","End":"07:03.235","Text":"so x minus 1 is the cube root which is minus 1,"},{"Start":"07:03.235 ","End":"07:06.800","Text":"and add 1 to both sides, x equals 0."},{"Start":"07:06.800 ","End":"07:12.240","Text":"We only have 1 eigenvalue, x equals 0."},{"Start":"07:12.240 ","End":"07:14.625","Text":"Now, the eigenvectors."},{"Start":"07:14.625 ","End":"07:16.780","Text":"We do for each eigenvalue separately,"},{"Start":"07:16.780 ","End":"07:20.440","Text":"but there is only 1 eigenvalue here for x equals 0 so we just"},{"Start":"07:20.440 ","End":"07:26.230","Text":"have to find its eigenvectors."},{"Start":"07:26.230 ","End":"07:35.190","Text":"We take the characteristic matrix and substitute x equals 0 in here,"},{"Start":"07:35.190 ","End":"07:41.400","Text":"and this substitution leads us to this matrix."},{"Start":"07:41.920 ","End":"07:46.150","Text":"This is the corresponding SLA,"},{"Start":"07:46.150 ","End":"07:48.330","Text":"system of linear equations,"},{"Start":"07:48.330 ","End":"07:52.160","Text":"and we want to find the solution space to this,"},{"Start":"07:52.160 ","End":"07:54.830","Text":"or rather, a basis for that solution space."},{"Start":"07:54.830 ","End":"08:01.710","Text":"As usual, we do row operations to bring to row echelon form."},{"Start":"08:02.390 ","End":"08:09.875","Text":"This, I got by adding the first row to the third row."},{"Start":"08:09.875 ","End":"08:11.420","Text":"Now, clearly,"},{"Start":"08:11.420 ","End":"08:15.970","Text":"the thing to do is to subtract the second row from the third row,"},{"Start":"08:15.970 ","End":"08:18.660","Text":"and that gives us this, and here,"},{"Start":"08:18.660 ","End":"08:19.740","Text":"we have a row of 0,"},{"Start":"08:19.740 ","End":"08:21.915","Text":"so we can just discard it."},{"Start":"08:21.915 ","End":"08:24.860","Text":"Now, this is the corresponding system of"},{"Start":"08:24.860 ","End":"08:30.350","Text":"linear equations where the free variable is the z,"},{"Start":"08:30.350 ","End":"08:33.840","Text":"and x and y are computed,"},{"Start":"08:33.840 ","End":"08:37.925","Text":"they\u0027re restricted, they are not free."},{"Start":"08:37.925 ","End":"08:40.400","Text":"Remember that technique we had,"},{"Start":"08:40.400 ","End":"08:42.515","Text":"I called it the wondering 1s."},{"Start":"08:42.515 ","End":"08:44.210","Text":"This case, there\u0027s only 1 free variable,"},{"Start":"08:44.210 ","End":"08:45.665","Text":"we let it be 1,"},{"Start":"08:45.665 ","End":"08:47.570","Text":"and if z is 1, from here,"},{"Start":"08:47.570 ","End":"08:49.940","Text":"we compute that y is minus 1,"},{"Start":"08:49.940 ","End":"08:51.770","Text":"and we put y is minus 1 here,"},{"Start":"08:51.770 ","End":"08:53.510","Text":"you get x equals 1."},{"Start":"08:53.510 ","End":"08:55.280","Text":"Now, just put them in the right order,"},{"Start":"08:55.280 ","End":"08:57.355","Text":"1 minus 1, 1,"},{"Start":"08:57.355 ","End":"09:01.775","Text":"and we get this vector which is a basis for"},{"Start":"09:01.775 ","End":"09:07.585","Text":"the solution space for our system of linear equations."},{"Start":"09:07.585 ","End":"09:12.424","Text":"That\u0027s an eigenvector corresponding to eigenvalue 0."},{"Start":"09:12.424 ","End":"09:14.705","Text":"That answers that part."},{"Start":"09:14.705 ","End":"09:18.845","Text":"We are done with part 1 of the exercise."},{"Start":"09:18.845 ","End":"09:25.430","Text":"Part 2 was to say whether the transformation T is diagonalizable,"},{"Start":"09:25.430 ","End":"09:33.860","Text":"and that\u0027s equivalent to asking if the representation matrix for it is diagonalizable."},{"Start":"09:33.860 ","End":"09:36.290","Text":"That could be relative to any basis,"},{"Start":"09:36.290 ","End":"09:37.790","Text":"including the standard basis."},{"Start":"09:37.790 ","End":"09:39.820","Text":"That\u0027s where A comes in."},{"Start":"09:39.820 ","End":"09:42.795","Text":"It represents the transformation."},{"Start":"09:42.795 ","End":"09:45.380","Text":"Now, we need this theorem we\u0027ve used before,"},{"Start":"09:45.380 ","End":"09:48.080","Text":"it talks generally about n by n matrices,"},{"Start":"09:48.080 ","End":"09:49.710","Text":"though, here, n will be 3."},{"Start":"09:49.710 ","End":"09:55.480","Text":"But it\u0027s diagonalizable if and only if it has n linearly independent eigenvectors."},{"Start":"09:55.480 ","End":"09:58.495","Text":"In our case, where n is 3,"},{"Start":"09:58.495 ","End":"10:00.380","Text":"we only have 1."},{"Start":"10:00.380 ","End":"10:03.005","Text":"We don\u0027t have 3 eigenvectors."},{"Start":"10:03.005 ","End":"10:05.150","Text":"Regardless of the linearly independent,"},{"Start":"10:05.150 ","End":"10:07.790","Text":"we just don\u0027t have 3 eigenvectors at all, we only have 1,"},{"Start":"10:07.790 ","End":"10:11.315","Text":"so it\u0027s not diagonalizable."},{"Start":"10:11.315 ","End":"10:14.010","Text":"We are done."}],"Thumbnail":null,"ID":14317},{"Watched":false,"Name":"Exercise 2","Duration":"4m 35s","ChapterTopicVideoID":13600,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"This exercise is a 2-in-1,"},{"Start":"00:04.065 ","End":"00:12.250","Text":"but in each part we have a transformation from R something to R something."},{"Start":"00:12.620 ","End":"00:17.055","Text":"In each case we have to represent"},{"Start":"00:17.055 ","End":"00:21.210","Text":"the transformation T as a matrix and we\u0027re"},{"Start":"00:21.210 ","End":"00:25.485","Text":"always going to use the standard basis of the appropriate R^n."},{"Start":"00:25.485 ","End":"00:27.240","Text":"R_2 has its standard basis,"},{"Start":"00:27.240 ","End":"00:28.890","Text":"R_3 has its standard basis,"},{"Start":"00:28.890 ","End":"00:31.335","Text":"R_4 has its standard basis."},{"Start":"00:31.335 ","End":"00:33.210","Text":"Also just to remark,"},{"Start":"00:33.210 ","End":"00:34.790","Text":"when we write this way,"},{"Start":"00:34.790 ","End":"00:36.980","Text":"it\u0027s for convenience as rows really,"},{"Start":"00:36.980 ","End":"00:38.400","Text":"if we spell it out,"},{"Start":"00:38.400 ","End":"00:41.500","Text":"you would write them as column vectors."},{"Start":"00:41.500 ","End":"00:48.590","Text":"Let\u0027s get to part A. I\u0027ll just clear some space here."},{"Start":"00:48.590 ","End":"00:54.755","Text":"This I just copied from above and here are the bases."},{"Start":"00:54.755 ","End":"01:02.670","Text":"Remember that we had T was going from R_2 to R_3."},{"Start":"01:02.670 ","End":"01:06.000","Text":"The standard basis for R_2 is this 1,"},{"Start":"01:06.000 ","End":"01:07.110","Text":"0 and 0, 1,"},{"Start":"01:07.110 ","End":"01:09.555","Text":"let\u0027s call it B_1, the basis."},{"Start":"01:09.555 ","End":"01:16.210","Text":"For R_3, these are the 3 basis elements and we\u0027ll call that B_2."},{"Start":"01:16.210 ","End":"01:19.179","Text":"Now using this formula twice,"},{"Start":"01:19.179 ","End":"01:24.250","Text":"I\u0027m going to apply it to each of the basis vectors."},{"Start":"01:24.250 ","End":"01:27.310","Text":"T of 1, 0, just plug it in."},{"Start":"01:27.310 ","End":"01:31.215","Text":"X plus y is 1,"},{"Start":"01:31.215 ","End":"01:39.300","Text":"y is 0, minus x is minus 1 and similarly for the other 1."},{"Start":"01:39.300 ","End":"01:43.285","Text":"Now, I want to express each of these as a linear combination"},{"Start":"01:43.285 ","End":"01:47.120","Text":"of the basis vectors but this basis,"},{"Start":"01:47.120 ","End":"01:48.880","Text":"we\u0027re now in R_3."},{"Start":"01:48.880 ","End":"01:51.310","Text":"Well its a standard basis, it\u0027s easy."},{"Start":"01:51.310 ","End":"01:53.330","Text":"We just copy these numbers, 1,"},{"Start":"01:53.330 ","End":"01:58.915","Text":"0 and minus 1 in front of each basis vector respectively."},{"Start":"01:58.915 ","End":"02:02.490","Text":"The other 1 same thing, the 1, 1,"},{"Start":"02:02.490 ","End":"02:06.495","Text":"0 so here\u0027s 1, 1, and 0."},{"Start":"02:06.495 ","End":"02:11.300","Text":"Now that we have these coefficients,"},{"Start":"02:11.300 ","End":"02:13.324","Text":"the ones that are in blue,"},{"Start":"02:13.324 ","End":"02:18.980","Text":"we just have to put those into a matrix but to transpose it."},{"Start":"02:18.980 ","End":"02:21.590","Text":"The rows become the columns,"},{"Start":"02:21.590 ","End":"02:23.045","Text":"1, 0 minus 1,"},{"Start":"02:23.045 ","End":"02:24.770","Text":"1, 0 minus 1, 1,"},{"Start":"02:24.770 ","End":"02:27.065","Text":"1, 0 is 1, 1, 0."},{"Start":"02:27.065 ","End":"02:29.930","Text":"That\u0027s the representation of T,"},{"Start":"02:29.930 ","End":"02:31.865","Text":"but to be more precise,"},{"Start":"02:31.865 ","End":"02:36.485","Text":"we might write it fully as transformation matrix of"},{"Start":"02:36.485 ","End":"02:43.240","Text":"T from basis B_1 to bases B_2."},{"Start":"02:44.020 ","End":"02:46.850","Text":"When the bases are clear,"},{"Start":"02:46.850 ","End":"02:49.680","Text":"we just can write it like this."},{"Start":"02:49.820 ","End":"02:53.685","Text":"That was part A, let\u0027s go on to part B."},{"Start":"02:53.685 ","End":"03:02.575","Text":"Here we are in part B and this was a T from R_4 to R_2 but this was the formula."},{"Start":"03:02.575 ","End":"03:06.200","Text":"Here we have the standard basis for R_4,"},{"Start":"03:06.200 ","End":"03:08.990","Text":"the standard basis for R_2."},{"Start":"03:08.990 ","End":"03:14.420","Text":"Now what we want to do is apply T to each of the vectors in"},{"Start":"03:14.420 ","End":"03:21.070","Text":"B_1 and then express them as linear combinations of vectors in B_2."},{"Start":"03:21.070 ","End":"03:22.755","Text":"Like the first 1,"},{"Start":"03:22.755 ","End":"03:24.070","Text":"1, 0, 0, 0,"},{"Start":"03:24.070 ","End":"03:29.270","Text":"we apply T to it according to this formula using these computations"},{"Start":"03:29.270 ","End":"03:35.530","Text":"and 4x minus y minus z plus t comes out to 4,"},{"Start":"03:35.530 ","End":"03:39.275","Text":"and the other 1 comes out to 1 and then we break it up."},{"Start":"03:39.275 ","End":"03:45.180","Text":"The 4, 1 just becomes 4 times this basis vector plus 1 times this."},{"Start":"03:45.950 ","End":"03:48.200","Text":"Here is the rest of them."},{"Start":"03:48.200 ","End":"03:51.350","Text":"You think you\u0027ve got the idea afterwards, you can check."},{"Start":"03:51.350 ","End":"03:56.280","Text":"In each case we apply T to the appropriate basis vector."},{"Start":"03:56.280 ","End":"03:58.190","Text":"Using this formula we got this,"},{"Start":"03:58.190 ","End":"03:59.315","Text":"this, this and this,"},{"Start":"03:59.315 ","End":"04:00.470","Text":"and then you break them up."},{"Start":"04:00.470 ","End":"04:02.240","Text":"You can see here, the minus 1,"},{"Start":"04:02.240 ","End":"04:04.735","Text":"4 is the minus 1, 4."},{"Start":"04:04.735 ","End":"04:09.515","Text":"Then at the end we take all these coefficients."},{"Start":"04:09.515 ","End":"04:11.345","Text":"There\u0027s 4 times 2 of them,"},{"Start":"04:11.345 ","End":"04:14.555","Text":"but you have to remember to do the transpose."},{"Start":"04:14.555 ","End":"04:16.520","Text":"It\u0027s not 4 by 2 matrix,"},{"Start":"04:16.520 ","End":"04:17.780","Text":"it\u0027s a 2 by 4 matrix."},{"Start":"04:17.780 ","End":"04:19.400","Text":"The 4, 1 is the first column,"},{"Start":"04:19.400 ","End":"04:21.545","Text":"minus 1, 1 second column."},{"Start":"04:21.545 ","End":"04:26.824","Text":"That gives us the matrix representation of T. To be precise,"},{"Start":"04:26.824 ","End":"04:28.550","Text":"we would write it this way,"},{"Start":"04:28.550 ","End":"04:31.670","Text":"although in most cases you know what the basis is and you just write it"},{"Start":"04:31.670 ","End":"04:36.030","Text":"simply like this. We\u0027re done."}],"Thumbnail":null,"ID":14318},{"Watched":false,"Name":"Exercise 3","Duration":"5m 8s","ChapterTopicVideoID":13601,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this exercise,"},{"Start":"00:02.160 ","End":"00:05.205","Text":"we have a linear transformation T,"},{"Start":"00:05.205 ","End":"00:09.570","Text":"and it\u0027s from R^3 to R^2."},{"Start":"00:09.570 ","End":"00:11.715","Text":"It\u0027s defined as follows."},{"Start":"00:11.715 ","End":"00:13.500","Text":"T of x,y,z."},{"Start":"00:13.500 ","End":"00:15.060","Text":"x,y,z is in R^3,"},{"Start":"00:15.060 ","End":"00:16.815","Text":"and we need to give 2 components."},{"Start":"00:16.815 ","End":"00:18.585","Text":"There\u0027s the 1st component,"},{"Start":"00:18.585 ","End":"00:21.435","Text":"there\u0027s the 3rd component, the formula."},{"Start":"00:21.435 ","End":"00:26.430","Text":"We want to compute the matrix of T,"},{"Start":"00:26.430 ","End":"00:31.290","Text":"the representation that has to be done relative to basis."},{"Start":"00:31.290 ","End":"00:33.810","Text":"For this R^3,"},{"Start":"00:33.810 ","End":"00:36.330","Text":"we take this basis,"},{"Start":"00:36.330 ","End":"00:40.905","Text":"and for R^2, we\u0027ll take this as a basis."},{"Start":"00:40.905 ","End":"00:43.925","Text":"You just take it on trust that these really are"},{"Start":"00:43.925 ","End":"00:46.625","Text":"basis they span, they\u0027re linearly independent."},{"Start":"00:46.625 ","End":"00:49.265","Text":"We\u0027re not going to spend time checking that."},{"Start":"00:49.265 ","End":"00:52.670","Text":"What we have to find is that matrix,"},{"Start":"00:52.670 ","End":"00:57.680","Text":"and the notation is the square brackets means representation"},{"Start":"00:57.680 ","End":"01:02.900","Text":"of T as a matrix in relative to the first base in the source,"},{"Start":"01:02.900 ","End":"01:06.335","Text":"and second base in the target."},{"Start":"01:06.335 ","End":"01:09.625","Text":"Also, I like to remark that,"},{"Start":"01:09.625 ","End":"01:13.610","Text":"really this thing should be written this way,"},{"Start":"01:13.610 ","End":"01:18.550","Text":"we should be working with column vectors."},{"Start":"01:18.620 ","End":"01:23.220","Text":"We just flatten them out for convenience."},{"Start":"01:23.530 ","End":"01:31.065","Text":"That\u0027s quite an involved question exercise,"},{"Start":"01:31.065 ","End":"01:33.030","Text":"and let\u0027s start solving it."},{"Start":"01:33.030 ","End":"01:36.230","Text":"To summarize the details we have so far,"},{"Start":"01:36.230 ","End":"01:39.650","Text":"we have T from R^3 to R^2,"},{"Start":"01:39.650 ","End":"01:41.225","Text":"and this is the formula,"},{"Start":"01:41.225 ","End":"01:42.710","Text":"and these are the 2 basis,"},{"Start":"01:42.710 ","End":"01:44.724","Text":"and we want to find the matrix."},{"Start":"01:44.724 ","End":"01:46.745","Text":"Here\u0027s how we go about it."},{"Start":"01:46.745 ","End":"01:51.990","Text":"We take the target space, the R^2."},{"Start":"01:56.930 ","End":"02:03.215","Text":"We have to find the coordinate vector relative to this basis."},{"Start":"02:03.215 ","End":"02:05.780","Text":"In other words, what is a typical vector,"},{"Start":"02:05.780 ","End":"02:09.845","Text":"x,y in terms of this basis B_2?"},{"Start":"02:09.845 ","End":"02:14.360","Text":"We do that by putting these as column vectors 1, 4 and 1,"},{"Start":"02:14.360 ","End":"02:18.315","Text":"5 separator then x,y,"},{"Start":"02:18.315 ","End":"02:24.935","Text":"and the task is to do row operations to bring this to the identity matrix."},{"Start":"02:24.935 ","End":"02:30.215","Text":"Subtract 4 times the top row from the bottom row."},{"Start":"02:30.215 ","End":"02:32.350","Text":"This is what we get."},{"Start":"02:32.350 ","End":"02:38.120","Text":"Then we subtract the second row from the first row,"},{"Start":"02:38.120 ","End":"02:39.725","Text":"and that gives us this."},{"Start":"02:39.725 ","End":"02:43.530","Text":"Here we have the identity matrix."},{"Start":"02:43.790 ","End":"02:47.299","Text":"These 2 quantities expressions,"},{"Start":"02:47.299 ","End":"02:51.710","Text":"they are the coordinates of x,y,"},{"Start":"02:51.710 ","End":"02:55.200","Text":"relative to the basis B_2."},{"Start":"02:55.200 ","End":"02:59.820","Text":"Each x,y will be 5x minus y times the 1,"},{"Start":"02:59.820 ","End":"03:04.900","Text":"4 plus y minus 4x times 1, 5."},{"Start":"03:05.120 ","End":"03:09.410","Text":"Sometimes it\u0027s a good idea to actually multiply this out and check,"},{"Start":"03:09.410 ","End":"03:11.480","Text":"but we\u0027ll assume that this is right."},{"Start":"03:11.480 ","End":"03:16.400","Text":"Next, we\u0027re going to do some work on each of these 3 vectors from the source."},{"Start":"03:16.400 ","End":"03:18.500","Text":"We\u0027ll take them 1 at a time."},{"Start":"03:18.500 ","End":"03:23.855","Text":"T, the first basis vector, 1, 1, 0."},{"Start":"03:23.855 ","End":"03:29.165","Text":"We use this formula for x plus y minus z comes out 5,"},{"Start":"03:29.165 ","End":"03:33.395","Text":"and x minus y plus z comes out 0."},{"Start":"03:33.395 ","End":"03:35.450","Text":"Now we\u0027re in R^2 space."},{"Start":"03:35.450 ","End":"03:39.200","Text":"Now we want to write this in terms of the basis B_2."},{"Start":"03:39.200 ","End":"03:49.035","Text":"We use this formula and we get 5x minus y is 5 times 5 minus 0 is 25,"},{"Start":"03:49.035 ","End":"03:55.890","Text":"and y minus 4x is 0 minus 4 times 5, so it\u0027s minus 20."},{"Start":"03:55.890 ","End":"03:58.635","Text":"These are the numbers we need."},{"Start":"03:58.635 ","End":"04:02.085","Text":"Now we do the other 2."},{"Start":"04:02.085 ","End":"04:04.700","Text":"Similarly, each in 2 steps."},{"Start":"04:04.700 ","End":"04:06.500","Text":"We first compute T,"},{"Start":"04:06.500 ","End":"04:10.700","Text":"the transformation what it does to this basis vector,"},{"Start":"04:10.700 ","End":"04:18.675","Text":"and then we convert it in terms of the basis B_2,"},{"Start":"04:18.675 ","End":"04:21.570","Text":"and similarly for the last ones."},{"Start":"04:21.570 ","End":"04:22.995","Text":"That\u0027s all 3 of them."},{"Start":"04:22.995 ","End":"04:28.060","Text":"Now that we\u0027ve got all these numbers that I\u0027ve put in blue,"},{"Start":"04:28.060 ","End":"04:34.080","Text":"we just have to put these into a matrix."},{"Start":"04:34.080 ","End":"04:37.380","Text":"Transposed like the first row 25,"},{"Start":"04:37.380 ","End":"04:41.370","Text":"20, is this 25, 20."},{"Start":"04:41.370 ","End":"04:43.650","Text":"Like here we have 0 0,"},{"Start":"04:43.650 ","End":"04:45.570","Text":"here we have 0 0,"},{"Start":"04:45.570 ","End":"04:50.670","Text":"minus 6 plus 5 minus 6, 5."},{"Start":"04:50.670 ","End":"04:53.250","Text":"Basically this is the answer."},{"Start":"04:53.250 ","End":"04:56.240","Text":"Really it\u0027s more correct to write it like this,"},{"Start":"04:56.240 ","End":"04:58.070","Text":"but this is cumbersome notation."},{"Start":"04:58.070 ","End":"05:01.150","Text":"Usually we just write the representation of T,"},{"Start":"05:01.150 ","End":"05:04.665","Text":"and we understand what the 2 basis are."},{"Start":"05:04.665 ","End":"05:08.290","Text":"That\u0027s the answer and we\u0027re done."}],"Thumbnail":null,"ID":14319},{"Watched":false,"Name":"Exercise 4","Duration":"7m 13s","ChapterTopicVideoID":13602,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.764","Text":"In this exercise, we take a linear transformation from"},{"Start":"00:06.764 ","End":"00:15.360","Text":"P_4 over R to P_3 over R. It\u0027s defined as follows."},{"Start":"00:15.360 ","End":"00:18.075","Text":"Everything needs to be explained."},{"Start":"00:18.075 ","End":"00:20.970","Text":"What is P_4 over R?"},{"Start":"00:20.970 ","End":"00:30.675","Text":"It\u0027s the polynomials of degree up to and including 4 with real number coefficients."},{"Start":"00:30.675 ","End":"00:35.855","Text":"P_3 is polynomials up to and including degree 3."},{"Start":"00:35.855 ","End":"00:41.925","Text":"Notice that this transformation is from P_4 to P_3,"},{"Start":"00:41.925 ","End":"00:48.435","Text":"because the derivative lowers the degree."},{"Start":"00:48.435 ","End":"00:53.240","Text":"For our fourth degree polynomial or less than we differentiate,"},{"Start":"00:53.240 ","End":"00:56.190","Text":"it will be third-degree or less."},{"Start":"00:56.510 ","End":"01:00.395","Text":"Now you might wonder, why didn\u0027t I use the letter T?"},{"Start":"01:00.395 ","End":"01:03.530","Text":"Because it\u0027s customary for derivative to use the letter"},{"Start":"01:03.530 ","End":"01:07.830","Text":"D. You shouldn\u0027t get used to the letter T all the time."},{"Start":"01:10.640 ","End":"01:18.110","Text":"Our task is to find the matrix that represents this transformation relative"},{"Start":"01:18.110 ","End":"01:24.770","Text":"to the standard bases E. Let me just say 1 more thing to be pedantic,"},{"Start":"01:24.770 ","End":"01:27.670","Text":"I really should explain why it\u0027s linear."},{"Start":"01:27.670 ","End":"01:31.550","Text":"Basically, it\u0027s linear because the derivative of the sum of"},{"Start":"01:31.550 ","End":"01:35.315","Text":"2 functions is the sum of the derivatives."},{"Start":"01:35.315 ","End":"01:37.760","Text":"If I multiply a function by a constant,"},{"Start":"01:37.760 ","End":"01:44.315","Text":"then the derivative is also multiplied by a constant and functions can be polynomials."},{"Start":"01:44.315 ","End":"01:46.850","Text":"Now we\u0027re ready to start."},{"Start":"01:46.850 ","End":"01:50.820","Text":"Just copied some stuff because I\u0027m going to scroll."},{"Start":"01:50.980 ","End":"01:57.300","Text":"This exercise is a bit different and it will need some explaining."},{"Start":"01:57.520 ","End":"01:59.540","Text":"I won\u0027t just read it."},{"Start":"01:59.540 ","End":"02:00.950","Text":"I\u0027ll explain."},{"Start":"02:00.950 ","End":"02:04.005","Text":"We have 2 vector spaces here."},{"Start":"02:04.005 ","End":"02:06.360","Text":"We have P_4 and P_3,"},{"Start":"02:06.360 ","End":"02:08.880","Text":"both over the real numbers."},{"Start":"02:08.880 ","End":"02:12.800","Text":"P_4 is polynomials up to and including degree 4,"},{"Start":"02:12.800 ","End":"02:17.185","Text":"and P_3 is polynomials up to and including degree 3."},{"Start":"02:17.185 ","End":"02:21.770","Text":"Then we usually use the letter T for linear transformation."},{"Start":"02:21.770 ","End":"02:26.570","Text":"But here we\u0027re going to use the letter D. What the transformation"},{"Start":"02:26.570 ","End":"02:33.020","Text":"does is it takes a polynomial and sends it to its derivative."},{"Start":"02:33.020 ","End":"02:38.090","Text":"Now, notice that this makes sense because when you differentiate a polynomial,"},{"Start":"02:38.090 ","End":"02:41.040","Text":"its degree is lowered."},{"Start":"02:41.040 ","End":"02:43.219","Text":"A degree 4 polynomial,"},{"Start":"02:43.219 ","End":"02:47.640","Text":"if we differentiate it will be degree 3 or less."},{"Start":"02:48.070 ","End":"02:52.440","Text":"Also D stands for derivative, of course."},{"Start":"02:53.210 ","End":"02:55.650","Text":"That\u0027s the transformation."},{"Start":"02:55.650 ","End":"02:58.235","Text":"We want the matrix that represents it."},{"Start":"02:58.235 ","End":"03:00.620","Text":"But we have to give a basis for here and for here."},{"Start":"03:00.620 ","End":"03:06.250","Text":"We take the standard bases of P_4 and of P_3."},{"Start":"03:06.250 ","End":"03:09.695","Text":"Everything\u0027s over the real numbers, like I said."},{"Start":"03:09.695 ","End":"03:13.690","Text":"I just copied some stuff so I can scroll."},{"Start":"03:13.690 ","End":"03:17.259","Text":"Now to be pedantic,"},{"Start":"03:17.259 ","End":"03:21.080","Text":"I really should explain why D is linear."},{"Start":"03:21.570 ","End":"03:24.280","Text":"It\u0027s just a technicality,"},{"Start":"03:24.280 ","End":"03:27.265","Text":"but basically, it\u0027s because,"},{"Start":"03:27.265 ","End":"03:32.245","Text":"if we have p plus q and we take its derivative,"},{"Start":"03:32.245 ","End":"03:35.800","Text":"it\u0027s the same as the derivative of p plus the derivative of q."},{"Start":"03:35.800 ","End":"03:40.930","Text":"Let\u0027s say if I have a constant a times p polynomial,"},{"Start":"03:40.930 ","End":"03:44.695","Text":"take the derivative, it\u0027s a times p derivative."},{"Start":"03:44.695 ","End":"03:46.615","Text":"This is just the idea,"},{"Start":"03:46.615 ","End":"03:48.580","Text":"the proof that we have to put D in it."},{"Start":"03:48.580 ","End":"03:52.735","Text":"But you can accept the fact that it\u0027s linear because the derivative"},{"Start":"03:52.735 ","End":"03:58.380","Text":"preserve sums and multiplication with a constant, it\u0027s linear."},{"Start":"03:58.380 ","End":"04:02.330","Text":"I have to say that to be pedantic,"},{"Start":"04:02.330 ","End":"04:06.450","Text":"but it doesn\u0027t affect the solution."},{"Start":"04:07.280 ","End":"04:15.170","Text":"Now the matter of the standard bases. For P_4, polynomials up to and including degree 4,"},{"Start":"04:15.170 ","End":"04:17.105","Text":"this is the standard bases, 1,"},{"Start":"04:17.105 ","End":"04:18.140","Text":"x, x squared,"},{"Start":"04:18.140 ","End":"04:19.880","Text":"x cubed, x to the 4."},{"Start":"04:19.880 ","End":"04:21.920","Text":"Then of x I write x to the 1,"},{"Start":"04:21.920 ","End":"04:25.710","Text":"could have also written here x to the power of 0."},{"Start":"04:26.450 ","End":"04:29.175","Text":"I called it E_3."},{"Start":"04:29.175 ","End":"04:33.925","Text":"The standard bases for P_3 is similar,"},{"Start":"04:33.925 ","End":"04:37.050","Text":"just goes up to x cubed."},{"Start":"04:37.190 ","End":"04:45.815","Text":"What I\u0027m going to do now is apply D to each of the bases members."},{"Start":"04:45.815 ","End":"04:47.210","Text":"Starting with 1,"},{"Start":"04:47.210 ","End":"04:51.230","Text":"d of 1 is the derivative of 1 is 0."},{"Start":"04:51.230 ","End":"04:56.900","Text":"D of the polynomial x is the derivative of x is 1."},{"Start":"04:56.900 ","End":"04:58.880","Text":"Here\u0027s the rest of them."},{"Start":"04:58.880 ","End":"05:00.110","Text":"D of x squared,"},{"Start":"05:00.110 ","End":"05:02.045","Text":"derivative of x squared is 2x,"},{"Start":"05:02.045 ","End":"05:05.850","Text":"x cubed gets sent to 3x squared,"},{"Start":"05:05.850 ","End":"05:09.240","Text":"x fourth gets sent to 4x cubed."},{"Start":"05:09.240 ","End":"05:11.245","Text":"That\u0027s 1 part."},{"Start":"05:11.245 ","End":"05:15.500","Text":"Now what I have to do is take each of these and write"},{"Start":"05:15.500 ","End":"05:20.770","Text":"it in terms of the bases here in P_3."},{"Start":"05:21.890 ","End":"05:28.654","Text":"The polynomial 0 clearly is just 0 times each 1 of these."},{"Start":"05:28.654 ","End":"05:31.585","Text":"The polynomial 1."},{"Start":"05:31.585 ","End":"05:35.490","Text":"I take 1 of these and 0 each of these."},{"Start":"05:35.490 ","End":"05:44.199","Text":"You see am writing each 1 of these as a linear combination of the bases elements."},{"Start":"05:44.199 ","End":"05:50.265","Text":"For 2x, I just take 2 of the X and the rest of them as 0."},{"Start":"05:50.265 ","End":"05:55.710","Text":"Similarly, 3x squared means I put a 3 in front of the x squared and 0 elsewhere,"},{"Start":"05:55.710 ","End":"05:58.870","Text":"and 4x cubed means I get a 4 here."},{"Start":"05:59.030 ","End":"06:01.580","Text":"Now what I have to do,"},{"Start":"06:01.580 ","End":"06:06.020","Text":"the last step is to take these numbers that are"},{"Start":"06:06.020 ","End":"06:11.465","Text":"in this color is blue and put them into a matrix,"},{"Start":"06:11.465 ","End":"06:15.050","Text":"but it needs to be transposed."},{"Start":"06:15.050 ","End":"06:17.870","Text":"Try not to forget the transpose."},{"Start":"06:17.870 ","End":"06:19.610","Text":"Look this first row,"},{"Start":"06:19.610 ","End":"06:24.304","Text":"the coefficients, these, and that would be the first column."},{"Start":"06:24.304 ","End":"06:27.245","Text":"The second row here, here, here,"},{"Start":"06:27.245 ","End":"06:31.180","Text":"and here, that would correspond to this column."},{"Start":"06:31.180 ","End":"06:33.360","Text":"Similarly for the other 3,"},{"Start":"06:33.360 ","End":"06:35.910","Text":"the rows become columns here with you 0,"},{"Start":"06:35.910 ","End":"06:37.920","Text":"2, 0, 0, 0, 2,"},{"Start":"06:37.920 ","End":"06:41.280","Text":"0, 0 that\u0027s the way it works."},{"Start":"06:41.280 ","End":"06:45.920","Text":"1 last thing I simplified the notation."},{"Start":"06:45.920 ","End":"06:48.965","Text":"I really should mention the bases."},{"Start":"06:48.965 ","End":"06:55.040","Text":"What we really would write would be the representation of D as"},{"Start":"06:55.040 ","End":"07:02.750","Text":"a matrix with respect to the bases E_4 and the bases E_3."},{"Start":"07:02.750 ","End":"07:04.460","Text":"We name the 2 bases,"},{"Start":"07:04.460 ","End":"07:08.780","Text":"the 1 that\u0027s going from on the bottom and the 1 you\u0027re going to at the top."},{"Start":"07:08.780 ","End":"07:10.325","Text":"But that\u0027s technical."},{"Start":"07:10.325 ","End":"07:14.610","Text":"This is the answer. We\u0027re done."}],"Thumbnail":null,"ID":14320},{"Watched":false,"Name":"Exercise 5","Duration":"8m 50s","ChapterTopicVideoID":13603,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:08.415","Text":"In this exercise, we are dealing with the space M_2 over R,"},{"Start":"00:08.415 ","End":"00:16.365","Text":"which means 2 by 2 matrices with entries that are all real numbers."},{"Start":"00:16.365 ","End":"00:20.340","Text":"We saw earlier that this is a vector space so"},{"Start":"00:20.340 ","End":"00:24.209","Text":"the matrices can be also looked upon as vectors."},{"Start":"00:24.209 ","End":"00:31.235","Text":"Now we\u0027re going to define a linear transformation from this space to itself."},{"Start":"00:31.235 ","End":"00:33.995","Text":"We could call that an operator."},{"Start":"00:33.995 ","End":"00:36.665","Text":"It\u0027s defined as follows,"},{"Start":"00:36.665 ","End":"00:38.585","Text":"A is the variable here."},{"Start":"00:38.585 ","End":"00:47.980","Text":"What T does to A is it multiplies A on the left by this fixed matrix 1, 2, 3, 4."},{"Start":"00:47.980 ","End":"00:54.020","Text":"By analogy, I mean I could define a function f from say,"},{"Start":"00:54.020 ","End":"00:57.050","Text":"the reals to the reals."},{"Start":"00:57.050 ","End":"01:03.150","Text":"I could say f of x equals 3x for example,"},{"Start":"01:03.150 ","End":"01:04.590","Text":"and that 3 is the constant."},{"Start":"01:04.590 ","End":"01:06.380","Text":"It\u0027s like this matrix here."},{"Start":"01:06.380 ","End":"01:10.945","Text":"Every matrix A gets multiplied on the left by this."},{"Start":"01:10.945 ","End":"01:16.550","Text":"It\u0027s not hard to show that this is indeed a linear transformation."},{"Start":"01:16.550 ","End":"01:21.015","Text":"Then we\u0027re given a basis B,"},{"Start":"01:21.015 ","End":"01:26.410","Text":"which is a set of 4 matrices"},{"Start":"01:26.410 ","End":"01:31.485","Text":"which can be looked upon as vectors also and they are a basis."},{"Start":"01:31.485 ","End":"01:41.149","Text":"We want the matrix that represents the transformation T with respect to the basis B."},{"Start":"01:41.149 ","End":"01:45.880","Text":"Now what we want to do is when we look upon a matrix as a vector,"},{"Start":"01:45.880 ","End":"01:47.590","Text":"we flatten it out."},{"Start":"01:47.590 ","End":"01:49.794","Text":"We\u0027ve done this before."},{"Start":"01:49.794 ","End":"01:51.700","Text":"I call it a snake."},{"Start":"01:51.700 ","End":"01:54.360","Text":"We do it like this,"},{"Start":"01:54.360 ","End":"01:56.000","Text":"we take them in a certain order,"},{"Start":"01:56.000 ","End":"01:58.655","Text":"1, 1, 0, 0."},{"Start":"01:58.655 ","End":"02:00.590","Text":"You could think of this as the letter Z,"},{"Start":"02:00.590 ","End":"02:04.835","Text":"but a snake is a better analogy because in 3 by 3 matrices,"},{"Start":"02:04.835 ","End":"02:09.860","Text":"you would have something more like that."},{"Start":"02:09.860 ","End":"02:17.230","Text":"Anyway, you go from left to right and the rows go from top to bottom."},{"Start":"02:18.020 ","End":"02:21.840","Text":"This 1 would be 0, 1, 1, 0,"},{"Start":"02:21.840 ","End":"02:23.700","Text":"0, 0, 1, 1,"},{"Start":"02:23.700 ","End":"02:25.425","Text":"0, 0, 0, 1."},{"Start":"02:25.425 ","End":"02:31.590","Text":"Now we place these in a matrix but they go in vertically."},{"Start":"02:31.590 ","End":"02:32.970","Text":"I mean like this 1, 1,"},{"Start":"02:32.970 ","End":"02:35.505","Text":"0, 0 is this."},{"Start":"02:35.505 ","End":"02:39.285","Text":"Zero, 1, 1, 0 is this 1."},{"Start":"02:39.285 ","End":"02:41.010","Text":"Then 0, 0, 1,"},{"Start":"02:41.010 ","End":"02:42.450","Text":"1, this 1 0,"},{"Start":"02:42.450 ","End":"02:45.645","Text":"0, 0, 1 is this 1."},{"Start":"02:45.645 ","End":"02:51.870","Text":"They go in vertically and then we put a separator."},{"Start":"02:51.870 ","End":"02:54.015","Text":"It\u0027s an augmented matrix then."},{"Start":"02:54.015 ","End":"02:57.660","Text":"For 4 variables we use x, y,"},{"Start":"02:57.660 ","End":"03:01.080","Text":"z, and t. I don\u0027t think you\u0027ll get more than 4."},{"Start":"03:01.080 ","End":"03:04.240","Text":"Otherwise you just have to keep using letters."},{"Start":"03:04.730 ","End":"03:07.245","Text":"What we want to do with this,"},{"Start":"03:07.245 ","End":"03:13.445","Text":"the technique is to do row operations on it until we get"},{"Start":"03:13.445 ","End":"03:22.985","Text":"the identity matrix on the restricted matrix path just to the left of the separator."},{"Start":"03:22.985 ","End":"03:26.740","Text":"The first thing I want to do is I want to have all 0s under this 1."},{"Start":"03:26.740 ","End":"03:29.435","Text":"It\u0027s just a matter of this second row."},{"Start":"03:29.435 ","End":"03:32.670","Text":"I\u0027ll subtract the first 1 from it."},{"Start":"03:32.930 ","End":"03:42.105","Text":"Then I get this here by subtracting."},{"Start":"03:42.105 ","End":"03:45.530","Text":"The only thing that changes is that this becomes a 0 and"},{"Start":"03:45.530 ","End":"03:49.750","Text":"also here we subtract the x from the y."},{"Start":"03:49.750 ","End":"03:56.425","Text":"Then here I subtracted the second row from the third row to get a 0 here."},{"Start":"03:56.425 ","End":"04:04.510","Text":"Also if you subtract z less y minus x it\u0027s z minus y plus x."},{"Start":"04:05.000 ","End":"04:08.175","Text":"Let\u0027s get some space."},{"Start":"04:08.175 ","End":"04:13.300","Text":"Then the last step was to subtract"},{"Start":"04:13.580 ","End":"04:20.820","Text":"this row from this row to get the 0 here and this is what we get,"},{"Start":"04:20.820 ","End":"04:24.465","Text":"t minus this and here we have a 0."},{"Start":"04:24.465 ","End":"04:28.235","Text":"Now we have the identity matrix here."},{"Start":"04:28.235 ","End":"04:35.930","Text":"What we do with these coefficients is that these become"},{"Start":"04:35.930 ","End":"04:44.490","Text":"the coordinate vector relative to the basis."},{"Start":"04:44.490 ","End":"04:47.580","Text":"V here is just x, y, z,"},{"Start":"04:47.580 ","End":"04:55.175","Text":"t. What it means is that in general,"},{"Start":"04:55.175 ","End":"04:59.090","Text":"any x, y, z, t,"},{"Start":"04:59.090 ","End":"05:02.570","Text":"any matrix 2-by-2 can be broken"},{"Start":"05:02.570 ","End":"05:09.500","Text":"up as a linear combination of the 4 basis vectors,"},{"Start":"05:09.500 ","End":"05:17.945","Text":"matrices and the coefficients or coordinates or whatever you want to call these,"},{"Start":"05:17.945 ","End":"05:19.775","Text":"are just taken from here."},{"Start":"05:19.775 ","End":"05:23.945","Text":"Get x here, y minus x here,"},{"Start":"05:23.945 ","End":"05:25.910","Text":"z minus y plus x here,"},{"Start":"05:25.910 ","End":"05:28.415","Text":"t minus z and plus y minus x here."},{"Start":"05:28.415 ","End":"05:33.560","Text":"This breaks up this as a linear combination of the basis vectors."},{"Start":"05:33.560 ","End":"05:35.750","Text":"If you multiplied this out,"},{"Start":"05:35.750 ","End":"05:38.915","Text":"it would verify that you really do get this."},{"Start":"05:38.915 ","End":"05:43.080","Text":"I want to go to a new page. Here we are."},{"Start":"05:43.080 ","End":"05:46.380","Text":"The result, we just got I put it in a box,"},{"Start":"05:46.380 ","End":"05:51.180","Text":"put some colors and bold and I also copy"},{"Start":"05:51.180 ","End":"06:01.380","Text":"the definition of T and the basis vectors matrices."},{"Start":"06:02.870 ","End":"06:14.590","Text":"The next step is to take T and apply it to each of the 4 basis."},{"Start":"06:14.750 ","End":"06:17.010","Text":"As I said you can call them vectors,"},{"Start":"06:17.010 ","End":"06:19.080","Text":"you can call them matrices."},{"Start":"06:19.080 ","End":"06:23.020","Text":"Let\u0027s do these 4 computations."},{"Start":"06:23.340 ","End":"06:30.070","Text":"Now we just apply the definition of T. What T does to A,"},{"Start":"06:30.070 ","End":"06:31.645","Text":"let\u0027s call it a matrix,"},{"Start":"06:31.645 ","End":"06:35.710","Text":"is to multiply it on the left by this 1, 2, 3, 4."},{"Start":"06:35.710 ","End":"06:37.210","Text":"I put 1, 2, 3,"},{"Start":"06:37.210 ","End":"06:38.890","Text":"4, 1, 2, 3, 4, 1, 2, 3,"},{"Start":"06:38.890 ","End":"06:42.070","Text":"4, 1, 2, 3, 4 in front of each of these."},{"Start":"06:42.070 ","End":"06:44.650","Text":"Now I have to do the multiplications,"},{"Start":"06:44.650 ","End":"06:46.700","Text":"there\u0027s 4 of them."},{"Start":"06:47.090 ","End":"06:50.900","Text":"These are the results of the multiplication."},{"Start":"06:50.900 ","End":"06:53.030","Text":"By the way, you could work differently."},{"Start":"06:53.030 ","End":"06:56.990","Text":"You could work and do each row separately,"},{"Start":"06:56.990 ","End":"06:59.960","Text":"work on each basis element."},{"Start":"06:59.960 ","End":"07:02.420","Text":"I just prefer to work in parallel,"},{"Start":"07:02.420 ","End":"07:04.430","Text":"do the same operation,"},{"Start":"07:04.430 ","End":"07:07.470","Text":"more like mass production."},{"Start":"07:08.180 ","End":"07:10.290","Text":"We\u0027ve got these 4."},{"Start":"07:10.290 ","End":"07:15.560","Text":"Now we\u0027re going to use what\u0027s in the box here to break each of these"},{"Start":"07:15.560 ","End":"07:22.920","Text":"up as a linear combination of the basis matrices, vectors."},{"Start":"07:23.330 ","End":"07:25.820","Text":"The first 1, for example,"},{"Start":"07:25.820 ","End":"07:28.100","Text":"we just look at what it says here."},{"Start":"07:28.100 ","End":"07:30.815","Text":"We have x then y minus x."},{"Start":"07:30.815 ","End":"07:34.670","Text":"Here x is 1, y minus x is"},{"Start":"07:34.670 ","End":"07:41.825","Text":"0 and z minus y plus x is this minus this plus this is 3."},{"Start":"07:41.825 ","End":"07:48.335","Text":"Then t minus z plus y minus x comes out 0."},{"Start":"07:48.335 ","End":"07:51.000","Text":"This is just in order, x,"},{"Start":"07:51.000 ","End":"07:52.125","Text":"y, z, t,"},{"Start":"07:52.125 ","End":"07:54.910","Text":"is following this recipe."},{"Start":"07:55.520 ","End":"07:58.185","Text":"Here are the other 3."},{"Start":"07:58.185 ","End":"08:03.170","Text":"You can pause or freeze frame and check the computations."},{"Start":"08:03.170 ","End":"08:08.515","Text":"I\u0027m not going to go through that, it\u0027s just technical."},{"Start":"08:08.515 ","End":"08:11.500","Text":"Now we can have"},{"Start":"08:14.210 ","End":"08:19.910","Text":"the matrix representation of T relative to the basis."},{"Start":"08:19.910 ","End":"08:24.440","Text":"We just take all these coefficients but transpose them."},{"Start":"08:24.440 ","End":"08:27.875","Text":"Look here, I have a row 1, 0, 3, 0."},{"Start":"08:27.875 ","End":"08:30.650","Text":"This is column 1, 0, 3, 0."},{"Start":"08:30.650 ","End":"08:32.060","Text":"Or you could look at it the other way."},{"Start":"08:32.060 ","End":"08:33.800","Text":"If I have a column 1, 2, 2,"},{"Start":"08:33.800 ","End":"08:36.695","Text":"0, that would become a row 1, 2, 2, 0."},{"Start":"08:36.695 ","End":"08:40.049","Text":"Transpose, we switch rows and columns."},{"Start":"08:40.570 ","End":"08:43.850","Text":"This is the representation of"},{"Start":"08:43.850 ","End":"08:51.150","Text":"T. That was what was required of us to find and so we are done."}],"Thumbnail":null,"ID":14321},{"Watched":false,"Name":"Exercise 6","Duration":"6m 24s","ChapterTopicVideoID":13604,"CourseChapterTopicPlaylistID":7319,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:05.160","Text":"In this exercise, we have the vector base R^3 and"},{"Start":"00:05.160 ","End":"00:09.675","Text":"we have 2 different bases, B_1 and B_2."},{"Start":"00:09.675 ","End":"00:14.115","Text":"We have T, which is a linear operator on R^3."},{"Start":"00:14.115 ","End":"00:17.145","Text":"In case you forgot what linear operator means,"},{"Start":"00:17.145 ","End":"00:25.215","Text":"it just means that it\u0027s a linear transformation from R^3 to itself, R^3 to R^3."},{"Start":"00:25.215 ","End":"00:28.380","Text":"Now we\u0027re given 2 things."},{"Start":"00:28.380 ","End":"00:33.390","Text":"We\u0027re given the change of basis matrix from B_1 to B_2 as this."},{"Start":"00:33.390 ","End":"00:41.540","Text":"We\u0027re given the representation matrix of T relative to basis B_1."},{"Start":"00:41.540 ","End":"00:45.320","Text":"We have to compute these 2 quantities."},{"Start":"00:45.320 ","End":"00:48.660","Text":"Let\u0027s take them 1 at a time."},{"Start":"00:49.640 ","End":"00:56.805","Text":"Here we are with the change of basis from B_1 to B_2."},{"Start":"00:56.805 ","End":"00:59.330","Text":"You just have to remember the formula."},{"Start":"00:59.330 ","End":"01:05.785","Text":"We\u0027ve done this in the previous exercise that if you do the opposite from B_2 to B_1,"},{"Start":"01:05.785 ","End":"01:10.564","Text":"all you have to do is take the inverse of this matrix."},{"Start":"01:10.564 ","End":"01:13.399","Text":"This will always be an invertible matrix."},{"Start":"01:13.399 ","End":"01:18.700","Text":"You have to remember how to compute inverse matrices."},{"Start":"01:18.700 ","End":"01:24.955","Text":"Just let me write that this is equal to this minus 1 means inverse."},{"Start":"01:24.955 ","End":"01:28.595","Text":"The standard technique is to take this matrix,"},{"Start":"01:28.595 ","End":"01:32.930","Text":"and alongside it we put the identity matrix with a separator,"},{"Start":"01:32.930 ","End":"01:35.095","Text":"a kind of augmented matrix."},{"Start":"01:35.095 ","End":"01:37.045","Text":"What we want to do,"},{"Start":"01:37.045 ","End":"01:44.810","Text":"is to do row operations until on the left side we get the identity,"},{"Start":"01:44.810 ","End":"01:51.225","Text":"and what we have left on the right will be the inverse."},{"Start":"01:51.225 ","End":"01:58.780","Text":"Let\u0027s start. We can get these 2 to be 0 by adding this row to this and this."},{"Start":"01:58.780 ","End":"02:03.380","Text":"Adding this to this and this does give us 0s here and this is what we get."},{"Start":"02:03.380 ","End":"02:07.940","Text":"Don\u0027t forget, we also have to do the right-hand part, of course."},{"Start":"02:07.940 ","End":"02:14.000","Text":"I\u0027m adding this to this and to this that\u0027s like the ones here and so on."},{"Start":"02:14.000 ","End":"02:16.820","Text":"Let\u0027s see what\u0027s next."},{"Start":"02:16.820 ","End":"02:19.370","Text":"We would like a 0 here,"},{"Start":"02:19.370 ","End":"02:22.550","Text":"then it will be in echelon form."},{"Start":"02:22.550 ","End":"02:29.330","Text":"Take 3 times this row and subtract 4 times this row,"},{"Start":"02:29.330 ","End":"02:32.210","Text":"then you\u0027ll get here 12 minus 12,"},{"Start":"02:32.210 ","End":"02:36.480","Text":"and that will be 0 or rather,"},{"Start":"02:36.480 ","End":"02:39.925","Text":"it\u0027s minus 12 plus 12 minus, minus."},{"Start":"02:39.925 ","End":"02:45.500","Text":"Here we are, but we don\u0027t stop when we have a row echelon form."},{"Start":"02:45.500 ","End":"02:48.880","Text":"We want to keep going until we have the identity here."},{"Start":"02:48.880 ","End":"02:52.425","Text":"Let\u0027s go for a 0 in this position."},{"Start":"02:52.425 ","End":"02:58.350","Text":"I\u0027m going to double the middle row and subtract the bottom row."},{"Start":"02:58.520 ","End":"03:00.815","Text":"That gives us this,"},{"Start":"03:00.815 ","End":"03:02.555","Text":"well, we\u0027re getting close."},{"Start":"03:02.555 ","End":"03:05.510","Text":"We want to get at least to a diagonal matrix,"},{"Start":"03:05.510 ","End":"03:08.170","Text":"we want to get rid of that minus 9."},{"Start":"03:08.170 ","End":"03:10.280","Text":"Here we are, move to a new page."},{"Start":"03:10.280 ","End":"03:12.050","Text":"Did I say get rid of the minus 9?"},{"Start":"03:12.050 ","End":"03:14.630","Text":"No, I mean just get rid of the 6."},{"Start":"03:14.630 ","End":"03:21.715","Text":"Let\u0027s take what, 4 times this row minus 6 times the last row."},{"Start":"03:21.715 ","End":"03:25.385","Text":"If we do that on the left and on the right,"},{"Start":"03:25.385 ","End":"03:30.335","Text":"then we get the 0 here. We\u0027re getting there."},{"Start":"03:30.335 ","End":"03:33.145","Text":"Got to get a 0 here and now."},{"Start":"03:33.145 ","End":"03:38.705","Text":"We can get that by taking this row and subtracting 6 times this row,"},{"Start":"03:38.705 ","End":"03:40.610","Text":"then we\u0027ll get minus 36,"},{"Start":"03:40.610 ","End":"03:42.740","Text":"less minus 36 will be 0."},{"Start":"03:42.740 ","End":"03:45.320","Text":"This is what we get. We\u0027re getting very close."},{"Start":"03:45.320 ","End":"03:46.760","Text":"Once we have a diagonal,"},{"Start":"03:46.760 ","End":"03:53.359","Text":"we just have to divide each row by the entry."},{"Start":"03:53.359 ","End":"03:55.520","Text":"I would divide this by minus 4,"},{"Start":"03:55.520 ","End":"03:58.445","Text":"this by minus 6 and this by 4."},{"Start":"03:58.445 ","End":"04:00.590","Text":"Well, instead of saying divide by 4,"},{"Start":"04:00.590 ","End":"04:02.825","Text":"I can say multiply by a 1/4, same thing."},{"Start":"04:02.825 ","End":"04:04.550","Text":"These are the operations."},{"Start":"04:04.550 ","End":"04:08.630","Text":"Now we get the identity matrix here,"},{"Start":"04:08.630 ","End":"04:13.825","Text":"which means that what we have here is our inverse."},{"Start":"04:13.825 ","End":"04:20.375","Text":"That inverse was precisely the change of basis matrix from B to B_1."},{"Start":"04:20.375 ","End":"04:24.350","Text":"That answers the first part of the question."},{"Start":"04:24.350 ","End":"04:33.470","Text":"The second part was to compute what is this transformation as a matrix relative to B_2,"},{"Start":"04:33.470 ","End":"04:37.050","Text":"the representation of T as a matrix."},{"Start":"04:37.490 ","End":"04:41.525","Text":"This is just a matter of getting the right formula."},{"Start":"04:41.525 ","End":"04:45.990","Text":"This was done in the previous exercise."},{"Start":"04:45.990 ","End":"04:48.020","Text":"I won\u0027t even read it out."},{"Start":"04:48.020 ","End":"04:51.765","Text":"Just you can see that what it is."},{"Start":"04:51.765 ","End":"04:53.750","Text":"We have all these quantities,"},{"Start":"04:53.750 ","End":"04:57.935","Text":"two of them are given and 1 of them we computed in the first part."},{"Start":"04:57.935 ","End":"05:04.325","Text":"This part here was what we just computed the moment ago, and that\u0027s this."},{"Start":"05:04.325 ","End":"05:11.075","Text":"The representation matrix of T relative to B_1 was given to us in the beginning as this."},{"Start":"05:11.075 ","End":"05:15.880","Text":"The reverse change of basis matrix from B_1 to B_2 this time"},{"Start":"05:15.880 ","End":"05:22.430","Text":"was also given in as part of the data of the question, That\u0027s this."},{"Start":"05:22.430 ","End":"05:25.880","Text":"We have all these 3 and what"},{"Start":"05:25.880 ","End":"05:30.200","Text":"remains to do is just multiply them together in the correct order,"},{"Start":"05:30.200 ","End":"05:32.640","Text":"and that\u0027s what we\u0027ll do."},{"Start":"05:35.210 ","End":"05:38.630","Text":"Here we are. They disappeared off screen,"},{"Start":"05:38.630 ","End":"05:45.170","Text":"but these were the 3 matrices that we had to multiply in this order."},{"Start":"05:45.170 ","End":"05:49.340","Text":"We can take whichever 2 we want to"},{"Start":"05:49.340 ","End":"05:52.850","Text":"multiply first and then multiply by the remaining 1."},{"Start":"05:52.850 ","End":"05:55.775","Text":"Either these 2 first or these 2 first."},{"Start":"05:55.775 ","End":"06:03.755","Text":"Doesn\u0027t much matter, but I multiply these 2 first and that\u0027s this first matrix here."},{"Start":"06:03.755 ","End":"06:05.730","Text":"I\u0027m not going to do all the computations,"},{"Start":"06:05.730 ","End":"06:07.460","Text":"you can just pause,"},{"Start":"06:07.460 ","End":"06:10.115","Text":"freeze-frame and verify this."},{"Start":"06:10.115 ","End":"06:11.510","Text":"Then the other 1 as it is."},{"Start":"06:11.510 ","End":"06:14.490","Text":"Now we have 1 more multiplication."},{"Start":"06:15.020 ","End":"06:17.540","Text":"Here\u0027s the result again,"},{"Start":"06:17.540 ","End":"06:19.580","Text":"you can check it on your own."},{"Start":"06:19.580 ","End":"06:21.304","Text":"This is the answer,"},{"Start":"06:21.304 ","End":"06:24.089","Text":"and that\u0027s the end of the question."}],"Thumbnail":null,"ID":14322}],"ID":7319}]

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