WEBVTT 00:00:07.170 --> 00:00:12.980 OK, this is the lecture on linear transformations. 00:00:12.980 --> 00:00:17.330 Actually, linear algebra courses used 00:00:17.330 --> 00:00:19.600 to begin with this lecture, so you 00:00:19.600 --> 00:00:22.590 could say I'm beginning this course again 00:00:22.590 --> 00:00:26.090 by talking about linear transformations. 00:00:30.090 --> 00:00:35.120 In a lot of courses, those come first before matrices. 00:00:35.120 --> 00:00:37.910 The idea of a linear transformation makes sense 00:00:37.910 --> 00:00:43.320 without a matrix, and physicists and other -- 00:00:43.320 --> 00:00:45.250 some people like it better that way. 00:00:45.250 --> 00:00:47.810 They don't like coordinates. 00:00:47.810 --> 00:00:50.380 They don't want those numbers. 00:00:50.380 --> 00:00:54.960 They want to see what's going on with the whole space. 00:00:54.960 --> 00:00:59.640 But, for most of us, in the end, if we're 00:00:59.640 --> 00:01:03.190 going to compute anything, we introduce coordinates, 00:01:03.190 --> 00:01:06.960 and then every linear transformation 00:01:06.960 --> 00:01:08.600 will lead us to a 00:01:08.600 --> 00:01:09.680 matrix. 00:01:09.680 --> 00:01:17.080 And then, to all the things that we've done about null space 00:01:17.080 --> 00:01:21.630 and row space, and determinant, and eigenvalues -- 00:01:21.630 --> 00:01:24.100 all will come from the matrix. 00:01:24.100 --> 00:01:26.590 But, behind it -- in other words, 00:01:26.590 --> 00:01:30.300 behind this is the idea of a linear transformation. 00:01:30.300 --> 00:01:32.746 Let me give an example of a linear transformation. 00:01:35.805 --> 00:01:36.305 So, example. 00:01:39.340 --> 00:01:41.600 Example one. 00:01:41.600 --> 00:01:42.145 A projection. 00:01:47.580 --> 00:01:53.520 I can describe a projection without telling you any matrix, 00:01:53.520 --> 00:01:55.190 anything about any matrix. 00:01:55.190 --> 00:01:58.050 I can describe a projection, say, 00:01:58.050 --> 00:02:02.960 this will be a linear transformation that takes, say, 00:02:02.960 --> 00:02:06.940 all of R^2, every vector in the plane, 00:02:06.940 --> 00:02:11.030 into a vector in the plane. 00:02:11.030 --> 00:02:17.770 And this is the way people describe, a mapping. 00:02:17.770 --> 00:02:21.230 It takes every vector, and so, by what rule? 00:02:21.230 --> 00:02:27.120 So, what's the rule, is, I take a -- so here's the plane, 00:02:27.120 --> 00:02:33.250 this is going to be my line, my line through my line, 00:02:33.250 --> 00:02:37.630 and I'm going to project every vector onto that line. 00:02:37.630 --> 00:02:42.310 So if I take a vector like b -- or let me call the vector v 00:02:42.310 --> 00:02:43.840 for the moment -- 00:02:43.840 --> 00:02:47.610 the projection -- the linear transformation is going 00:02:47.610 --> 00:02:51.800 to produce this vector as T(v). 00:02:51.800 --> 00:02:54.730 So T -- it's like a function. 00:02:54.730 --> 00:02:56.500 Exactly like a function. 00:02:56.500 --> 00:03:00.280 You give me an input, the transformation 00:03:00.280 --> 00:03:02.390 produces the output. 00:03:02.390 --> 00:03:07.840 So transformation, sometimes the word map, or mapping is used. 00:03:07.840 --> 00:03:12.980 A map between inputs and outputs. 00:03:12.980 --> 00:03:17.670 So this is one particular map, this is one example, 00:03:17.670 --> 00:03:20.690 a projection that takes every vector -- here, 00:03:20.690 --> 00:03:24.870 let me do another vector v, or let me do this vector w, 00:03:24.870 --> 00:03:26.701 what is T(w)? 00:03:26.701 --> 00:03:27.200 You see? 00:03:27.200 --> 00:03:29.530 There are no coordinates here. 00:03:29.530 --> 00:03:32.640 I've drawn those axes, but I'm sorry I drew them, 00:03:32.640 --> 00:03:35.190 I'm going to remove them, that's the whole point, 00:03:35.190 --> 00:03:41.040 is that we don't need axes, we just need -- so guts -- 00:03:41.040 --> 00:03:45.430 get it out of there, I'm not a physicist, 00:03:45.430 --> 00:03:47.290 so I draw those axes. 00:03:47.290 --> 00:03:52.610 So the input is w, the output of the projection 00:03:52.610 --> 00:03:56.990 is, project on that line, T(w). 00:03:56.990 --> 00:03:57.580 OK. 00:03:57.580 --> 00:04:01.900 Now, I could think of a lot of transformations T. 00:04:01.900 --> 00:04:04.360 But, in this linear algebra course, 00:04:04.360 --> 00:04:07.170 I want it to be a linear transformation. 00:04:07.170 --> 00:04:11.415 So here are the rules for a linear transformation. 00:04:14.940 --> 00:04:17.430 Here, see, exactly, the two operations 00:04:17.430 --> 00:04:21.120 that we can do on vectors, adding and multiplying 00:04:21.120 --> 00:04:25.140 by scalars, the transformation does something special 00:04:25.140 --> 00:04:27.330 with respect to those operations. 00:04:27.330 --> 00:04:33.460 So, for example, the projection is a linear transformation 00:04:33.460 --> 00:04:35.230 because -- 00:04:35.230 --> 00:04:38.070 for example, if I wanted to check that one, 00:04:38.070 --> 00:04:42.880 if I took v to be twice as long, the projection 00:04:42.880 --> 00:04:44.540 would be twice as long. 00:04:44.540 --> 00:04:48.210 If I took v to be minus -- 00:04:48.210 --> 00:04:51.130 if I changed from v to minus v, the projection 00:04:51.130 --> 00:04:52.780 would change to a minus. 00:04:52.780 --> 00:04:57.470 So c equal to two, c equal minus one, any c is OK. 00:04:57.470 --> 00:05:02.530 So you see that actually, those combine, I can combine those 00:05:02.530 --> 00:05:06.040 into one statement. 00:05:06.040 --> 00:05:11.020 What the transformation does to any linear combination, 00:05:11.020 --> 00:05:19.550 it must produce the same combination of T(v) and T(w). 00:05:19.550 --> 00:05:22.290 Let's think about some -- 00:05:22.290 --> 00:05:26.120 I mean, it's like, not hard to decide, 00:05:26.120 --> 00:05:31.090 is a transformation linear or is it not. 00:05:31.090 --> 00:05:37.150 Let me give you an example so you can tell me the answer. 00:05:37.150 --> 00:05:41.815 Suppose my transformation is -- here's another example two. 00:05:47.470 --> 00:05:49.545 Shift the whole plane. 00:05:56.860 --> 00:06:01.940 So here are all my vectors, my plane, and every vector 00:06:01.940 --> 00:06:06.000 v in the plane, I shift it over by, 00:06:06.000 --> 00:06:11.640 let's say, three by some vector v0. 00:06:11.640 --> 00:06:14.770 Shift whole plane by v0. 00:06:14.770 --> 00:06:18.370 So every vector in the plane -- 00:06:18.370 --> 00:06:22.830 this was v, T(v) will be v+v0. 00:06:22.830 --> 00:06:23.770 There's T(v). 00:06:23.770 --> 00:06:25.360 Here's v0. 00:06:25.360 --> 00:06:27.390 There's the typical v. 00:06:27.390 --> 00:06:30.110 And there's T(v). 00:06:30.110 --> 00:06:33.090 You see what this transformation does? 00:06:33.090 --> 00:06:37.140 Takes this vector and adds to it. 00:06:37.140 --> 00:06:39.980 Adds a fixed vector to it. 00:06:39.980 --> 00:06:43.770 Well, that seems like a pretty reasonable, 00:06:43.770 --> 00:06:48.500 simple transformation, but is it linear? 00:06:48.500 --> 00:06:52.040 The answer is no, it's not linear. 00:06:52.040 --> 00:06:55.010 Which law is broken? 00:06:55.010 --> 00:06:57.770 Maybe both laws are broken. 00:06:57.770 --> 00:06:58.390 Let's see. 00:07:00.930 --> 00:07:09.470 If I double the length of v, does the transformation produce 00:07:09.470 --> 00:07:12.070 something double -- do I double T(v)? 00:07:12.070 --> 00:07:14.010 No. 00:07:14.010 --> 00:07:16.960 If I double the length of v, in this transformation, 00:07:16.960 --> 00:07:21.720 I'm just adding on the same one -- same v0, not two v0s, 00:07:21.720 --> 00:07:24.660 but only one v0 for every vector, 00:07:24.660 --> 00:07:28.830 so I don't get two times the transform. 00:07:28.830 --> 00:07:30.480 Do you see what I'm saying? 00:07:30.480 --> 00:07:35.000 That if I double this, then the transformation 00:07:35.000 --> 00:07:40.360 starts there and only goes one v0 out and doesn't double T(v). 00:07:40.360 --> 00:07:47.030 In fact, a linear transformation -- what is T of 00:07:47.030 --> 00:07:48.060 zero? 00:07:48.060 --> 00:07:53.610 That's just like a special case, but really worth noticing. 00:07:53.610 --> 00:07:56.650 The zero vector in a linear transformation 00:07:56.650 --> 00:08:00.740 must get transformed to zero. 00:08:00.740 --> 00:08:10.670 It can't move, because, take any vector V here -- 00:08:10.670 --> 00:08:15.170 well, so you can see why T of zero is zero. 00:08:15.170 --> 00:08:20.640 Take v to be the zero vector, take c to be three. 00:08:20.640 --> 00:08:23.480 Then we'd have T of zero vector equaling 00:08:23.480 --> 00:08:29.131 three T of zero vector, the T of zero has to be zero. 00:08:29.131 --> 00:08:29.630 OK. 00:08:29.630 --> 00:08:34.307 So, this example is really a non-example. 00:08:37.169 --> 00:08:42.520 Shifting the whole plane is not a linear transformation. 00:08:42.520 --> 00:08:47.180 Or if I cooked up some formula that involved squaring, 00:08:47.180 --> 00:08:55.450 or the transformation that, also non-example, 00:08:55.450 --> 00:09:01.540 how about the transformation that, takes any vector 00:09:01.540 --> 00:09:04.805 and produces its length? 00:09:08.440 --> 00:09:11.550 So there's a transformation that takes any vector, say, 00:09:11.550 --> 00:09:14.360 any vector in R^3, let me just -- 00:09:14.360 --> 00:09:18.070 I'll just get a chance to use this notation again. 00:09:18.070 --> 00:09:22.490 Suppose I think of the transformation that takes any 00:09:22.490 --> 00:09:28.700 vector in R^3 and produces this number. 00:09:28.700 --> 00:09:33.660 So that, I could say, is a member of R^1, for example, 00:09:33.660 --> 00:09:34.960 if I wanted. 00:09:34.960 --> 00:09:36.180 Or just real numbers. 00:09:38.740 --> 00:09:41.480 That's certainly not linear. 00:09:41.480 --> 00:09:45.760 It's true that the zero vector goes to zero. 00:09:45.760 --> 00:09:49.260 But if I double a vector, it does double the length, 00:09:49.260 --> 00:09:50.920 that's true. 00:09:50.920 --> 00:09:55.810 But suppose I multiply a vector by minus two. 00:09:55.810 --> 00:09:58.180 What happens to its length? 00:09:58.180 --> 00:09:59.790 It just doubles. 00:09:59.790 --> 00:10:02.140 It doesn't get multiplied by minus two. 00:10:02.140 --> 00:10:06.690 So when c is minus two in my requirement, 00:10:06.690 --> 00:10:11.010 I'm not satisfying that requirement. 00:10:11.010 --> 00:10:16.320 So T of minus v is not minus v -- minus, the length, 00:10:16.320 --> 00:10:18.170 it's just the length. 00:10:18.170 --> 00:10:20.630 OK, so that's another non-example. 00:10:20.630 --> 00:10:24.350 Projection was an example, let me give you another example. 00:10:24.350 --> 00:10:30.810 I can stay here and have a -- this will be an example that is 00:10:30.810 --> 00:10:35.650 a linear transformation, a rotation. 00:10:35.650 --> 00:10:38.940 Rotation by -- what shall we say? 00:10:38.940 --> 00:10:42.050 By 45 degrees. 00:10:42.050 --> 00:10:42.860 OK? 00:10:42.860 --> 00:10:47.750 So again, let me choose this, this will be a mapping, 00:10:47.750 --> 00:10:54.410 from the whole plane of vectors, into the whole plane 00:10:54.410 --> 00:10:59.320 of vectors, and it just -- 00:10:59.320 --> 00:11:05.930 here is the input vector v, and the output vector foam this 45 00:11:05.930 --> 00:11:13.410 degree rotation is just rotate that thing by 45 degrees, T(v). 00:11:13.410 --> 00:11:15.770 So every vector got rotated. 00:11:15.770 --> 00:11:19.510 You see that I can describe this without any coordinates. 00:11:22.940 --> 00:11:27.280 And see that it's linear. 00:11:27.280 --> 00:11:33.430 If I doubled v, the rotation would just be twice as far out. 00:11:33.430 --> 00:11:39.220 If I had v+w, and if I rotated each of them and added, 00:11:39.220 --> 00:11:43.800 the answer's the same as if I add and then rotate. 00:11:43.800 --> 00:11:46.480 That's what the linear transformation is. 00:11:46.480 --> 00:11:49.350 OK, so those are two examples. 00:11:49.350 --> 00:11:52.900 Two examples, projection and rotation, and I 00:11:52.900 --> 00:12:00.790 could invent more that are linear transformations where I 00:12:00.790 --> 00:12:04.080 haven't told you a matrix yet. 00:12:04.080 --> 00:12:11.920 Actually, the book has a picture of the action of linear 00:12:11.920 --> 00:12:16.950 transformations -- 00:12:16.950 --> 00:12:19.930 actually, the cover of the book has it. 00:12:19.930 --> 00:12:25.060 So, in this section seven point one, we can think of a -- 00:12:25.060 --> 00:12:29.280 actually, here let's take this linear transformation, 00:12:29.280 --> 00:12:33.780 rotation, suppose I have, as the cover of the book has, 00:12:33.780 --> 00:12:37.020 a house in R^2. 00:12:37.020 --> 00:12:46.600 So instead of this, let me take a small house in R^2. 00:12:46.600 --> 00:12:49.200 So that's a whole lot of points. 00:12:49.200 --> 00:12:51.830 The idea is, with this linear transformation, 00:12:51.830 --> 00:12:55.300 that I can see what it does to everything at once. 00:12:55.300 --> 00:12:57.970 I don't have to just take one vector at a time 00:12:57.970 --> 00:13:00.810 and see what T of V is, I can take 00:13:00.810 --> 00:13:05.610 all the vectors on the outline of the house, 00:13:05.610 --> 00:13:09.020 and see where they all go. 00:13:09.020 --> 00:13:12.470 In fact, that will show me where the whole house goes. 00:13:12.470 --> 00:13:17.340 So what will happen with this particular linear 00:13:17.340 --> 00:13:19.470 transformation? 00:13:19.470 --> 00:13:25.410 The whole house will rotate, so the result, if I can draw it, 00:13:25.410 --> 00:13:29.510 will be, the house will be sitting there. 00:13:29.510 --> 00:13:31.930 OK. 00:13:31.930 --> 00:13:36.150 And, but suppose I give some other examples. 00:13:36.150 --> 00:13:38.900 Oh, let me give some examples that involve a matrix. 00:13:38.900 --> 00:13:45.250 Example three -- and this is important -- 00:13:45.250 --> 00:13:50.880 coming from a matrix at -- we always call A. 00:13:50.880 --> 00:13:59.316 So the transformation will be, multiply by A. 00:13:59.316 --> 00:14:00.690 There is a linear transformation. 00:14:04.110 --> 00:14:07.870 And a whole family of them, because every matrix 00:14:07.870 --> 00:14:11.280 produces a transformation by this simple rule, 00:14:11.280 --> 00:14:16.190 just multiply every vector by that matrix, and it's linear, 00:14:16.190 --> 00:14:16.900 right? 00:14:16.900 --> 00:14:20.510 Linear, I have to check that A(v) -- 00:14:20.510 --> 00:14:25.820 A times v plus w equals Av plus A w, which is fine, 00:14:25.820 --> 00:14:31.630 and I have to check that A times vc equals c A(v). 00:14:31.630 --> 00:14:32.580 Check. 00:14:32.580 --> 00:14:34.620 Those are fine. 00:14:34.620 --> 00:14:37.460 So there is a linear transformation. 00:14:37.460 --> 00:14:42.380 And if I take my favorite matrix A, 00:14:42.380 --> 00:14:46.220 and I apply it to all vectors in the plane, 00:14:46.220 --> 00:14:49.950 it will produce a bunch of outputs. 00:14:49.950 --> 00:14:52.020 See, the idea is now worth thinking 00:14:52.020 --> 00:14:53.490 of, like, the big picture. 00:14:53.490 --> 00:15:01.670 The whole plane is transformed by matrix multiplication. 00:15:01.670 --> 00:15:04.990 Every vector in the plane gets multiplied by A. 00:15:04.990 --> 00:15:08.600 Let's take an example, and see what happens 00:15:08.600 --> 00:15:10.390 to the vectors of the house. 00:15:10.390 --> 00:15:13.110 So this is still a transformation from plane 00:15:13.110 --> 00:15:17.900 to plane, and let me take a particular matrix A -- 00:15:17.900 --> 00:15:22.710 well, if I cooked up a rotation matrix, 00:15:22.710 --> 00:15:24.820 this would be the right picture. 00:15:24.820 --> 00:15:27.220 If I cooked up a projection matrix, 00:15:27.220 --> 00:15:29.550 the projection would be the picture. 00:15:29.550 --> 00:15:31.890 Let me just take some other matrix. 00:15:31.890 --> 00:15:35.920 Let me take the matrix one zero zero minus one. 00:15:39.960 --> 00:15:46.520 What happens to the house, to all vectors, and in particular, 00:15:46.520 --> 00:15:50.150 we can sort of visualize it if we look at the house -- 00:15:50.150 --> 00:15:54.530 so the house is not rotated any more, what do I get? 00:15:57.240 --> 00:16:02.160 What happens to all the vectors if I do this transformation? 00:16:02.160 --> 00:16:04.090 I multiply by this matrix. 00:16:04.090 --> 00:16:07.660 Well, of course, it's an easy matrix, it's diagonal. 00:16:07.660 --> 00:16:13.790 The x component stays the same, the y component reverses sign, 00:16:13.790 --> 00:16:17.510 so that like the roof of that house, 00:16:17.510 --> 00:16:22.890 the point, the tip of the roof, has an x component which 00:16:22.890 --> 00:16:26.800 stays the same, but its y component reverses, 00:16:26.800 --> 00:16:28.340 and it's down here. 00:16:28.340 --> 00:16:31.170 And, of course, what we get is, the house 00:16:31.170 --> 00:16:33.510 is, like, upside down. 00:16:33.510 --> 00:16:36.270 Now, I have to put -- where does the door go? 00:16:36.270 --> 00:16:40.710 I guess the door goes upside down there, right? 00:16:40.710 --> 00:16:47.280 So here's the input, here's the input house, 00:16:47.280 --> 00:16:48.960 and this is the output. 00:16:52.610 --> 00:16:54.120 OK. 00:16:54.120 --> 00:16:57.220 This idea of a linear transformation 00:16:57.220 --> 00:17:00.440 is like kind of the abstract description 00:17:00.440 --> 00:17:04.030 of matrix multiplication. 00:17:04.030 --> 00:17:07.560 And what's our goal here? 00:17:07.560 --> 00:17:11.060 Our goal is to understand linear transformations, 00:17:11.060 --> 00:17:15.200 and the way to understand them is 00:17:15.200 --> 00:17:19.380 to find the matrix that lies behind them. 00:17:19.380 --> 00:17:21.280 That's really the idea. 00:17:21.280 --> 00:17:23.349 Find the matrix that lies behind them. 00:17:23.349 --> 00:17:29.500 Um, and to do that, we have to bring in coordinates. 00:17:29.500 --> 00:17:31.630 We have to choose a basis. 00:17:31.630 --> 00:17:37.540 So let me point out what's the story -- 00:17:37.540 --> 00:17:39.810 if we have a linear transformation -- 00:17:39.810 --> 00:17:41.675 so start with -- 00:17:41.675 --> 00:17:42.175 start. 00:17:48.950 --> 00:17:52.030 Suppose we have a linear transformation. 00:17:52.030 --> 00:17:55.790 Let -- from now on, let T stand for linear transformations. 00:17:55.790 --> 00:17:58.490 I won't be interested in the nonlinear ones. 00:17:58.490 --> 00:18:01.110 Only linear transformations I'm interested in. 00:18:01.110 --> 00:18:01.830 OK. 00:18:01.830 --> 00:18:05.530 I start with a linear transformation T. 00:18:05.530 --> 00:18:11.855 Let's suppose its inputs are vectors in R^3. 00:18:14.381 --> 00:18:14.880 OK? 00:18:14.880 --> 00:18:21.270 And suppose its outputs are vectors in R^2, for example. 00:18:21.270 --> 00:18:22.210 OK. 00:18:22.210 --> 00:18:25.260 What's an example of such a transformation, 00:18:25.260 --> 00:18:26.470 just before I go any further? 00:18:29.570 --> 00:18:32.930 Any matrix of the right size will do this. 00:18:32.930 --> 00:18:35.830 So what would be the right shape of a matrix? 00:18:35.830 --> 00:18:37.396 So, for example -- 00:18:43.180 --> 00:18:44.870 I'm wanting to give you an example, 00:18:44.870 --> 00:18:50.260 just because, here, I'm thinking of transformations 00:18:50.260 --> 00:18:55.800 that take three-dimensional space to two-dimensional space. 00:18:55.800 --> 00:19:01.150 And I want them to be linear, and the easy way to invent them 00:19:01.150 --> 00:19:04.810 is a matrix multiplication. 00:19:04.810 --> 00:19:10.630 So example, T of v should be any A 00:19:10.630 --> 00:19:13.270 v. Those transformations are linear, 00:19:13.270 --> 00:19:15.410 that's what 18.06 is about. 00:19:15.410 --> 00:19:20.420 And A should be what size, what shape of matrix should that be? 00:19:20.420 --> 00:19:23.310 I want V to have three components, 00:19:23.310 --> 00:19:25.340 because this is what the inputs have -- 00:19:25.340 --> 00:19:39.220 so here's the input in R^3, and here's the output in R^2. 00:19:39.220 --> 00:19:41.860 So what shape of matrix? 00:19:41.860 --> 00:19:50.270 So this should be, I guess, a two by three matrix? 00:19:50.270 --> 00:19:50.770 Right? 00:19:53.860 --> 00:19:57.190 A two by three matrix. 00:19:57.190 --> 00:20:00.760 A two by three matrix, we'll multiply a vector in R^3 -- 00:20:00.760 --> 00:20:04.680 you see I'm moving to coordinates so quickly, 00:20:04.680 --> 00:20:09.150 I'm not a true physicist here. 00:20:09.150 --> 00:20:13.060 A two by three matrix, we'll multiply a vector in R^3 00:20:13.060 --> 00:20:16.470 an produce an output in R^2, and it will be a linear 00:20:16.470 --> 00:20:20.660 transformation, and OK. 00:20:20.660 --> 00:20:23.840 So there's a whole lot of examples, 00:20:23.840 --> 00:20:26.670 every two by three matrix give me an example, 00:20:26.670 --> 00:20:29.230 and basically, I want to show you that there 00:20:29.230 --> 00:20:30.520 are no other examples. 00:20:30.520 --> 00:20:34.540 Every linear transformation is associated with a matrix. 00:20:34.540 --> 00:20:38.430 Now, let me come back to the idea of linear transformation. 00:20:42.160 --> 00:20:48.030 Suppose I've got this linear transformation in my mind, 00:20:48.030 --> 00:20:50.450 and I want to tell you what it is. 00:20:53.220 --> 00:20:56.430 Suppose I tell you what the transformation does 00:20:56.430 --> 00:20:58.220 to one vector. 00:20:58.220 --> 00:20:58.720 OK. 00:20:58.720 --> 00:21:00.190 You know one thing, then. 00:21:00.190 --> 00:21:01.090 All right. 00:21:01.090 --> 00:21:05.260 So this is like the -- what I'm speaking about now is, 00:21:05.260 --> 00:21:20.980 how much information is needed to know the transformation? 00:21:20.980 --> 00:21:24.300 By knowing T, I -- 00:21:24.300 --> 00:21:28.930 to know T of v for all v. 00:21:28.930 --> 00:21:30.670 All inputs. 00:21:30.670 --> 00:21:33.520 How much information do I have to give you 00:21:33.520 --> 00:21:35.930 so that you know what the transformation does 00:21:35.930 --> 00:21:38.380 to every vector? 00:21:38.380 --> 00:21:40.950 OK, I could tell you what the transformation -- 00:21:40.950 --> 00:21:47.700 so I could take a vector v1, one particular vector, 00:21:47.700 --> 00:21:53.320 tell you what the transformation does to it -- 00:21:53.320 --> 00:21:54.980 fine. 00:21:54.980 --> 00:21:57.730 But now you only know what the transformation does to one 00:21:57.730 --> 00:21:59.170 vector. 00:21:59.170 --> 00:22:02.600 So you say, OK, that's not enough, 00:22:02.600 --> 00:22:05.390 tell me what it does to another vector. 00:22:05.390 --> 00:22:10.470 So I say, OK, give me a vector, you give me a vector v2, 00:22:10.470 --> 00:22:14.805 and we see, what does the transformation do to v2? 00:22:17.510 --> 00:22:21.570 Now, you only know -- or do you only know what 00:22:21.570 --> 00:22:23.650 the transformation does to two vectors? 00:22:23.650 --> 00:22:27.890 Have I got to ask you -- answer you about every vector 00:22:27.890 --> 00:22:32.570 in the whole input space, or can you, 00:22:32.570 --> 00:22:35.022 knowing what it does to v1 and v2, 00:22:35.022 --> 00:22:37.105 how much do you now know about the transformation? 00:22:39.730 --> 00:22:42.370 You know what the transformation does 00:22:42.370 --> 00:22:47.860 to a larger bunch of vectors than just these two, 00:22:47.860 --> 00:22:54.440 because you know what it does to every linear combination. 00:22:54.440 --> 00:23:00.520 You know what it does, now, to the whole plane of vectors, 00:23:00.520 --> 00:23:03.260 with bases v1 and v2. 00:23:03.260 --> 00:23:07.310 I'm assuming v1 and v2 were independent. 00:23:07.310 --> 00:23:12.100 If they were dependent, if v2 was six times v1, 00:23:12.100 --> 00:23:15.440 then I didn't give you any new information in T of v2, 00:23:15.440 --> 00:23:20.680 you already knew it would be six times T of v1. 00:23:20.680 --> 00:23:23.660 So you can see what I'd headed for. 00:23:23.660 --> 00:23:27.180 If I know what the transformation does 00:23:27.180 --> 00:23:32.150 to every vector in a basis, then I know everything. 00:23:32.150 --> 00:23:37.110 So the information needed to know T of v for all inputs is T 00:23:37.110 --> 00:23:47.890 of v1, T of v2, up to T of vm, let's say, or vn, 00:23:47.890 --> 00:23:50.278 for any basis -- 00:23:53.630 --> 00:23:58.280 for a basis v1 up to vn. 00:23:58.280 --> 00:24:02.030 This is a base for any -- 00:24:02.030 --> 00:24:04.220 can I call it an input basis? 00:24:04.220 --> 00:24:08.330 It's a basis for the space of inputs. 00:24:08.330 --> 00:24:11.860 The things that T is acting on. 00:24:11.860 --> 00:24:19.210 You see this point, that if I have a basis for the input 00:24:19.210 --> 00:24:22.550 space, and I tell you what the transformation does 00:24:22.550 --> 00:24:25.620 to every one of those basis vectors, that 00:24:25.620 --> 00:24:31.110 is all I'm allowed to tell you, and it's enough to know T of v 00:24:31.110 --> 00:24:33.400 for all v-s, because why? 00:24:33.400 --> 00:24:39.870 Because every v is some combination of these basis 00:24:39.870 --> 00:24:47.430 vectors, c1v1+...+cnvn, that's what a basis is, right? 00:24:47.430 --> 00:24:49.740 It spans the space. 00:24:49.740 --> 00:24:56.310 And if I know what T does to this, and what T does to v2, 00:24:56.310 --> 00:25:05.220 and what T does to vn, then I know what T does to V. 00:25:05.220 --> 00:25:13.130 By this linearity, it has to be c1 T of v1 plus O one 00:25:13.130 --> 00:25:15.907 plus cn T of vn. 00:25:19.855 --> 00:25:20.605 There's no choice. 00:25:26.450 --> 00:25:31.380 So, the point of this comment is that if I 00:25:31.380 --> 00:25:37.280 know what T does to a basis, to each vector in a basis, then 00:25:37.280 --> 00:25:39.130 I know the linear transformation. 00:25:39.130 --> 00:25:43.980 The property of linearity tells me all the other vectors. 00:25:43.980 --> 00:25:46.420 All the other outputs. 00:25:46.420 --> 00:25:47.180 OK. 00:25:47.180 --> 00:25:54.340 So now, we got -- so that light we now see, 00:25:54.340 --> 00:25:56.930 what do we really need in a linear transformation, 00:25:56.930 --> 00:25:59.720 and we're ready to go to a 00:25:59.720 --> 00:26:00.740 matrix. 00:26:00.740 --> 00:26:01.470 OK. 00:26:01.470 --> 00:26:04.410 What's the step now that takes us 00:26:04.410 --> 00:26:07.510 from a linear transformation that's 00:26:07.510 --> 00:26:14.700 free of coordinates to a matrix that's been created 00:26:14.700 --> 00:26:16.950 with respect to coordinates? 00:26:16.950 --> 00:26:20.190 The matrix is going to come from the coordinate system. 00:26:20.190 --> 00:26:21.730 These are the coordinates. 00:26:21.730 --> 00:26:26.170 Coordinates mean a basis is decided. 00:26:26.170 --> 00:26:29.570 Once you decide on a basis -- 00:26:29.570 --> 00:26:30.850 this is where coordinates come 00:26:30.850 --> 00:26:31.470 from. 00:26:31.470 --> 00:26:36.280 You decide on a basis, then every vector, 00:26:36.280 --> 00:26:41.090 these are the coordinates in that basis. 00:26:41.090 --> 00:26:46.730 There is one and only one way to express v 00:26:46.730 --> 00:26:49.380 as a combination of the basis vectors, 00:26:49.380 --> 00:26:52.520 and the numbers you need in that combination 00:26:52.520 --> 00:26:53.430 are the coordinates. 00:26:53.430 --> 00:26:55.080 Let me write that down. 00:26:55.080 --> 00:26:56.480 So what are coordinates? 00:26:56.480 --> 00:27:05.030 Coordinates come from a basis. 00:27:10.320 --> 00:27:13.690 Coordinates come from a basis. 00:27:13.690 --> 00:27:17.710 The coordinates of v, the coordinates of v 00:27:17.710 --> 00:27:32.140 are these numbers that tell you how much of each basis vector 00:27:32.140 --> 00:27:33.580 is in v. 00:27:33.580 --> 00:27:37.580 If I change the basis, I change the coordinates, right? 00:27:37.580 --> 00:27:40.270 Now, we have always been assuming 00:27:40.270 --> 00:27:44.260 that were working with a standard basis, right? 00:27:44.260 --> 00:27:48.110 The basis we don't even think about this stuff, 00:27:48.110 --> 00:27:55.870 because if I give you the vector v equals three two four, 00:27:55.870 --> 00:27:59.120 you have been assuming completely -- 00:27:59.120 --> 00:28:04.050 and probably rightly -- that I had in mind the standard basis, 00:28:04.050 --> 00:28:11.910 that this vector was three times the first coordinate vector, 00:28:11.910 --> 00:28:16.382 and two times the second, and four times the third. 00:28:22.130 --> 00:28:25.010 But you're not entitled -- 00:28:25.010 --> 00:28:27.390 I might have had some other basis in mind. 00:28:27.390 --> 00:28:30.520 This is like the standard basis. 00:28:30.520 --> 00:28:33.770 And then the coordinates are sitting right there 00:28:33.770 --> 00:28:35.150 in the vector. 00:28:35.150 --> 00:28:37.210 But I could have chosen a different basis, 00:28:37.210 --> 00:28:42.400 like I might have had eigenvectors of a matrix, 00:28:42.400 --> 00:28:45.470 and I might have said, OK, that's a great basis, 00:28:45.470 --> 00:28:49.110 I'll use the eigenvectors of this matrix 00:28:49.110 --> 00:28:52.280 as my basis vectors. 00:28:52.280 --> 00:28:55.541 Which are not necessarily these three, but some other basis. 00:28:58.130 --> 00:29:03.180 So that was an example, this is the real thing, 00:29:03.180 --> 00:29:05.060 the coordinates are these numbers, 00:29:05.060 --> 00:29:08.490 I'll circle them again, the amounts of each basis. 00:29:08.490 --> 00:29:11.110 OK. 00:29:11.110 --> 00:29:15.710 So, if I want to create a matrix that 00:29:15.710 --> 00:29:17.600 describes a linear transformation, 00:29:17.600 --> 00:29:19.380 now I'm ready to do that. 00:29:19.380 --> 00:29:20.940 OK, OK. 00:29:20.940 --> 00:29:33.630 So now what I plan to do is construct the matrix A 00:29:33.630 --> 00:29:42.840 that represents, or tells me about, a linear transformation, 00:29:42.840 --> 00:29:47.230 linear transformation T. OK. 00:29:47.230 --> 00:29:51.060 So I really start with the transformation -- 00:29:51.060 --> 00:29:53.280 whether it's a projection or a rotation, 00:29:53.280 --> 00:29:57.100 or some strange movement of this house in the plane, 00:29:57.100 --> 00:30:02.760 or some transformation from n-dimensional space to -- 00:30:02.760 --> 00:30:05.260 or m-dimensional space to n-dimensional space. 00:30:08.200 --> 00:30:09.840 n to m, I guess. 00:30:09.840 --> 00:30:14.990 Usually, we'll have T, we'll somehow transform n-dimensional 00:30:14.990 --> 00:30:21.660 space to m-dimensional space, and the whole point is that 00:30:21.660 --> 00:30:25.390 if I have a basis for n-dimensional space -- 00:30:25.390 --> 00:30:28.360 I guess I need two bases, really. 00:30:28.360 --> 00:30:31.810 I need an input basis to describe the inputs, 00:30:31.810 --> 00:30:36.090 and I need an output basis to give me coordinates -- 00:30:36.090 --> 00:30:39.110 to give me some numbers for the output. 00:30:39.110 --> 00:30:41.240 So I've got to choose two bases. 00:30:41.240 --> 00:30:51.590 Choose a basis v1 up to vn for the inputs, 00:30:51.590 --> 00:30:54.950 for the inputs in -- 00:30:54.950 --> 00:30:57.550 they came from R^n. 00:30:57.550 --> 00:31:03.080 So the transformation is taking every n-dimensional vector 00:31:03.080 --> 00:31:05.150 into some m-dimensional vector. 00:31:05.150 --> 00:31:12.520 And I have to choose a basis, and I'll call them w1 up to wn, 00:31:12.520 --> 00:31:13.620 for the outputs. 00:31:17.120 --> 00:31:18.550 Those are guys in R^m. 00:31:22.380 --> 00:31:26.960 Once I've chosen the basis, that settles the matrix -- 00:31:26.960 --> 00:31:29.660 I now working with coordinates. 00:31:29.660 --> 00:31:35.190 Every vector in R^n, every input vector has some coordinates. 00:31:35.190 --> 00:31:38.910 So here's what I do, here's what I do. 00:31:38.910 --> 00:31:41.850 Can I say it in words? 00:31:41.850 --> 00:31:45.180 I take a vector v. 00:31:45.180 --> 00:31:48.214 I express it in its basis, in the basis, 00:31:48.214 --> 00:31:49.255 so I get its coordinates. 00:31:51.820 --> 00:31:55.120 Then I'm going to multiply those coordinates by the right matrix 00:31:55.120 --> 00:32:00.020 A, and that will give me the coordinates of the output 00:32:00.020 --> 00:32:01.830 in the output basis. 00:32:01.830 --> 00:32:05.320 I'd better write that down, that was a mouthful. 00:32:05.320 --> 00:32:06.870 What I want -- 00:32:11.000 --> 00:32:23.080 I want a matrix A that does what the linear transformation does. 00:32:23.080 --> 00:32:29.680 And it does it with respecting these bases. 00:32:29.680 --> 00:32:35.000 So I want the matrix to be -- well, let's suppose -- look, 00:32:35.000 --> 00:32:37.790 let me take an example. 00:32:37.790 --> 00:32:40.150 Let me take the projection example. 00:32:40.150 --> 00:32:42.260 The projection example. 00:32:42.260 --> 00:32:45.190 Suppose I take -- 00:32:45.190 --> 00:32:47.260 because we've got that -- 00:32:47.260 --> 00:32:49.240 we've got that projection in mind -- 00:32:49.240 --> 00:32:50.650 I can fit in here. 00:32:50.650 --> 00:32:52.100 Here's the projection example. 00:32:52.100 --> 00:32:58.800 So the projection example, I'm thinking of n and m as two. 00:32:58.800 --> 00:33:01.570 The transformation takes the plane, 00:33:01.570 --> 00:33:07.530 takes every vector in the plane, and, let me draw the plane, 00:33:07.530 --> 00:33:10.600 just so we remember it's a plane -- 00:33:10.600 --> 00:33:15.180 and there's the thing that I'm projecting onto, 00:33:15.180 --> 00:33:17.800 that's the line I'm projecting onto -- 00:33:17.800 --> 00:33:21.620 so the transformation takes every vector in the plane 00:33:21.620 --> 00:33:24.360 and projects it onto that line. 00:33:24.360 --> 00:33:28.570 So this is projection, so I'm going to do projection. 00:33:28.570 --> 00:33:29.280 OK. 00:33:29.280 --> 00:33:37.330 But, I'm going to choose a basis that I like better 00:33:37.330 --> 00:33:39.680 than the standard basis. 00:33:39.680 --> 00:33:44.600 My basis -- in fact, I'll choose the same basis for inputs 00:33:44.600 --> 00:33:49.260 and for outputs, and the basis will be -- 00:33:49.260 --> 00:33:53.860 my first basis vector will be right on the line. 00:33:53.860 --> 00:33:55.330 There's my first basis vector. 00:33:55.330 --> 00:33:57.510 Say, a unit vector, on the line. 00:33:57.510 --> 00:34:01.660 And my second basis vector will be a unit vector perpendicular 00:34:01.660 --> 00:34:02.700 to that line. 00:34:02.700 --> 00:34:05.270 And I'm going to choose that as the output basis, 00:34:05.270 --> 00:34:06.610 also. 00:34:06.610 --> 00:34:11.010 And I'm going to ask you, what's the matrix? 00:34:11.010 --> 00:34:14.139 What's the matrix? 00:34:14.139 --> 00:34:17.610 How do I describe this transformation of projection 00:34:17.610 --> 00:34:20.130 with respect to this basis? 00:34:20.130 --> 00:34:20.760 OK? 00:34:20.760 --> 00:34:22.050 So what's the rule? 00:34:22.050 --> 00:34:26.110 I take any vector v, it's some combination 00:34:26.110 --> 00:34:31.690 of the first basis ve- vector, and the second basis vector. 00:34:31.690 --> 00:34:33.420 Now, what is T of v? 00:34:38.080 --> 00:34:45.179 Suppose the input is -- well, suppose the input is v1. 00:34:45.179 --> 00:34:48.150 What's the output? 00:34:48.150 --> 00:34:49.870 v1, right? 00:34:49.870 --> 00:34:53.429 The projection leaves this one alone. 00:34:53.429 --> 00:34:56.790 So we know what the projection does to this first basis 00:34:56.790 --> 00:35:00.140 vector, this guy, it leaves it. 00:35:00.140 --> 00:35:04.240 What does the projection do to the second basis vector? 00:35:04.240 --> 00:35:08.160 It kills it, sends it to zero. 00:35:08.160 --> 00:35:11.060 So what does the projection do to a combination? 00:35:15.600 --> 00:35:20.060 It kills this part, and this part, it leaves alone. 00:35:23.580 --> 00:35:27.210 Now, all I want to do is find the matrix. 00:35:27.210 --> 00:35:29.630 I now want to find the matrix that 00:35:29.630 --> 00:35:35.560 takes an input, c1 c2, the coordinates, 00:35:35.560 --> 00:35:39.350 and gives me the output, c1 0. 00:35:39.350 --> 00:35:44.630 You see that in this basis, the coordinates of the input 00:35:44.630 --> 00:35:51.840 were c1, c2, and the coordinates of the output are c1, 00:35:51.840 --> 00:35:56.590 And of course, not hard to find a matrix that will do that. 00:35:56.590 --> 00:36:02.470 The matrix that will do that is the matrix one, zero, zero, 00:36:02.470 --> 00:36:04.250 zero. 00:36:04.250 --> 00:36:10.560 Because if I multiply input by that matrix A -- 00:36:10.560 --> 00:36:16.130 this is A times input coordinates -- 00:36:16.130 --> 00:36:18.380 and I'm hoping to get the output coordinates. 00:36:23.450 --> 00:36:25.610 And what do I get from that multiplication? 00:36:25.610 --> 00:36:27.750 I get the right answer, c1 and zero. 00:36:30.450 --> 00:36:31.970 So what's the point? 00:36:31.970 --> 00:36:36.840 So the first point is, there's a matrix that does the job. 00:36:36.840 --> 00:36:39.250 If there's a linear transformation out there, 00:36:39.250 --> 00:36:42.350 coordinate-free, no coordinates, and then I 00:36:42.350 --> 00:36:45.360 choose a basis for the inputs, and I 00:36:45.360 --> 00:36:47.790 choose a basis for the outputs, then 00:36:47.790 --> 00:36:51.820 there's a matrix that does And what's the job? 00:36:51.820 --> 00:36:52.570 the job. 00:36:52.570 --> 00:36:56.740 It multiplies the input coordinates and produces 00:36:56.740 --> 00:36:58.790 the output coordinates. 00:36:58.790 --> 00:37:00.730 Now, in this example -- let me repeat, 00:37:00.730 --> 00:37:05.380 I chose the input basis was the same as the output basis. 00:37:05.380 --> 00:37:09.860 The input basis and output basis were both along the line, 00:37:09.860 --> 00:37:12.510 and perpendicular to the line. 00:37:12.510 --> 00:37:16.670 They're actually the eigenvectors of the projection. 00:37:16.670 --> 00:37:20.370 And, as a result, the matrix came out diagonal. 00:37:20.370 --> 00:37:24.450 In fact, it came out to be lambda. 00:37:24.450 --> 00:37:27.650 This is like, the good basis. 00:37:27.650 --> 00:37:37.640 So the good -- the eigenvector basis is the good basis, 00:37:37.640 --> 00:37:43.210 it leads to the matrix -- 00:37:43.210 --> 00:37:49.230 the diagonal matrix of eigenvalues lambda, 00:37:49.230 --> 00:37:54.260 and just as in this example, the eigenvectors and eigenvalues 00:37:54.260 --> 00:38:00.120 of this linear transformation were along the line, 00:38:00.120 --> 00:38:01.630 and perpendicular. 00:38:01.630 --> 00:38:04.740 The eigenvalues were one and zero, 00:38:04.740 --> 00:38:07.380 and that's the matrix that we got. 00:38:07.380 --> 00:38:08.450 OK. 00:38:08.450 --> 00:38:12.290 So that's a, like, the great choice of matrix, 00:38:12.290 --> 00:38:16.110 that's the choice a physicist would do when he had to finally 00:38:16.110 --> 00:38:19.330 -- he or she had to finally bring coordinates 00:38:19.330 --> 00:38:24.740 in unwillingly, the coordinates to be chosen, 00:38:24.740 --> 00:38:27.240 the good coordinates are the eigenvectors, 00:38:27.240 --> 00:38:32.540 because, if I did this projection in the standard 00:38:32.540 --> 00:38:34.190 basis -- 00:38:34.190 --> 00:38:35.910 which I could do, right? 00:38:35.910 --> 00:38:40.240 I could do the whole thing in the standard basis -- 00:38:40.240 --> 00:38:42.790 I better try, if I can do that. 00:38:42.790 --> 00:38:45.390 What are we calling -- 00:38:45.390 --> 00:38:49.400 so I'll have to tell you now which line we're projecting on. 00:38:49.400 --> 00:38:51.640 Say, the 45 degree line. 00:38:51.640 --> 00:38:59.200 So say we're projecting onto 45 degree line, 00:38:59.200 --> 00:39:04.340 and we use not the eigenvector basis, but the standard basis. 00:39:07.240 --> 00:39:15.730 The standard basis, v1, is one, zero, and v2 is zero, one. 00:39:15.730 --> 00:39:18.720 And again, I'll use the same basis for the outputs. 00:39:22.110 --> 00:39:24.660 Then I have to do this -- 00:39:24.660 --> 00:39:29.990 I can find a matrix, it will be the matrix 00:39:29.990 --> 00:39:31.510 that we would always think of, it 00:39:31.510 --> 00:39:33.600 would be the projection matrix. 00:39:33.600 --> 00:39:39.610 It will be, actually, it's the matrix that we learned about 00:39:39.610 --> 00:39:47.220 in chapter four, it's what I call the matrix -- 00:39:47.220 --> 00:39:52.920 do you remember, P was A, A transpose over A transpose A? 00:39:52.920 --> 00:39:55.800 And I think, in this example, it will come out, 00:39:55.800 --> 00:39:58.875 one-half, one-half, one-half, one-half. 00:40:04.240 --> 00:40:07.940 I believe that's the matrix that comes from our formula. 00:40:07.940 --> 00:40:10.126 And that's the matrix that will do the job. 00:40:13.460 --> 00:40:19.020 If I give you this input, one, zero, what's the output? 00:40:19.020 --> 00:40:20.700 The output is one-half, one-half. 00:40:24.120 --> 00:40:28.900 And that should be the right projection. 00:40:28.900 --> 00:40:31.010 And if I give you the input zero, one, 00:40:31.010 --> 00:40:34.270 the output is, again, one-half, one-half, again the projection. 00:40:37.580 --> 00:40:40.950 So that's the matrix, but not diagonal of course, 00:40:40.950 --> 00:40:43.660 because we didn't choose a great basis, 00:40:43.660 --> 00:40:46.700 we just chose the handiest basis. 00:40:46.700 --> 00:40:49.040 Well, so the course has practically 00:40:49.040 --> 00:40:53.760 been about the handiest basis, and just dealing 00:40:53.760 --> 00:40:55.210 with the matrix that we got. 00:40:55.210 --> 00:40:59.400 And it's not that bad a matrix, it's symmetric, 00:40:59.400 --> 00:41:02.310 and it has this P squared equal P property, 00:41:02.310 --> 00:41:03.870 all those things are good. 00:41:03.870 --> 00:41:11.630 But in the best basis, it's easy to see that P squared equals P, 00:41:11.630 --> 00:41:15.460 and it's symmetric, and it's diagonal. 00:41:15.460 --> 00:41:19.480 So that's the idea then, is, do you 00:41:19.480 --> 00:41:24.770 see now how I'm associating a matrix to the transformation? 00:41:24.770 --> 00:41:28.910 I'd better write the rule down, I'd better write the rule down. 00:41:28.910 --> 00:41:39.230 The rule to find the matrix A. 00:41:39.230 --> 00:41:40.370 All right, first column. 00:41:40.370 --> 00:41:50.070 So, a rule to find A, we're given the bases. 00:41:50.070 --> 00:41:52.420 Of course, we don't -- because there's no way we could 00:41:52.420 --> 00:41:55.390 construct the matrix until we're told what the bases are. 00:41:55.390 --> 00:42:01.290 So we're given the input basis, and the output basis, v1 to vn, 00:42:01.290 --> 00:42:03.660 w1 to wm. 00:42:03.660 --> 00:42:05.530 Those are given. 00:42:05.530 --> 00:42:10.550 Now, in the first column of A, how do I find that column? 00:42:10.550 --> 00:42:13.780 The first column of the matrix. 00:42:13.780 --> 00:42:18.650 So that should tell me what happens to the first basis 00:42:18.650 --> 00:42:19.910 vector. 00:42:19.910 --> 00:42:28.070 So the rule is, apply the linear transformation to v1. 00:42:28.070 --> 00:42:32.280 To the first basis vector. 00:42:32.280 --> 00:42:37.010 And then, I'll write it -- so that's the output, right? 00:42:37.010 --> 00:42:40.830 The input is v1, what's the output? 00:42:40.830 --> 00:42:42.920 The output is in the output space, 00:42:42.920 --> 00:42:45.290 it's some combination of these guys, 00:42:45.290 --> 00:42:50.230 and it's that combination that goes into the first column -- 00:42:50.230 --> 00:42:51.230 so, let me -- 00:42:51.230 --> 00:42:57.510 I'll put this word -- right, I'll say it in words again. 00:42:57.510 --> 00:42:59.750 How to find this matrix. 00:42:59.750 --> 00:43:01.810 Take the first basis vector. 00:43:01.810 --> 00:43:04.330 Apply the transformation, then it's 00:43:04.330 --> 00:43:06.710 in the output space, T of v1, so it's 00:43:06.710 --> 00:43:11.440 some combination of these outputs, this output basis. 00:43:11.440 --> 00:43:14.870 So that combination, the coefficients in that 00:43:14.870 --> 00:43:19.680 combination will be the first column -- 00:43:19.680 --> 00:43:32.580 so a1, a row 2, column 1, w2, am1, wm. 00:43:32.580 --> 00:43:38.920 There are the numbers in the first column of the matrix. 00:43:38.920 --> 00:43:41.890 Let me make the point by doing the second column. 00:43:41.890 --> 00:43:47.960 Second column of A. 00:43:47.960 --> 00:43:49.550 What's the idea, now? 00:43:49.550 --> 00:43:54.820 I take the second basis vector, I apply the transformation 00:43:54.820 --> 00:43:58.510 to it, that's in -- now I get an output, 00:43:58.510 --> 00:44:01.570 so it's some combination in the output basis -- 00:44:01.570 --> 00:44:06.830 and that combination is the bunch of numbers that should go 00:44:06.830 --> 00:44:15.470 in the second column of the matrix. 00:44:15.470 --> 00:44:16.740 OK. 00:44:16.740 --> 00:44:18.510 And so forth. 00:44:18.510 --> 00:44:22.560 So I get a matrix, and the matrix I get 00:44:22.560 --> 00:44:24.870 does the right job. 00:44:24.870 --> 00:44:29.120 Now, the matrix constructed that way, and following the rules 00:44:29.120 --> 00:44:31.680 of matrix multiplication. 00:44:31.680 --> 00:44:37.310 The result will be that if I give you the input coordinates, 00:44:37.310 --> 00:44:42.810 and I multiply by the matrix, so the outcome of all this 00:44:42.810 --> 00:44:52.370 is A times the input coordinates correctly reproduces 00:44:52.370 --> 00:44:53.380 the output coordinates. 00:44:59.190 --> 00:45:01.520 Why is this right? 00:45:01.520 --> 00:45:04.240 Let me just check the first column. 00:45:04.240 --> 00:45:08.697 Suppose the input coordinates are one and all zeros. 00:45:08.697 --> 00:45:09.530 What does that mean? 00:45:09.530 --> 00:45:10.900 What's the input? 00:45:10.900 --> 00:45:13.080 If the input coordinates are one and other -- 00:45:13.080 --> 00:45:19.610 and the rest zeros, then the input is v1, right? 00:45:19.610 --> 00:45:23.550 That's the vector that has coordinates one and all zeros. 00:45:23.550 --> 00:45:24.270 OK? 00:45:24.270 --> 00:45:27.800 When I multiply A by the one and all zeros, 00:45:27.800 --> 00:45:32.270 I'll get the first column of A, I'll get these numbers. 00:45:32.270 --> 00:45:37.070 And, sure enough, those are the output coordinates for T of v1. 00:45:37.070 --> 00:45:39.860 So we made it right on the first column, 00:45:39.860 --> 00:45:41.940 we made it right on the second column, 00:45:41.940 --> 00:45:44.810 we made it right on all the basis vectors, 00:45:44.810 --> 00:45:50.380 and then it has to be right on every vector. 00:45:50.380 --> 00:45:56.890 So there is a picture of the matrix for a linear 00:45:56.890 --> 00:45:57.710 OK. transformation. 00:45:57.710 --> 00:46:01.610 Finally, let me give you another -- 00:46:01.610 --> 00:46:04.190 a different linear transformation. 00:46:04.190 --> 00:46:07.485 The linear transformation that takes the derivative. 00:46:10.110 --> 00:46:13.560 That's a linear transformation. 00:46:13.560 --> 00:46:18.590 Suppose the input space is all combination 00:46:18.590 --> 00:46:24.510 c1 plus c2x plus c3 x squared. 00:46:24.510 --> 00:46:32.620 So the basis is these simple functions. 00:46:32.620 --> 00:46:33.695 Then what's the output? 00:46:38.210 --> 00:46:39.770 Is the derivative. 00:46:39.770 --> 00:46:48.910 The output is the derivative, so the output is c2+2c3 x. 00:46:48.910 --> 00:46:55.100 And let's take as output basis, the vectors one and x. 00:46:55.100 --> 00:46:57.930 So we're going from a three-dimensional space 00:46:57.930 --> 00:47:00.660 of inputs to a two-dimensional space 00:47:00.660 --> 00:47:04.900 of outputs by the derivative. 00:47:04.900 --> 00:47:06.780 And I don't know if you ever thought 00:47:06.780 --> 00:47:09.375 that the derivative is linear. 00:47:13.520 --> 00:47:16.360 But if it weren't linear, taking derivatives 00:47:16.360 --> 00:47:19.380 would take forever, right? 00:47:19.380 --> 00:47:22.980 We are able to compute derivatives of functions 00:47:22.980 --> 00:47:27.030 exactly because we know it's a linear transformation, 00:47:27.030 --> 00:47:30.400 so that if we learn the derivatives of a few functions, 00:47:30.400 --> 00:47:32.960 like sine x and cos x and e to the x, 00:47:32.960 --> 00:47:35.730 and another little short list, then we 00:47:35.730 --> 00:47:37.500 can take all their combinations and we 00:47:37.500 --> 00:47:40.320 can do all the derivatives. 00:47:40.320 --> 00:47:43.010 OK, now what's the matrix? 00:47:43.010 --> 00:47:44.290 What's the matrix? 00:47:44.290 --> 00:47:50.400 So I want the matrix to multiply these input vectors -- 00:47:50.400 --> 00:47:57.200 input coordinates, and give these output coordinates. 00:47:57.200 --> 00:48:00.250 So I just think, OK, what's the matrix that does it? 00:48:00.250 --> 00:48:02.490 I can follow my rule of construction, 00:48:02.490 --> 00:48:05.320 or I can see what the matrix is. 00:48:05.320 --> 00:48:10.950 It should be a two by three matrix, right? 00:48:10.950 --> 00:48:13.230 And the matrix -- 00:48:13.230 --> 00:48:15.490 so I'm just figuring out, what do I want? 00:48:15.490 --> 00:48:17.580 No, I'll -- let me write it here. 00:48:17.580 --> 00:48:18.995 What do I want from my matrix? 00:48:22.550 --> 00:48:23.740 What should that matrix do? 00:48:23.740 --> 00:48:26.200 Well, I want to get c2 in the first output, 00:48:26.200 --> 00:48:28.920 so zero, one, zero will do it. 00:48:28.920 --> 00:48:34.080 I want to get two c3, so zero, zero, two will do it. 00:48:34.080 --> 00:48:40.030 That's the matrix for this linear transformation 00:48:40.030 --> 00:48:44.440 with those bases and those coordinates. 00:48:44.440 --> 00:48:50.330 You see, it just clicks, and the whole point is that the inverse 00:48:50.330 --> 00:48:53.880 matrix gives the inverse to the linear transformation, 00:48:53.880 --> 00:48:58.150 that the product of two matrices gives the right matrix 00:48:58.150 --> 00:49:00.760 for the product of two transformations -- 00:49:00.760 --> 00:49:04.930 matrix multiplication really came from linear 00:49:04.930 --> 00:49:05.860 transformations. 00:49:05.860 --> 00:49:10.660 I'd better pick up on that theme Monday after Thanksgiving. 00:49:10.660 --> 00:49:13.900 And I hope you have a great holiday. 00:49:13.900 --> 00:49:16.260 I hope Indian summer keeps going. 00:49:16.260 --> 00:49:18.670 OK, see you on Monday.