1 00:00:07,170 --> 00:00:12,980 OK, this is the lecture on linear transformations. 2 00:00:12,980 --> 00:00:17,330 Actually, linear algebra courses used 3 00:00:17,330 --> 00:00:19,600 to begin with this lecture, so you 4 00:00:19,600 --> 00:00:22,590 could say I'm beginning this course again 5 00:00:22,590 --> 00:00:26,090 by talking about linear transformations. 6 00:00:30,090 --> 00:00:35,120 In a lot of courses, those come first before matrices. 7 00:00:35,120 --> 00:00:37,910 The idea of a linear transformation makes sense 8 00:00:37,910 --> 00:00:43,320 without a matrix, and physicists and other -- 9 00:00:43,320 --> 00:00:45,250 some people like it better that way. 10 00:00:45,250 --> 00:00:47,810 They don't like coordinates. 11 00:00:47,810 --> 00:00:50,380 They don't want those numbers. 12 00:00:50,380 --> 00:00:54,960 They want to see what's going on with the whole space. 13 00:00:54,960 --> 00:00:59,640 But, for most of us, in the end, if we're 14 00:00:59,640 --> 00:01:03,190 going to compute anything, we introduce coordinates, 15 00:01:03,190 --> 00:01:06,960 and then every linear transformation 16 00:01:06,960 --> 00:01:08,600 will lead us to a 17 00:01:08,600 --> 00:01:09,680 matrix. 18 00:01:09,680 --> 00:01:17,080 And then, to all the things that we've done about null space 19 00:01:17,080 --> 00:01:21,630 and row space, and determinant, and eigenvalues -- 20 00:01:21,630 --> 00:01:24,100 all will come from the matrix. 21 00:01:24,100 --> 00:01:26,590 But, behind it -- in other words, 22 00:01:26,590 --> 00:01:30,300 behind this is the idea of a linear transformation. 23 00:01:30,300 --> 00:01:32,746 Let me give an example of a linear transformation. 24 00:01:35,805 --> 00:01:36,305 So, example. 25 00:01:39,340 --> 00:01:41,600 Example one. 26 00:01:41,600 --> 00:01:42,145 A projection. 27 00:01:47,580 --> 00:01:53,520 I can describe a projection without telling you any matrix, 28 00:01:53,520 --> 00:01:55,190 anything about any matrix. 29 00:01:55,190 --> 00:01:58,050 I can describe a projection, say, 30 00:01:58,050 --> 00:02:02,960 this will be a linear transformation that takes, say, 31 00:02:02,960 --> 00:02:06,940 all of R^2, every vector in the plane, 32 00:02:06,940 --> 00:02:11,030 into a vector in the plane. 33 00:02:11,030 --> 00:02:17,770 And this is the way people describe, a mapping. 34 00:02:17,770 --> 00:02:21,230 It takes every vector, and so, by what rule? 35 00:02:21,230 --> 00:02:27,120 So, what's the rule, is, I take a -- so here's the plane, 36 00:02:27,120 --> 00:02:33,250 this is going to be my line, my line through my line, 37 00:02:33,250 --> 00:02:37,630 and I'm going to project every vector onto that line. 38 00:02:37,630 --> 00:02:42,310 So if I take a vector like b -- or let me call the vector v 39 00:02:42,310 --> 00:02:43,840 for the moment -- 40 00:02:43,840 --> 00:02:47,610 the projection -- the linear transformation is going 41 00:02:47,610 --> 00:02:51,800 to produce this vector as T(v). 42 00:02:51,800 --> 00:02:54,730 So T -- it's like a function. 43 00:02:54,730 --> 00:02:56,500 Exactly like a function. 44 00:02:56,500 --> 00:03:00,280 You give me an input, the transformation 45 00:03:00,280 --> 00:03:02,390 produces the output. 46 00:03:02,390 --> 00:03:07,840 So transformation, sometimes the word map, or mapping is used. 47 00:03:07,840 --> 00:03:12,980 A map between inputs and outputs. 48 00:03:12,980 --> 00:03:17,670 So this is one particular map, this is one example, 49 00:03:17,670 --> 00:03:20,690 a projection that takes every vector -- here, 50 00:03:20,690 --> 00:03:24,870 let me do another vector v, or let me do this vector w, 51 00:03:24,870 --> 00:03:26,701 what is T(w)? 52 00:03:26,701 --> 00:03:27,200 You see? 53 00:03:27,200 --> 00:03:29,530 There are no coordinates here. 54 00:03:29,530 --> 00:03:32,640 I've drawn those axes, but I'm sorry I drew them, 55 00:03:32,640 --> 00:03:35,190 I'm going to remove them, that's the whole point, 56 00:03:35,190 --> 00:03:41,040 is that we don't need axes, we just need -- so guts -- 57 00:03:41,040 --> 00:03:45,430 get it out of there, I'm not a physicist, 58 00:03:45,430 --> 00:03:47,290 so I draw those axes. 59 00:03:47,290 --> 00:03:52,610 So the input is w, the output of the projection 60 00:03:52,610 --> 00:03:56,990 is, project on that line, T(w). 61 00:03:56,990 --> 00:03:57,580 OK. 62 00:03:57,580 --> 00:04:01,900 Now, I could think of a lot of transformations T. 63 00:04:01,900 --> 00:04:04,360 But, in this linear algebra course, 64 00:04:04,360 --> 00:04:07,170 I want it to be a linear transformation. 65 00:04:07,170 --> 00:04:11,415 So here are the rules for a linear transformation. 66 00:04:14,940 --> 00:04:17,430 Here, see, exactly, the two operations 67 00:04:17,430 --> 00:04:21,120 that we can do on vectors, adding and multiplying 68 00:04:21,120 --> 00:04:25,140 by scalars, the transformation does something special 69 00:04:25,140 --> 00:04:27,330 with respect to those operations. 70 00:04:27,330 --> 00:04:33,460 So, for example, the projection is a linear transformation 71 00:04:33,460 --> 00:04:35,230 because -- 72 00:04:35,230 --> 00:04:38,070 for example, if I wanted to check that one, 73 00:04:38,070 --> 00:04:42,880 if I took v to be twice as long, the projection 74 00:04:42,880 --> 00:04:44,540 would be twice as long. 75 00:04:44,540 --> 00:04:48,210 If I took v to be minus -- 76 00:04:48,210 --> 00:04:51,130 if I changed from v to minus v, the projection 77 00:04:51,130 --> 00:04:52,780 would change to a minus. 78 00:04:52,780 --> 00:04:57,470 So c equal to two, c equal minus one, any c is OK. 79 00:04:57,470 --> 00:05:02,530 So you see that actually, those combine, I can combine those 80 00:05:02,530 --> 00:05:06,040 into one statement. 81 00:05:06,040 --> 00:05:11,020 What the transformation does to any linear combination, 82 00:05:11,020 --> 00:05:19,550 it must produce the same combination of T(v) and T(w). 83 00:05:19,550 --> 00:05:22,290 Let's think about some -- 84 00:05:22,290 --> 00:05:26,120 I mean, it's like, not hard to decide, 85 00:05:26,120 --> 00:05:31,090 is a transformation linear or is it not. 86 00:05:31,090 --> 00:05:37,150 Let me give you an example so you can tell me the answer. 87 00:05:37,150 --> 00:05:41,815 Suppose my transformation is -- here's another example two. 88 00:05:47,470 --> 00:05:49,545 Shift the whole plane. 89 00:05:56,860 --> 00:06:01,940 So here are all my vectors, my plane, and every vector 90 00:06:01,940 --> 00:06:06,000 v in the plane, I shift it over by, 91 00:06:06,000 --> 00:06:11,640 let's say, three by some vector v0. 92 00:06:11,640 --> 00:06:14,770 Shift whole plane by v0. 93 00:06:14,770 --> 00:06:18,370 So every vector in the plane -- 94 00:06:18,370 --> 00:06:22,830 this was v, T(v) will be v+v0. 95 00:06:22,830 --> 00:06:23,770 There's T(v). 96 00:06:23,770 --> 00:06:25,360 Here's v0. 97 00:06:25,360 --> 00:06:27,390 There's the typical v. 98 00:06:27,390 --> 00:06:30,110 And there's T(v). 99 00:06:30,110 --> 00:06:33,090 You see what this transformation does? 100 00:06:33,090 --> 00:06:37,140 Takes this vector and adds to it. 101 00:06:37,140 --> 00:06:39,980 Adds a fixed vector to it. 102 00:06:39,980 --> 00:06:43,770 Well, that seems like a pretty reasonable, 103 00:06:43,770 --> 00:06:48,500 simple transformation, but is it linear? 104 00:06:48,500 --> 00:06:52,040 The answer is no, it's not linear. 105 00:06:52,040 --> 00:06:55,010 Which law is broken? 106 00:06:55,010 --> 00:06:57,770 Maybe both laws are broken. 107 00:06:57,770 --> 00:06:58,390 Let's see. 108 00:07:00,930 --> 00:07:09,470 If I double the length of v, does the transformation produce 109 00:07:09,470 --> 00:07:12,070 something double -- do I double T(v)? 110 00:07:12,070 --> 00:07:14,010 No. 111 00:07:14,010 --> 00:07:16,960 If I double the length of v, in this transformation, 112 00:07:16,960 --> 00:07:21,720 I'm just adding on the same one -- same v0, not two v0s, 113 00:07:21,720 --> 00:07:24,660 but only one v0 for every vector, 114 00:07:24,660 --> 00:07:28,830 so I don't get two times the transform. 115 00:07:28,830 --> 00:07:30,480 Do you see what I'm saying? 116 00:07:30,480 --> 00:07:35,000 That if I double this, then the transformation 117 00:07:35,000 --> 00:07:40,360 starts there and only goes one v0 out and doesn't double T(v). 118 00:07:40,360 --> 00:07:47,030 In fact, a linear transformation -- what is T of 119 00:07:47,030 --> 00:07:48,060 zero? 120 00:07:48,060 --> 00:07:53,610 That's just like a special case, but really worth noticing. 121 00:07:53,610 --> 00:07:56,650 The zero vector in a linear transformation 122 00:07:56,650 --> 00:08:00,740 must get transformed to zero. 123 00:08:00,740 --> 00:08:10,670 It can't move, because, take any vector V here -- 124 00:08:10,670 --> 00:08:15,170 well, so you can see why T of zero is zero. 125 00:08:15,170 --> 00:08:20,640 Take v to be the zero vector, take c to be three. 126 00:08:20,640 --> 00:08:23,480 Then we'd have T of zero vector equaling 127 00:08:23,480 --> 00:08:29,131 three T of zero vector, the T of zero has to be zero. 128 00:08:29,131 --> 00:08:29,630 OK. 129 00:08:29,630 --> 00:08:34,307 So, this example is really a non-example. 130 00:08:37,169 --> 00:08:42,520 Shifting the whole plane is not a linear transformation. 131 00:08:42,520 --> 00:08:47,180 Or if I cooked up some formula that involved squaring, 132 00:08:47,180 --> 00:08:55,450 or the transformation that, also non-example, 133 00:08:55,450 --> 00:09:01,540 how about the transformation that, takes any vector 134 00:09:01,540 --> 00:09:04,805 and produces its length? 135 00:09:08,440 --> 00:09:11,550 So there's a transformation that takes any vector, say, 136 00:09:11,550 --> 00:09:14,360 any vector in R^3, let me just -- 137 00:09:14,360 --> 00:09:18,070 I'll just get a chance to use this notation again. 138 00:09:18,070 --> 00:09:22,490 Suppose I think of the transformation that takes any 139 00:09:22,490 --> 00:09:28,700 vector in R^3 and produces this number. 140 00:09:28,700 --> 00:09:33,660 So that, I could say, is a member of R^1, for example, 141 00:09:33,660 --> 00:09:34,960 if I wanted. 142 00:09:34,960 --> 00:09:36,180 Or just real numbers. 143 00:09:38,740 --> 00:09:41,480 That's certainly not linear. 144 00:09:41,480 --> 00:09:45,760 It's true that the zero vector goes to zero. 145 00:09:45,760 --> 00:09:49,260 But if I double a vector, it does double the length, 146 00:09:49,260 --> 00:09:50,920 that's true. 147 00:09:50,920 --> 00:09:55,810 But suppose I multiply a vector by minus two. 148 00:09:55,810 --> 00:09:58,180 What happens to its length? 149 00:09:58,180 --> 00:09:59,790 It just doubles. 150 00:09:59,790 --> 00:10:02,140 It doesn't get multiplied by minus two. 151 00:10:02,140 --> 00:10:06,690 So when c is minus two in my requirement, 152 00:10:06,690 --> 00:10:11,010 I'm not satisfying that requirement. 153 00:10:11,010 --> 00:10:16,320 So T of minus v is not minus v -- minus, the length, 154 00:10:16,320 --> 00:10:18,170 it's just the length. 155 00:10:18,170 --> 00:10:20,630 OK, so that's another non-example. 156 00:10:20,630 --> 00:10:24,350 Projection was an example, let me give you another example. 157 00:10:24,350 --> 00:10:30,810 I can stay here and have a -- this will be an example that is 158 00:10:30,810 --> 00:10:35,650 a linear transformation, a rotation. 159 00:10:35,650 --> 00:10:38,940 Rotation by -- what shall we say? 160 00:10:38,940 --> 00:10:42,050 By 45 degrees. 161 00:10:42,050 --> 00:10:42,860 OK? 162 00:10:42,860 --> 00:10:47,750 So again, let me choose this, this will be a mapping, 163 00:10:47,750 --> 00:10:54,410 from the whole plane of vectors, into the whole plane 164 00:10:54,410 --> 00:10:59,320 of vectors, and it just -- 165 00:10:59,320 --> 00:11:05,930 here is the input vector v, and the output vector foam this 45 166 00:11:05,930 --> 00:11:13,410 degree rotation is just rotate that thing by 45 degrees, T(v). 167 00:11:13,410 --> 00:11:15,770 So every vector got rotated. 168 00:11:15,770 --> 00:11:19,510 You see that I can describe this without any coordinates. 169 00:11:22,940 --> 00:11:27,280 And see that it's linear. 170 00:11:27,280 --> 00:11:33,430 If I doubled v, the rotation would just be twice as far out. 171 00:11:33,430 --> 00:11:39,220 If I had v+w, and if I rotated each of them and added, 172 00:11:39,220 --> 00:11:43,800 the answer's the same as if I add and then rotate. 173 00:11:43,800 --> 00:11:46,480 That's what the linear transformation is. 174 00:11:46,480 --> 00:11:49,350 OK, so those are two examples. 175 00:11:49,350 --> 00:11:52,900 Two examples, projection and rotation, and I 176 00:11:52,900 --> 00:12:00,790 could invent more that are linear transformations where I 177 00:12:00,790 --> 00:12:04,080 haven't told you a matrix yet. 178 00:12:04,080 --> 00:12:11,920 Actually, the book has a picture of the action of linear 179 00:12:11,920 --> 00:12:16,950 transformations -- 180 00:12:16,950 --> 00:12:19,930 actually, the cover of the book has it. 181 00:12:19,930 --> 00:12:25,060 So, in this section seven point one, we can think of a -- 182 00:12:25,060 --> 00:12:29,280 actually, here let's take this linear transformation, 183 00:12:29,280 --> 00:12:33,780 rotation, suppose I have, as the cover of the book has, 184 00:12:33,780 --> 00:12:37,020 a house in R^2. 185 00:12:37,020 --> 00:12:46,600 So instead of this, let me take a small house in R^2. 186 00:12:46,600 --> 00:12:49,200 So that's a whole lot of points. 187 00:12:49,200 --> 00:12:51,830 The idea is, with this linear transformation, 188 00:12:51,830 --> 00:12:55,300 that I can see what it does to everything at once. 189 00:12:55,300 --> 00:12:57,970 I don't have to just take one vector at a time 190 00:12:57,970 --> 00:13:00,810 and see what T of V is, I can take 191 00:13:00,810 --> 00:13:05,610 all the vectors on the outline of the house, 192 00:13:05,610 --> 00:13:09,020 and see where they all go. 193 00:13:09,020 --> 00:13:12,470 In fact, that will show me where the whole house goes. 194 00:13:12,470 --> 00:13:17,340 So what will happen with this particular linear 195 00:13:17,340 --> 00:13:19,470 transformation? 196 00:13:19,470 --> 00:13:25,410 The whole house will rotate, so the result, if I can draw it, 197 00:13:25,410 --> 00:13:29,510 will be, the house will be sitting there. 198 00:13:29,510 --> 00:13:31,930 OK. 199 00:13:31,930 --> 00:13:36,150 And, but suppose I give some other examples. 200 00:13:36,150 --> 00:13:38,900 Oh, let me give some examples that involve a matrix. 201 00:13:38,900 --> 00:13:45,250 Example three -- and this is important -- 202 00:13:45,250 --> 00:13:50,880 coming from a matrix at -- we always call A. 203 00:13:50,880 --> 00:13:59,316 So the transformation will be, multiply by A. 204 00:13:59,316 --> 00:14:00,690 There is a linear transformation. 205 00:14:04,110 --> 00:14:07,870 And a whole family of them, because every matrix 206 00:14:07,870 --> 00:14:11,280 produces a transformation by this simple rule, 207 00:14:11,280 --> 00:14:16,190 just multiply every vector by that matrix, and it's linear, 208 00:14:16,190 --> 00:14:16,900 right? 209 00:14:16,900 --> 00:14:20,510 Linear, I have to check that A(v) -- 210 00:14:20,510 --> 00:14:25,820 A times v plus w equals Av plus A w, which is fine, 211 00:14:25,820 --> 00:14:31,630 and I have to check that A times vc equals c A(v). 212 00:14:31,630 --> 00:14:32,580 Check. 213 00:14:32,580 --> 00:14:34,620 Those are fine. 214 00:14:34,620 --> 00:14:37,460 So there is a linear transformation. 215 00:14:37,460 --> 00:14:42,380 And if I take my favorite matrix A, 216 00:14:42,380 --> 00:14:46,220 and I apply it to all vectors in the plane, 217 00:14:46,220 --> 00:14:49,950 it will produce a bunch of outputs. 218 00:14:49,950 --> 00:14:52,020 See, the idea is now worth thinking 219 00:14:52,020 --> 00:14:53,490 of, like, the big picture. 220 00:14:53,490 --> 00:15:01,670 The whole plane is transformed by matrix multiplication. 221 00:15:01,670 --> 00:15:04,990 Every vector in the plane gets multiplied by A. 222 00:15:04,990 --> 00:15:08,600 Let's take an example, and see what happens 223 00:15:08,600 --> 00:15:10,390 to the vectors of the house. 224 00:15:10,390 --> 00:15:13,110 So this is still a transformation from plane 225 00:15:13,110 --> 00:15:17,900 to plane, and let me take a particular matrix A -- 226 00:15:17,900 --> 00:15:22,710 well, if I cooked up a rotation matrix, 227 00:15:22,710 --> 00:15:24,820 this would be the right picture. 228 00:15:24,820 --> 00:15:27,220 If I cooked up a projection matrix, 229 00:15:27,220 --> 00:15:29,550 the projection would be the picture. 230 00:15:29,550 --> 00:15:31,890 Let me just take some other matrix. 231 00:15:31,890 --> 00:15:35,920 Let me take the matrix one zero zero minus one. 232 00:15:39,960 --> 00:15:46,520 What happens to the house, to all vectors, and in particular, 233 00:15:46,520 --> 00:15:50,150 we can sort of visualize it if we look at the house -- 234 00:15:50,150 --> 00:15:54,530 so the house is not rotated any more, what do I get? 235 00:15:57,240 --> 00:16:02,160 What happens to all the vectors if I do this transformation? 236 00:16:02,160 --> 00:16:04,090 I multiply by this matrix. 237 00:16:04,090 --> 00:16:07,660 Well, of course, it's an easy matrix, it's diagonal. 238 00:16:07,660 --> 00:16:13,790 The x component stays the same, the y component reverses sign, 239 00:16:13,790 --> 00:16:17,510 so that like the roof of that house, 240 00:16:17,510 --> 00:16:22,890 the point, the tip of the roof, has an x component which 241 00:16:22,890 --> 00:16:26,800 stays the same, but its y component reverses, 242 00:16:26,800 --> 00:16:28,340 and it's down here. 243 00:16:28,340 --> 00:16:31,170 And, of course, what we get is, the house 244 00:16:31,170 --> 00:16:33,510 is, like, upside down. 245 00:16:33,510 --> 00:16:36,270 Now, I have to put -- where does the door go? 246 00:16:36,270 --> 00:16:40,710 I guess the door goes upside down there, right? 247 00:16:40,710 --> 00:16:47,280 So here's the input, here's the input house, 248 00:16:47,280 --> 00:16:48,960 and this is the output. 249 00:16:52,610 --> 00:16:54,120 OK. 250 00:16:54,120 --> 00:16:57,220 This idea of a linear transformation 251 00:16:57,220 --> 00:17:00,440 is like kind of the abstract description 252 00:17:00,440 --> 00:17:04,030 of matrix multiplication. 253 00:17:04,030 --> 00:17:07,560 And what's our goal here? 254 00:17:07,560 --> 00:17:11,060 Our goal is to understand linear transformations, 255 00:17:11,060 --> 00:17:15,200 and the way to understand them is 256 00:17:15,200 --> 00:17:19,380 to find the matrix that lies behind them. 257 00:17:19,380 --> 00:17:21,280 That's really the idea. 258 00:17:21,280 --> 00:17:23,349 Find the matrix that lies behind them. 259 00:17:23,349 --> 00:17:29,500 Um, and to do that, we have to bring in coordinates. 260 00:17:29,500 --> 00:17:31,630 We have to choose a basis. 261 00:17:31,630 --> 00:17:37,540 So let me point out what's the story -- 262 00:17:37,540 --> 00:17:39,810 if we have a linear transformation -- 263 00:17:39,810 --> 00:17:41,675 so start with -- 264 00:17:41,675 --> 00:17:42,175 start. 265 00:17:48,950 --> 00:17:52,030 Suppose we have a linear transformation. 266 00:17:52,030 --> 00:17:55,790 Let -- from now on, let T stand for linear transformations. 267 00:17:55,790 --> 00:17:58,490 I won't be interested in the nonlinear ones. 268 00:17:58,490 --> 00:18:01,110 Only linear transformations I'm interested in. 269 00:18:01,110 --> 00:18:01,830 OK. 270 00:18:01,830 --> 00:18:05,530 I start with a linear transformation T. 271 00:18:05,530 --> 00:18:11,855 Let's suppose its inputs are vectors in R^3. 272 00:18:14,381 --> 00:18:14,880 OK? 273 00:18:14,880 --> 00:18:21,270 And suppose its outputs are vectors in R^2, for example. 274 00:18:21,270 --> 00:18:22,210 OK. 275 00:18:22,210 --> 00:18:25,260 What's an example of such a transformation, 276 00:18:25,260 --> 00:18:26,470 just before I go any further? 277 00:18:29,570 --> 00:18:32,930 Any matrix of the right size will do this. 278 00:18:32,930 --> 00:18:35,830 So what would be the right shape of a matrix? 279 00:18:35,830 --> 00:18:37,396 So, for example -- 280 00:18:43,180 --> 00:18:44,870 I'm wanting to give you an example, 281 00:18:44,870 --> 00:18:50,260 just because, here, I'm thinking of transformations 282 00:18:50,260 --> 00:18:55,800 that take three-dimensional space to two-dimensional space. 283 00:18:55,800 --> 00:19:01,150 And I want them to be linear, and the easy way to invent them 284 00:19:01,150 --> 00:19:04,810 is a matrix multiplication. 285 00:19:04,810 --> 00:19:10,630 So example, T of v should be any A 286 00:19:10,630 --> 00:19:13,270 v. Those transformations are linear, 287 00:19:13,270 --> 00:19:15,410 that's what 18.06 is about. 288 00:19:15,410 --> 00:19:20,420 And A should be what size, what shape of matrix should that be? 289 00:19:20,420 --> 00:19:23,310 I want V to have three components, 290 00:19:23,310 --> 00:19:25,340 because this is what the inputs have -- 291 00:19:25,340 --> 00:19:39,220 so here's the input in R^3, and here's the output in R^2. 292 00:19:39,220 --> 00:19:41,860 So what shape of matrix? 293 00:19:41,860 --> 00:19:50,270 So this should be, I guess, a two by three matrix? 294 00:19:50,270 --> 00:19:50,770 Right? 295 00:19:53,860 --> 00:19:57,190 A two by three matrix. 296 00:19:57,190 --> 00:20:00,760 A two by three matrix, we'll multiply a vector in R^3 -- 297 00:20:00,760 --> 00:20:04,680 you see I'm moving to coordinates so quickly, 298 00:20:04,680 --> 00:20:09,150 I'm not a true physicist here. 299 00:20:09,150 --> 00:20:13,060 A two by three matrix, we'll multiply a vector in R^3 300 00:20:13,060 --> 00:20:16,470 an produce an output in R^2, and it will be a linear 301 00:20:16,470 --> 00:20:20,660 transformation, and OK. 302 00:20:20,660 --> 00:20:23,840 So there's a whole lot of examples, 303 00:20:23,840 --> 00:20:26,670 every two by three matrix give me an example, 304 00:20:26,670 --> 00:20:29,230 and basically, I want to show you that there 305 00:20:29,230 --> 00:20:30,520 are no other examples. 306 00:20:30,520 --> 00:20:34,540 Every linear transformation is associated with a matrix. 307 00:20:34,540 --> 00:20:38,430 Now, let me come back to the idea of linear transformation. 308 00:20:42,160 --> 00:20:48,030 Suppose I've got this linear transformation in my mind, 309 00:20:48,030 --> 00:20:50,450 and I want to tell you what it is. 310 00:20:53,220 --> 00:20:56,430 Suppose I tell you what the transformation does 311 00:20:56,430 --> 00:20:58,220 to one vector. 312 00:20:58,220 --> 00:20:58,720 OK. 313 00:20:58,720 --> 00:21:00,190 You know one thing, then. 314 00:21:00,190 --> 00:21:01,090 All right. 315 00:21:01,090 --> 00:21:05,260 So this is like the -- what I'm speaking about now is, 316 00:21:05,260 --> 00:21:20,980 how much information is needed to know the transformation? 317 00:21:20,980 --> 00:21:24,300 By knowing T, I -- 318 00:21:24,300 --> 00:21:28,930 to know T of v for all v. 319 00:21:28,930 --> 00:21:30,670 All inputs. 320 00:21:30,670 --> 00:21:33,520 How much information do I have to give you 321 00:21:33,520 --> 00:21:35,930 so that you know what the transformation does 322 00:21:35,930 --> 00:21:38,380 to every vector? 323 00:21:38,380 --> 00:21:40,950 OK, I could tell you what the transformation -- 324 00:21:40,950 --> 00:21:47,700 so I could take a vector v1, one particular vector, 325 00:21:47,700 --> 00:21:53,320 tell you what the transformation does to it -- 326 00:21:53,320 --> 00:21:54,980 fine. 327 00:21:54,980 --> 00:21:57,730 But now you only know what the transformation does to one 328 00:21:57,730 --> 00:21:59,170 vector. 329 00:21:59,170 --> 00:22:02,600 So you say, OK, that's not enough, 330 00:22:02,600 --> 00:22:05,390 tell me what it does to another vector. 331 00:22:05,390 --> 00:22:10,470 So I say, OK, give me a vector, you give me a vector v2, 332 00:22:10,470 --> 00:22:14,805 and we see, what does the transformation do to v2? 333 00:22:17,510 --> 00:22:21,570 Now, you only know -- or do you only know what 334 00:22:21,570 --> 00:22:23,650 the transformation does to two vectors? 335 00:22:23,650 --> 00:22:27,890 Have I got to ask you -- answer you about every vector 336 00:22:27,890 --> 00:22:32,570 in the whole input space, or can you, 337 00:22:32,570 --> 00:22:35,022 knowing what it does to v1 and v2, 338 00:22:35,022 --> 00:22:37,105 how much do you now know about the transformation? 339 00:22:39,730 --> 00:22:42,370 You know what the transformation does 340 00:22:42,370 --> 00:22:47,860 to a larger bunch of vectors than just these two, 341 00:22:47,860 --> 00:22:54,440 because you know what it does to every linear combination. 342 00:22:54,440 --> 00:23:00,520 You know what it does, now, to the whole plane of vectors, 343 00:23:00,520 --> 00:23:03,260 with bases v1 and v2. 344 00:23:03,260 --> 00:23:07,310 I'm assuming v1 and v2 were independent. 345 00:23:07,310 --> 00:23:12,100 If they were dependent, if v2 was six times v1, 346 00:23:12,100 --> 00:23:15,440 then I didn't give you any new information in T of v2, 347 00:23:15,440 --> 00:23:20,680 you already knew it would be six times T of v1. 348 00:23:20,680 --> 00:23:23,660 So you can see what I'd headed for. 349 00:23:23,660 --> 00:23:27,180 If I know what the transformation does 350 00:23:27,180 --> 00:23:32,150 to every vector in a basis, then I know everything. 351 00:23:32,150 --> 00:23:37,110 So the information needed to know T of v for all inputs is T 352 00:23:37,110 --> 00:23:47,890 of v1, T of v2, up to T of vm, let's say, or vn, 353 00:23:47,890 --> 00:23:50,278 for any basis -- 354 00:23:53,630 --> 00:23:58,280 for a basis v1 up to vn. 355 00:23:58,280 --> 00:24:02,030 This is a base for any -- 356 00:24:02,030 --> 00:24:04,220 can I call it an input basis? 357 00:24:04,220 --> 00:24:08,330 It's a basis for the space of inputs. 358 00:24:08,330 --> 00:24:11,860 The things that T is acting on. 359 00:24:11,860 --> 00:24:19,210 You see this point, that if I have a basis for the input 360 00:24:19,210 --> 00:24:22,550 space, and I tell you what the transformation does 361 00:24:22,550 --> 00:24:25,620 to every one of those basis vectors, that 362 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 363 00:24:31,110 --> 00:24:33,400 for all v-s, because why? 364 00:24:33,400 --> 00:24:39,870 Because every v is some combination of these basis 365 00:24:39,870 --> 00:24:47,430 vectors, c1v1+...+cnvn, that's what a basis is, right? 366 00:24:47,430 --> 00:24:49,740 It spans the space. 367 00:24:49,740 --> 00:24:56,310 And if I know what T does to this, and what T does to v2, 368 00:24:56,310 --> 00:25:05,220 and what T does to vn, then I know what T does to V. 369 00:25:05,220 --> 00:25:13,130 By this linearity, it has to be c1 T of v1 plus O one 370 00:25:13,130 --> 00:25:15,907 plus cn T of vn. 371 00:25:19,855 --> 00:25:20,605 There's no choice. 372 00:25:26,450 --> 00:25:31,380 So, the point of this comment is that if I 373 00:25:31,380 --> 00:25:37,280 know what T does to a basis, to each vector in a basis, then 374 00:25:37,280 --> 00:25:39,130 I know the linear transformation. 375 00:25:39,130 --> 00:25:43,980 The property of linearity tells me all the other vectors. 376 00:25:43,980 --> 00:25:46,420 All the other outputs. 377 00:25:46,420 --> 00:25:47,180 OK. 378 00:25:47,180 --> 00:25:54,340 So now, we got -- so that light we now see, 379 00:25:54,340 --> 00:25:56,930 what do we really need in a linear transformation, 380 00:25:56,930 --> 00:25:59,720 and we're ready to go to a 381 00:25:59,720 --> 00:26:00,740 matrix. 382 00:26:00,740 --> 00:26:01,470 OK. 383 00:26:01,470 --> 00:26:04,410 What's the step now that takes us 384 00:26:04,410 --> 00:26:07,510 from a linear transformation that's 385 00:26:07,510 --> 00:26:14,700 free of coordinates to a matrix that's been created 386 00:26:14,700 --> 00:26:16,950 with respect to coordinates? 387 00:26:16,950 --> 00:26:20,190 The matrix is going to come from the coordinate system. 388 00:26:20,190 --> 00:26:21,730 These are the coordinates. 389 00:26:21,730 --> 00:26:26,170 Coordinates mean a basis is decided. 390 00:26:26,170 --> 00:26:29,570 Once you decide on a basis -- 391 00:26:29,570 --> 00:26:30,850 this is where coordinates come 392 00:26:30,850 --> 00:26:31,470 from. 393 00:26:31,470 --> 00:26:36,280 You decide on a basis, then every vector, 394 00:26:36,280 --> 00:26:41,090 these are the coordinates in that basis. 395 00:26:41,090 --> 00:26:46,730 There is one and only one way to express v 396 00:26:46,730 --> 00:26:49,380 as a combination of the basis vectors, 397 00:26:49,380 --> 00:26:52,520 and the numbers you need in that combination 398 00:26:52,520 --> 00:26:53,430 are the coordinates. 399 00:26:53,430 --> 00:26:55,080 Let me write that down. 400 00:26:55,080 --> 00:26:56,480 So what are coordinates? 401 00:26:56,480 --> 00:27:05,030 Coordinates come from a basis. 402 00:27:10,320 --> 00:27:13,690 Coordinates come from a basis. 403 00:27:13,690 --> 00:27:17,710 The coordinates of v, the coordinates of v 404 00:27:17,710 --> 00:27:32,140 are these numbers that tell you how much of each basis vector 405 00:27:32,140 --> 00:27:33,580 is in v. 406 00:27:33,580 --> 00:27:37,580 If I change the basis, I change the coordinates, right? 407 00:27:37,580 --> 00:27:40,270 Now, we have always been assuming 408 00:27:40,270 --> 00:27:44,260 that were working with a standard basis, right? 409 00:27:44,260 --> 00:27:48,110 The basis we don't even think about this stuff, 410 00:27:48,110 --> 00:27:55,870 because if I give you the vector v equals three two four, 411 00:27:55,870 --> 00:27:59,120 you have been assuming completely -- 412 00:27:59,120 --> 00:28:04,050 and probably rightly -- that I had in mind the standard basis, 413 00:28:04,050 --> 00:28:11,910 that this vector was three times the first coordinate vector, 414 00:28:11,910 --> 00:28:16,382 and two times the second, and four times the third. 415 00:28:22,130 --> 00:28:25,010 But you're not entitled -- 416 00:28:25,010 --> 00:28:27,390 I might have had some other basis in mind. 417 00:28:27,390 --> 00:28:30,520 This is like the standard basis. 418 00:28:30,520 --> 00:28:33,770 And then the coordinates are sitting right there 419 00:28:33,770 --> 00:28:35,150 in the vector. 420 00:28:35,150 --> 00:28:37,210 But I could have chosen a different basis, 421 00:28:37,210 --> 00:28:42,400 like I might have had eigenvectors of a matrix, 422 00:28:42,400 --> 00:28:45,470 and I might have said, OK, that's a great basis, 423 00:28:45,470 --> 00:28:49,110 I'll use the eigenvectors of this matrix 424 00:28:49,110 --> 00:28:52,280 as my basis vectors. 425 00:28:52,280 --> 00:28:55,541 Which are not necessarily these three, but some other basis. 426 00:28:58,130 --> 00:29:03,180 So that was an example, this is the real thing, 427 00:29:03,180 --> 00:29:05,060 the coordinates are these numbers, 428 00:29:05,060 --> 00:29:08,490 I'll circle them again, the amounts of each basis. 429 00:29:08,490 --> 00:29:11,110 OK. 430 00:29:11,110 --> 00:29:15,710 So, if I want to create a matrix that 431 00:29:15,710 --> 00:29:17,600 describes a linear transformation, 432 00:29:17,600 --> 00:29:19,380 now I'm ready to do that. 433 00:29:19,380 --> 00:29:20,940 OK, OK. 434 00:29:20,940 --> 00:29:33,630 So now what I plan to do is construct the matrix A 435 00:29:33,630 --> 00:29:42,840 that represents, or tells me about, a linear transformation, 436 00:29:42,840 --> 00:29:47,230 linear transformation T. OK. 437 00:29:47,230 --> 00:29:51,060 So I really start with the transformation -- 438 00:29:51,060 --> 00:29:53,280 whether it's a projection or a rotation, 439 00:29:53,280 --> 00:29:57,100 or some strange movement of this house in the plane, 440 00:29:57,100 --> 00:30:02,760 or some transformation from n-dimensional space to -- 441 00:30:02,760 --> 00:30:05,260 or m-dimensional space to n-dimensional space. 442 00:30:08,200 --> 00:30:09,840 n to m, I guess. 443 00:30:09,840 --> 00:30:14,990 Usually, we'll have T, we'll somehow transform n-dimensional 444 00:30:14,990 --> 00:30:21,660 space to m-dimensional space, and the whole point is that 445 00:30:21,660 --> 00:30:25,390 if I have a basis for n-dimensional space -- 446 00:30:25,390 --> 00:30:28,360 I guess I need two bases, really. 447 00:30:28,360 --> 00:30:31,810 I need an input basis to describe the inputs, 448 00:30:31,810 --> 00:30:36,090 and I need an output basis to give me coordinates -- 449 00:30:36,090 --> 00:30:39,110 to give me some numbers for the output. 450 00:30:39,110 --> 00:30:41,240 So I've got to choose two bases. 451 00:30:41,240 --> 00:30:51,590 Choose a basis v1 up to vn for the inputs, 452 00:30:51,590 --> 00:30:54,950 for the inputs in -- 453 00:30:54,950 --> 00:30:57,550 they came from R^n. 454 00:30:57,550 --> 00:31:03,080 So the transformation is taking every n-dimensional vector 455 00:31:03,080 --> 00:31:05,150 into some m-dimensional vector. 456 00:31:05,150 --> 00:31:12,520 And I have to choose a basis, and I'll call them w1 up to wn, 457 00:31:12,520 --> 00:31:13,620 for the outputs. 458 00:31:17,120 --> 00:31:18,550 Those are guys in R^m. 459 00:31:22,380 --> 00:31:26,960 Once I've chosen the basis, that settles the matrix -- 460 00:31:26,960 --> 00:31:29,660 I now working with coordinates. 461 00:31:29,660 --> 00:31:35,190 Every vector in R^n, every input vector has some coordinates. 462 00:31:35,190 --> 00:31:38,910 So here's what I do, here's what I do. 463 00:31:38,910 --> 00:31:41,850 Can I say it in words? 464 00:31:41,850 --> 00:31:45,180 I take a vector v. 465 00:31:45,180 --> 00:31:48,214 I express it in its basis, in the basis, 466 00:31:48,214 --> 00:31:49,255 so I get its coordinates. 467 00:31:51,820 --> 00:31:55,120 Then I'm going to multiply those coordinates by the right matrix 468 00:31:55,120 --> 00:32:00,020 A, and that will give me the coordinates of the output 469 00:32:00,020 --> 00:32:01,830 in the output basis. 470 00:32:01,830 --> 00:32:05,320 I'd better write that down, that was a mouthful. 471 00:32:05,320 --> 00:32:06,870 What I want -- 472 00:32:11,000 --> 00:32:23,080 I want a matrix A that does what the linear transformation does. 473 00:32:23,080 --> 00:32:29,680 And it does it with respecting these bases. 474 00:32:29,680 --> 00:32:35,000 So I want the matrix to be -- well, let's suppose -- look, 475 00:32:35,000 --> 00:32:37,790 let me take an example. 476 00:32:37,790 --> 00:32:40,150 Let me take the projection example. 477 00:32:40,150 --> 00:32:42,260 The projection example. 478 00:32:42,260 --> 00:32:45,190 Suppose I take -- 479 00:32:45,190 --> 00:32:47,260 because we've got that -- 480 00:32:47,260 --> 00:32:49,240 we've got that projection in mind -- 481 00:32:49,240 --> 00:32:50,650 I can fit in here. 482 00:32:50,650 --> 00:32:52,100 Here's the projection example. 483 00:32:52,100 --> 00:32:58,800 So the projection example, I'm thinking of n and m as two. 484 00:32:58,800 --> 00:33:01,570 The transformation takes the plane, 485 00:33:01,570 --> 00:33:07,530 takes every vector in the plane, and, let me draw the plane, 486 00:33:07,530 --> 00:33:10,600 just so we remember it's a plane -- 487 00:33:10,600 --> 00:33:15,180 and there's the thing that I'm projecting onto, 488 00:33:15,180 --> 00:33:17,800 that's the line I'm projecting onto -- 489 00:33:17,800 --> 00:33:21,620 so the transformation takes every vector in the plane 490 00:33:21,620 --> 00:33:24,360 and projects it onto that line. 491 00:33:24,360 --> 00:33:28,570 So this is projection, so I'm going to do projection. 492 00:33:28,570 --> 00:33:29,280 OK. 493 00:33:29,280 --> 00:33:37,330 But, I'm going to choose a basis that I like better 494 00:33:37,330 --> 00:33:39,680 than the standard basis. 495 00:33:39,680 --> 00:33:44,600 My basis -- in fact, I'll choose the same basis for inputs 496 00:33:44,600 --> 00:33:49,260 and for outputs, and the basis will be -- 497 00:33:49,260 --> 00:33:53,860 my first basis vector will be right on the line. 498 00:33:53,860 --> 00:33:55,330 There's my first basis vector. 499 00:33:55,330 --> 00:33:57,510 Say, a unit vector, on the line. 500 00:33:57,510 --> 00:34:01,660 And my second basis vector will be a unit vector perpendicular 501 00:34:01,660 --> 00:34:02,700 to that line. 502 00:34:02,700 --> 00:34:05,270 And I'm going to choose that as the output basis, 503 00:34:05,270 --> 00:34:06,610 also. 504 00:34:06,610 --> 00:34:11,010 And I'm going to ask you, what's the matrix? 505 00:34:11,010 --> 00:34:14,139 What's the matrix? 506 00:34:14,139 --> 00:34:17,610 How do I describe this transformation of projection 507 00:34:17,610 --> 00:34:20,130 with respect to this basis? 508 00:34:20,130 --> 00:34:20,760 OK? 509 00:34:20,760 --> 00:34:22,050 So what's the rule? 510 00:34:22,050 --> 00:34:26,110 I take any vector v, it's some combination 511 00:34:26,110 --> 00:34:31,690 of the first basis ve- vector, and the second basis vector. 512 00:34:31,690 --> 00:34:33,420 Now, what is T of v? 513 00:34:38,080 --> 00:34:45,179 Suppose the input is -- well, suppose the input is v1. 514 00:34:45,179 --> 00:34:48,150 What's the output? 515 00:34:48,150 --> 00:34:49,870 v1, right? 516 00:34:49,870 --> 00:34:53,429 The projection leaves this one alone. 517 00:34:53,429 --> 00:34:56,790 So we know what the projection does to this first basis 518 00:34:56,790 --> 00:35:00,140 vector, this guy, it leaves it. 519 00:35:00,140 --> 00:35:04,240 What does the projection do to the second basis vector? 520 00:35:04,240 --> 00:35:08,160 It kills it, sends it to zero. 521 00:35:08,160 --> 00:35:11,060 So what does the projection do to a combination? 522 00:35:15,600 --> 00:35:20,060 It kills this part, and this part, it leaves alone. 523 00:35:23,580 --> 00:35:27,210 Now, all I want to do is find the matrix. 524 00:35:27,210 --> 00:35:29,630 I now want to find the matrix that 525 00:35:29,630 --> 00:35:35,560 takes an input, c1 c2, the coordinates, 526 00:35:35,560 --> 00:35:39,350 and gives me the output, c1 0. 527 00:35:39,350 --> 00:35:44,630 You see that in this basis, the coordinates of the input 528 00:35:44,630 --> 00:35:51,840 were c1, c2, and the coordinates of the output are c1, 529 00:35:51,840 --> 00:35:56,590 And of course, not hard to find a matrix that will do that. 530 00:35:56,590 --> 00:36:02,470 The matrix that will do that is the matrix one, zero, zero, 531 00:36:02,470 --> 00:36:04,250 zero. 532 00:36:04,250 --> 00:36:10,560 Because if I multiply input by that matrix A -- 533 00:36:10,560 --> 00:36:16,130 this is A times input coordinates -- 534 00:36:16,130 --> 00:36:18,380 and I'm hoping to get the output coordinates. 535 00:36:23,450 --> 00:36:25,610 And what do I get from that multiplication? 536 00:36:25,610 --> 00:36:27,750 I get the right answer, c1 and zero. 537 00:36:30,450 --> 00:36:31,970 So what's the point? 538 00:36:31,970 --> 00:36:36,840 So the first point is, there's a matrix that does the job. 539 00:36:36,840 --> 00:36:39,250 If there's a linear transformation out there, 540 00:36:39,250 --> 00:36:42,350 coordinate-free, no coordinates, and then I 541 00:36:42,350 --> 00:36:45,360 choose a basis for the inputs, and I 542 00:36:45,360 --> 00:36:47,790 choose a basis for the outputs, then 543 00:36:47,790 --> 00:36:51,820 there's a matrix that does And what's the job? 544 00:36:51,820 --> 00:36:52,570 the job. 545 00:36:52,570 --> 00:36:56,740 It multiplies the input coordinates and produces 546 00:36:56,740 --> 00:36:58,790 the output coordinates. 547 00:36:58,790 --> 00:37:00,730 Now, in this example -- let me repeat, 548 00:37:00,730 --> 00:37:05,380 I chose the input basis was the same as the output basis. 549 00:37:05,380 --> 00:37:09,860 The input basis and output basis were both along the line, 550 00:37:09,860 --> 00:37:12,510 and perpendicular to the line. 551 00:37:12,510 --> 00:37:16,670 They're actually the eigenvectors of the projection. 552 00:37:16,670 --> 00:37:20,370 And, as a result, the matrix came out diagonal. 553 00:37:20,370 --> 00:37:24,450 In fact, it came out to be lambda. 554 00:37:24,450 --> 00:37:27,650 This is like, the good basis. 555 00:37:27,650 --> 00:37:37,640 So the good -- the eigenvector basis is the good basis, 556 00:37:37,640 --> 00:37:43,210 it leads to the matrix -- 557 00:37:43,210 --> 00:37:49,230 the diagonal matrix of eigenvalues lambda, 558 00:37:49,230 --> 00:37:54,260 and just as in this example, the eigenvectors and eigenvalues 559 00:37:54,260 --> 00:38:00,120 of this linear transformation were along the line, 560 00:38:00,120 --> 00:38:01,630 and perpendicular. 561 00:38:01,630 --> 00:38:04,740 The eigenvalues were one and zero, 562 00:38:04,740 --> 00:38:07,380 and that's the matrix that we got. 563 00:38:07,380 --> 00:38:08,450 OK. 564 00:38:08,450 --> 00:38:12,290 So that's a, like, the great choice of matrix, 565 00:38:12,290 --> 00:38:16,110 that's the choice a physicist would do when he had to finally 566 00:38:16,110 --> 00:38:19,330 -- he or she had to finally bring coordinates 567 00:38:19,330 --> 00:38:24,740 in unwillingly, the coordinates to be chosen, 568 00:38:24,740 --> 00:38:27,240 the good coordinates are the eigenvectors, 569 00:38:27,240 --> 00:38:32,540 because, if I did this projection in the standard 570 00:38:32,540 --> 00:38:34,190 basis -- 571 00:38:34,190 --> 00:38:35,910 which I could do, right? 572 00:38:35,910 --> 00:38:40,240 I could do the whole thing in the standard basis -- 573 00:38:40,240 --> 00:38:42,790 I better try, if I can do that. 574 00:38:42,790 --> 00:38:45,390 What are we calling -- 575 00:38:45,390 --> 00:38:49,400 so I'll have to tell you now which line we're projecting on. 576 00:38:49,400 --> 00:38:51,640 Say, the 45 degree line. 577 00:38:51,640 --> 00:38:59,200 So say we're projecting onto 45 degree line, 578 00:38:59,200 --> 00:39:04,340 and we use not the eigenvector basis, but the standard basis. 579 00:39:07,240 --> 00:39:15,730 The standard basis, v1, is one, zero, and v2 is zero, one. 580 00:39:15,730 --> 00:39:18,720 And again, I'll use the same basis for the outputs. 581 00:39:22,110 --> 00:39:24,660 Then I have to do this -- 582 00:39:24,660 --> 00:39:29,990 I can find a matrix, it will be the matrix 583 00:39:29,990 --> 00:39:31,510 that we would always think of, it 584 00:39:31,510 --> 00:39:33,600 would be the projection matrix. 585 00:39:33,600 --> 00:39:39,610 It will be, actually, it's the matrix that we learned about 586 00:39:39,610 --> 00:39:47,220 in chapter four, it's what I call the matrix -- 587 00:39:47,220 --> 00:39:52,920 do you remember, P was A, A transpose over A transpose A? 588 00:39:52,920 --> 00:39:55,800 And I think, in this example, it will come out, 589 00:39:55,800 --> 00:39:58,875 one-half, one-half, one-half, one-half. 590 00:40:04,240 --> 00:40:07,940 I believe that's the matrix that comes from our formula. 591 00:40:07,940 --> 00:40:10,126 And that's the matrix that will do the job. 592 00:40:13,460 --> 00:40:19,020 If I give you this input, one, zero, what's the output? 593 00:40:19,020 --> 00:40:20,700 The output is one-half, one-half. 594 00:40:24,120 --> 00:40:28,900 And that should be the right projection. 595 00:40:28,900 --> 00:40:31,010 And if I give you the input zero, one, 596 00:40:31,010 --> 00:40:34,270 the output is, again, one-half, one-half, again the projection. 597 00:40:37,580 --> 00:40:40,950 So that's the matrix, but not diagonal of course, 598 00:40:40,950 --> 00:40:43,660 because we didn't choose a great basis, 599 00:40:43,660 --> 00:40:46,700 we just chose the handiest basis. 600 00:40:46,700 --> 00:40:49,040 Well, so the course has practically 601 00:40:49,040 --> 00:40:53,760 been about the handiest basis, and just dealing 602 00:40:53,760 --> 00:40:55,210 with the matrix that we got. 603 00:40:55,210 --> 00:40:59,400 And it's not that bad a matrix, it's symmetric, 604 00:40:59,400 --> 00:41:02,310 and it has this P squared equal P property, 605 00:41:02,310 --> 00:41:03,870 all those things are good. 606 00:41:03,870 --> 00:41:11,630 But in the best basis, it's easy to see that P squared equals P, 607 00:41:11,630 --> 00:41:15,460 and it's symmetric, and it's diagonal. 608 00:41:15,460 --> 00:41:19,480 So that's the idea then, is, do you 609 00:41:19,480 --> 00:41:24,770 see now how I'm associating a matrix to the transformation? 610 00:41:24,770 --> 00:41:28,910 I'd better write the rule down, I'd better write the rule down. 611 00:41:28,910 --> 00:41:39,230 The rule to find the matrix A. 612 00:41:39,230 --> 00:41:40,370 All right, first column. 613 00:41:40,370 --> 00:41:50,070 So, a rule to find A, we're given the bases. 614 00:41:50,070 --> 00:41:52,420 Of course, we don't -- because there's no way we could 615 00:41:52,420 --> 00:41:55,390 construct the matrix until we're told what the bases are. 616 00:41:55,390 --> 00:42:01,290 So we're given the input basis, and the output basis, v1 to vn, 617 00:42:01,290 --> 00:42:03,660 w1 to wm. 618 00:42:03,660 --> 00:42:05,530 Those are given. 619 00:42:05,530 --> 00:42:10,550 Now, in the first column of A, how do I find that column? 620 00:42:10,550 --> 00:42:13,780 The first column of the matrix. 621 00:42:13,780 --> 00:42:18,650 So that should tell me what happens to the first basis 622 00:42:18,650 --> 00:42:19,910 vector. 623 00:42:19,910 --> 00:42:28,070 So the rule is, apply the linear transformation to v1. 624 00:42:28,070 --> 00:42:32,280 To the first basis vector. 625 00:42:32,280 --> 00:42:37,010 And then, I'll write it -- so that's the output, right? 626 00:42:37,010 --> 00:42:40,830 The input is v1, what's the output? 627 00:42:40,830 --> 00:42:42,920 The output is in the output space, 628 00:42:42,920 --> 00:42:45,290 it's some combination of these guys, 629 00:42:45,290 --> 00:42:50,230 and it's that combination that goes into the first column -- 630 00:42:50,230 --> 00:42:51,230 so, let me -- 631 00:42:51,230 --> 00:42:57,510 I'll put this word -- right, I'll say it in words again. 632 00:42:57,510 --> 00:42:59,750 How to find this matrix. 633 00:42:59,750 --> 00:43:01,810 Take the first basis vector. 634 00:43:01,810 --> 00:43:04,330 Apply the transformation, then it's 635 00:43:04,330 --> 00:43:06,710 in the output space, T of v1, so it's 636 00:43:06,710 --> 00:43:11,440 some combination of these outputs, this output basis. 637 00:43:11,440 --> 00:43:14,870 So that combination, the coefficients in that 638 00:43:14,870 --> 00:43:19,680 combination will be the first column -- 639 00:43:19,680 --> 00:43:32,580 so a1, a row 2, column 1, w2, am1, wm. 640 00:43:32,580 --> 00:43:38,920 There are the numbers in the first column of the matrix. 641 00:43:38,920 --> 00:43:41,890 Let me make the point by doing the second column. 642 00:43:41,890 --> 00:43:47,960 Second column of A. 643 00:43:47,960 --> 00:43:49,550 What's the idea, now? 644 00:43:49,550 --> 00:43:54,820 I take the second basis vector, I apply the transformation 645 00:43:54,820 --> 00:43:58,510 to it, that's in -- now I get an output, 646 00:43:58,510 --> 00:44:01,570 so it's some combination in the output basis -- 647 00:44:01,570 --> 00:44:06,830 and that combination is the bunch of numbers that should go 648 00:44:06,830 --> 00:44:15,470 in the second column of the matrix. 649 00:44:15,470 --> 00:44:16,740 OK. 650 00:44:16,740 --> 00:44:18,510 And so forth. 651 00:44:18,510 --> 00:44:22,560 So I get a matrix, and the matrix I get 652 00:44:22,560 --> 00:44:24,870 does the right job. 653 00:44:24,870 --> 00:44:29,120 Now, the matrix constructed that way, and following the rules 654 00:44:29,120 --> 00:44:31,680 of matrix multiplication. 655 00:44:31,680 --> 00:44:37,310 The result will be that if I give you the input coordinates, 656 00:44:37,310 --> 00:44:42,810 and I multiply by the matrix, so the outcome of all this 657 00:44:42,810 --> 00:44:52,370 is A times the input coordinates correctly reproduces 658 00:44:52,370 --> 00:44:53,380 the output coordinates. 659 00:44:59,190 --> 00:45:01,520 Why is this right? 660 00:45:01,520 --> 00:45:04,240 Let me just check the first column. 661 00:45:04,240 --> 00:45:08,697 Suppose the input coordinates are one and all zeros. 662 00:45:08,697 --> 00:45:09,530 What does that mean? 663 00:45:09,530 --> 00:45:10,900 What's the input? 664 00:45:10,900 --> 00:45:13,080 If the input coordinates are one and other -- 665 00:45:13,080 --> 00:45:19,610 and the rest zeros, then the input is v1, right? 666 00:45:19,610 --> 00:45:23,550 That's the vector that has coordinates one and all zeros. 667 00:45:23,550 --> 00:45:24,270 OK? 668 00:45:24,270 --> 00:45:27,800 When I multiply A by the one and all zeros, 669 00:45:27,800 --> 00:45:32,270 I'll get the first column of A, I'll get these numbers. 670 00:45:32,270 --> 00:45:37,070 And, sure enough, those are the output coordinates for T of v1. 671 00:45:37,070 --> 00:45:39,860 So we made it right on the first column, 672 00:45:39,860 --> 00:45:41,940 we made it right on the second column, 673 00:45:41,940 --> 00:45:44,810 we made it right on all the basis vectors, 674 00:45:44,810 --> 00:45:50,380 and then it has to be right on every vector. 675 00:45:50,380 --> 00:45:56,890 So there is a picture of the matrix for a linear 676 00:45:56,890 --> 00:45:57,710 OK. transformation. 677 00:45:57,710 --> 00:46:01,610 Finally, let me give you another -- 678 00:46:01,610 --> 00:46:04,190 a different linear transformation. 679 00:46:04,190 --> 00:46:07,485 The linear transformation that takes the derivative. 680 00:46:10,110 --> 00:46:13,560 That's a linear transformation. 681 00:46:13,560 --> 00:46:18,590 Suppose the input space is all combination 682 00:46:18,590 --> 00:46:24,510 c1 plus c2x plus c3 x squared. 683 00:46:24,510 --> 00:46:32,620 So the basis is these simple functions. 684 00:46:32,620 --> 00:46:33,695 Then what's the output? 685 00:46:38,210 --> 00:46:39,770 Is the derivative. 686 00:46:39,770 --> 00:46:48,910 The output is the derivative, so the output is c2+2c3 x. 687 00:46:48,910 --> 00:46:55,100 And let's take as output basis, the vectors one and x. 688 00:46:55,100 --> 00:46:57,930 So we're going from a three-dimensional space 689 00:46:57,930 --> 00:47:00,660 of inputs to a two-dimensional space 690 00:47:00,660 --> 00:47:04,900 of outputs by the derivative. 691 00:47:04,900 --> 00:47:06,780 And I don't know if you ever thought 692 00:47:06,780 --> 00:47:09,375 that the derivative is linear. 693 00:47:13,520 --> 00:47:16,360 But if it weren't linear, taking derivatives 694 00:47:16,360 --> 00:47:19,380 would take forever, right? 695 00:47:19,380 --> 00:47:22,980 We are able to compute derivatives of functions 696 00:47:22,980 --> 00:47:27,030 exactly because we know it's a linear transformation, 697 00:47:27,030 --> 00:47:30,400 so that if we learn the derivatives of a few functions, 698 00:47:30,400 --> 00:47:32,960 like sine x and cos x and e to the x, 699 00:47:32,960 --> 00:47:35,730 and another little short list, then we 700 00:47:35,730 --> 00:47:37,500 can take all their combinations and we 701 00:47:37,500 --> 00:47:40,320 can do all the derivatives. 702 00:47:40,320 --> 00:47:43,010 OK, now what's the matrix? 703 00:47:43,010 --> 00:47:44,290 What's the matrix? 704 00:47:44,290 --> 00:47:50,400 So I want the matrix to multiply these input vectors -- 705 00:47:50,400 --> 00:47:57,200 input coordinates, and give these output coordinates. 706 00:47:57,200 --> 00:48:00,250 So I just think, OK, what's the matrix that does it? 707 00:48:00,250 --> 00:48:02,490 I can follow my rule of construction, 708 00:48:02,490 --> 00:48:05,320 or I can see what the matrix is. 709 00:48:05,320 --> 00:48:10,950 It should be a two by three matrix, right? 710 00:48:10,950 --> 00:48:13,230 And the matrix -- 711 00:48:13,230 --> 00:48:15,490 so I'm just figuring out, what do I want? 712 00:48:15,490 --> 00:48:17,580 No, I'll -- let me write it here. 713 00:48:17,580 --> 00:48:18,995 What do I want from my matrix? 714 00:48:22,550 --> 00:48:23,740 What should that matrix do? 715 00:48:23,740 --> 00:48:26,200 Well, I want to get c2 in the first output, 716 00:48:26,200 --> 00:48:28,920 so zero, one, zero will do it. 717 00:48:28,920 --> 00:48:34,080 I want to get two c3, so zero, zero, two will do it. 718 00:48:34,080 --> 00:48:40,030 That's the matrix for this linear transformation 719 00:48:40,030 --> 00:48:44,440 with those bases and those coordinates. 720 00:48:44,440 --> 00:48:50,330 You see, it just clicks, and the whole point is that the inverse 721 00:48:50,330 --> 00:48:53,880 matrix gives the inverse to the linear transformation, 722 00:48:53,880 --> 00:48:58,150 that the product of two matrices gives the right matrix 723 00:48:58,150 --> 00:49:00,760 for the product of two transformations -- 724 00:49:00,760 --> 00:49:04,930 matrix multiplication really came from linear 725 00:49:04,930 --> 00:49:05,860 transformations. 726 00:49:05,860 --> 00:49:10,660 I'd better pick up on that theme Monday after Thanksgiving. 727 00:49:10,660 --> 00:49:13,900 And I hope you have a great holiday. 728 00:49:13,900 --> 00:49:16,260 I hope Indian summer keeps going. 729 00:49:16,260 --> 00:49:18,670 OK, see you on Monday.