1 00:00:00,000 --> 00:00:01,025 2 00:00:01,025 --> 00:00:03,108 The following content is provided under a Creative 3 00:00:03,108 --> 00:00:03,774 Commons license. 4 00:00:03,774 --> 00:00:06,740 Your support will help MIT OpenCourseWare 5 00:00:06,740 --> 00:00:09,340 continue to offer high quality educational resources for free. 6 00:00:09,340 --> 00:00:13,380 To make a donation, or to view additional materials 7 00:00:13,380 --> 00:00:18,690 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:18,690 --> 00:00:21,820 at ocw.mit.edu. 9 00:00:21,820 --> 00:00:27,190 PROFESSOR STRANG: This is the one and only review, you 10 00:00:27,190 --> 00:00:30,740 could say, of linear algebra. 11 00:00:30,740 --> 00:00:33,660 I just think linear algebra is very important. 12 00:00:33,660 --> 00:00:35,420 You may have got that idea. 13 00:00:35,420 --> 00:00:43,430 And my website even has a little essay called Too Much Calculus. 14 00:00:43,430 --> 00:00:49,660 Because I think it's crazy for all the U.S. universities 15 00:00:49,660 --> 00:00:51,890 do this pretty much, you get semester 16 00:00:51,890 --> 00:00:55,460 after semester in differential calculus, integral calculus, 17 00:00:55,460 --> 00:00:57,130 ultimately differential equations. 18 00:00:57,130 --> 00:01:00,050 You run out of steam before the good stuff, 19 00:01:00,050 --> 00:01:03,340 before you run out of time. 20 00:01:03,340 --> 00:01:08,010 And anybody who computes, who's living in the real world, 21 00:01:08,010 --> 00:01:10,830 is using linear algebra. 22 00:01:10,830 --> 00:01:13,030 You're taking a differential equation, 23 00:01:13,030 --> 00:01:16,230 you're taking your model, making it discrete 24 00:01:16,230 --> 00:01:20,830 and computing with matrices. 25 00:01:20,830 --> 00:01:27,080 The world's digital now, not analog. 26 00:01:27,080 --> 00:01:33,870 I hope it's okay to start the course with linear algebra. 27 00:01:33,870 --> 00:01:38,200 But many engineering curricula don't fully 28 00:01:38,200 --> 00:01:40,170 recognize that and so if you haven't 29 00:01:40,170 --> 00:01:46,770 had an official course, linear algebra, stay with 18.085. 30 00:01:46,770 --> 00:01:48,280 This is a good way to learn it. 31 00:01:48,280 --> 00:01:51,330 You're sort of learning what's important. 32 00:01:51,330 --> 00:01:58,340 So my review would be-- And then this is-- Future Wednesdays 33 00:01:58,340 --> 00:02:02,190 will be in our regular room for homework, review, 34 00:02:02,190 --> 00:02:06,830 questions of all kinds, and today questions too. 35 00:02:06,830 --> 00:02:10,990 Shall I just fire away for the first half of the time 36 00:02:10,990 --> 00:02:15,350 to give you a sense of how I see the subject, 37 00:02:15,350 --> 00:02:17,950 or at least within that limited time. 38 00:02:17,950 --> 00:02:22,440 And then questions are totally welcome. 39 00:02:22,440 --> 00:02:24,031 Always welcome, actually. 40 00:02:24,031 --> 00:02:24,530 Right? 41 00:02:24,530 --> 00:02:26,590 So I'll just start right up. 42 00:02:26,590 --> 00:02:31,130 So essentially linear algebra progresses 43 00:02:31,130 --> 00:02:41,040 starting with vectors to matrices and then finally 44 00:02:41,040 --> 00:02:42,510 to subspaces. 45 00:02:42,510 --> 00:02:45,240 So that's, like, the abstraction. 46 00:02:45,240 --> 00:02:49,440 You could say abstraction, but it's not difficult, 47 00:02:49,440 --> 00:02:50,680 that you want to see. 48 00:02:50,680 --> 00:02:53,290 Until you see the idea of a subspace, 49 00:02:53,290 --> 00:02:56,130 you haven't really got linear algebra. 50 00:02:56,130 --> 00:02:58,460 Okay, so I'll start at the beginning. 51 00:02:58,460 --> 00:03:01,190 What do you do with vectors? 52 00:03:01,190 --> 00:03:04,190 Answer: you take their linear combinations. 53 00:03:04,190 --> 00:03:06,160 That's what you can do with a vector. 54 00:03:06,160 --> 00:03:11,760 You can multiply it by a number and you can add or subtract. 55 00:03:11,760 --> 00:03:13,790 So that's the key operation. 56 00:03:13,790 --> 00:03:18,120 Suppose I have vectors u, v and w. 57 00:03:18,120 --> 00:03:20,850 Let me take three of them. 58 00:03:20,850 --> 00:03:22,710 So I can take their combinations. 59 00:03:22,710 --> 00:03:28,170 So some combination will be, say some number times 60 00:03:28,170 --> 00:03:33,230 u plus some number times v plus some number times w. 61 00:03:33,230 --> 00:03:36,490 So these numbers are called scalars. 62 00:03:36,490 --> 00:03:39,640 So these would be called scalars. 63 00:03:39,640 --> 00:03:44,270 And the whole thing is a linear combination. 64 00:03:44,270 --> 00:03:48,180 Let me abbreviate those words, linear combination. 65 00:03:48,180 --> 00:03:54,390 And you get some answer, say b. 66 00:03:54,390 --> 00:04:00,600 But let's put it down, make this whole discussion specific. 67 00:04:00,600 --> 00:04:05,120 Yeah, I started a little early, I think. 68 00:04:05,120 --> 00:04:08,640 I'm going to take three vectors, u, v and w, 69 00:04:08,640 --> 00:04:12,640 and take their combinations. 70 00:04:12,640 --> 00:04:14,920 They're carefully chosen. 71 00:04:14,920 --> 00:04:20,440 My u is going to be [1, -1, 0]. 72 00:04:20,440 --> 00:04:23,580 And I'll take vectors in three dimensions. 73 00:04:23,580 --> 00:04:26,550 So that means their combinations will be in three dimensions, 74 00:04:26,550 --> 00:04:29,920 R^3, three-dimensional space. 75 00:04:29,920 --> 00:04:37,900 So that'll be u and then v, let's take 0, I think, 76 00:04:37,900 --> 00:04:40,530 1 and -1. 77 00:04:40,530 --> 00:04:41,240 Okay. 78 00:04:41,240 --> 00:04:45,170 Suppose I stopped there and took their linear combinations. 79 00:04:45,170 --> 00:04:47,820 It's very helpful to see a picture 80 00:04:47,820 --> 00:04:49,940 in three-dimensional space. 81 00:04:49,940 --> 00:04:52,100 I mean the great thing about linear algebra, 82 00:04:52,100 --> 00:04:55,000 it moves into n-dimensional space, 83 00:04:55,000 --> 00:04:58,320 10-dimensional, 100-dimensional, where we can't visualize, 84 00:04:58,320 --> 00:05:02,020 but yet, our instinct is right if we just follow. 85 00:05:02,020 --> 00:05:06,980 So what's your instinct if I took those two vectors, 86 00:05:06,980 --> 00:05:09,290 and notice they're not on the same line, 87 00:05:09,290 --> 00:05:11,180 one isn't a multiple of the other, 88 00:05:11,180 --> 00:05:13,220 they go in different directions. 89 00:05:13,220 --> 00:05:21,340 If I took their combinations, say x_1*u + x_2*v. Oh now, 90 00:05:21,340 --> 00:05:23,930 let me push, this is a serious question. 91 00:05:23,930 --> 00:05:26,630 If I took all their combinations. 92 00:05:26,630 --> 00:05:28,630 So let me try to draw a little bit. 93 00:05:28,630 --> 00:05:29,130 Okay. 94 00:05:29,130 --> 00:05:33,540 I'm in three-dimensional space and u goes somewhere, 95 00:05:33,540 --> 00:05:39,030 maybe there and v goes somewhere, maybe here. 96 00:05:39,030 --> 00:05:41,980 Now suppose I take all the combinations, 97 00:05:41,980 --> 00:05:44,040 so I could multiply that first guy 98 00:05:44,040 --> 00:05:46,740 by any number, that would fill the line. 99 00:05:46,740 --> 00:05:50,780 I can multiply that second guy, v. So this was u 100 00:05:50,780 --> 00:05:54,360 and this was v. I can multiply that by any number x_2, that 101 00:05:54,360 --> 00:05:56,290 would fill its line. 102 00:05:56,290 --> 00:05:58,070 Each of those lines I would later 103 00:05:58,070 --> 00:06:01,500 call a one-dimensional subspace, just a line. 104 00:06:01,500 --> 00:06:06,730 But now, what happens if I take all combinations of the two? 105 00:06:06,730 --> 00:06:08,550 What do you think? 106 00:06:08,550 --> 00:06:09,880 You got a plane. 107 00:06:09,880 --> 00:06:11,060 Get a plane. 108 00:06:11,060 --> 00:06:14,460 If I take anything on this line and anything on this line 109 00:06:14,460 --> 00:06:19,460 and add them up you can see that I'm not going to fill 3-D. 110 00:06:19,460 --> 00:06:23,690 But I'm going to fill a plane and that maybe 111 00:06:23,690 --> 00:06:25,640 takes a little thinking. 112 00:06:25,640 --> 00:06:28,450 It just, then it becomes sort of, 113 00:06:28,450 --> 00:06:31,230 you see that that's what it has to be. 114 00:06:31,230 --> 00:06:34,670 Okay, now I'm going to have a third vector. 115 00:06:34,670 --> 00:06:38,680 Okay, my third vector will be [0, 0, 1]. 116 00:06:38,680 --> 00:06:43,210 Okay, so that [0, 0, 1] is 0 in the x, 0 in the y 117 00:06:43,210 --> 00:06:44,660 and 1 in the z direction. 118 00:06:44,660 --> 00:06:48,740 So there's w. 119 00:06:48,740 --> 00:06:51,810 Now I want to take their combinations. 120 00:06:51,810 --> 00:06:54,530 So let me do that very specifically. 121 00:06:54,530 --> 00:06:57,310 How do I take combinations? 122 00:06:57,310 --> 00:06:58,540 This is important. 123 00:06:58,540 --> 00:07:01,550 Seems it's very simple, but important. 124 00:07:01,550 --> 00:07:07,770 I like to think of taking the combinations of some vectors, 125 00:07:07,770 --> 00:07:11,920 I'm always putting vectors into the columns of a matrix. 126 00:07:11,920 --> 00:07:15,120 So now I'm going to move to step two: matrix. 127 00:07:15,120 --> 00:07:19,090 I'm going to move to step two and maybe I'll put it-- 128 00:07:19,090 --> 00:07:20,770 well no, I better put it here. 129 00:07:20,770 --> 00:07:26,410 Okay, step two is the matrix has those vectors in its columns. 130 00:07:26,410 --> 00:07:31,790 So in this case, it's three by three. 131 00:07:31,790 --> 00:07:38,240 That's my matrix and I'm going to call it A. 132 00:07:38,240 --> 00:07:44,870 How do I take combinations of vectors? 133 00:07:44,870 --> 00:07:47,870 I should have maybe done it in detail here, 134 00:07:47,870 --> 00:07:52,790 but I'll just do it with a matrix here. 135 00:07:52,790 --> 00:07:54,850 Watch this now. 136 00:07:54,850 --> 00:08:05,030 If I multiply A by the vector of x's, what that does, 137 00:08:05,030 --> 00:08:08,080 so this is now A times x, so very important, 138 00:08:08,080 --> 00:08:10,090 a matrix times a vector. 139 00:08:10,090 --> 00:08:12,510 What does it do? 140 00:08:12,510 --> 00:08:15,730 The output is just what I want. 141 00:08:15,730 --> 00:08:17,840 This is the output. 142 00:08:17,840 --> 00:08:21,510 It takes x_1 times the first column 143 00:08:21,510 --> 00:08:24,460 plus x_2 times the second plus x_3 times the third. 144 00:08:24,460 --> 00:08:27,390 That's the way matrix multiplication works, 145 00:08:27,390 --> 00:08:28,870 by columns. 146 00:08:28,870 --> 00:08:30,870 And you don't always see that. 147 00:08:30,870 --> 00:08:32,140 Because what do you see? 148 00:08:32,140 --> 00:08:34,870 You probably know how to multiply that matrix 149 00:08:34,870 --> 00:08:35,950 by that vector. 150 00:08:35,950 --> 00:08:38,700 Let me ask you to do it. 151 00:08:38,700 --> 00:08:40,520 What do you get? 152 00:08:40,520 --> 00:08:43,770 So everybody does it a component at a time. 153 00:08:43,770 --> 00:08:48,730 So what's the first component of the answer? x_1, yeah. 154 00:08:48,730 --> 00:08:49,800 How do you get that? 155 00:08:49,800 --> 00:08:52,850 It's row times the vector. 156 00:08:52,850 --> 00:08:56,610 And when I say "times", I really mean that dot product. 157 00:08:56,610 --> 00:09:00,000 This plus this plus this is x_1. 158 00:09:00,000 --> 00:09:05,130 And what about the second row? 159 00:09:05,130 --> 00:09:06,520 -x_1 + x_2. 160 00:09:06,520 --> 00:09:09,730 Or I'll just say x_2 - x_1. 161 00:09:09,730 --> 00:09:12,480 And the third guy, the third component 162 00:09:12,480 --> 00:09:22,050 would be x_3 - x_2, right? 163 00:09:22,050 --> 00:09:25,050 So right away I'm going to say, I'm 164 00:09:25,050 --> 00:09:28,100 going to call this matrix A a difference matrix. 165 00:09:28,100 --> 00:09:31,370 It always helps to give names to things. 166 00:09:31,370 --> 00:09:35,880 So this A is a difference matrix because it takes differences 167 00:09:35,880 --> 00:09:37,430 of the x's. 168 00:09:37,430 --> 00:09:40,440 And I would even say a first difference matrix 169 00:09:40,440 --> 00:09:43,850 because it's just the straightforward difference 170 00:09:43,850 --> 00:09:48,770 and we'll see second differences in class Friday. 171 00:09:48,770 --> 00:09:50,120 So that's what A does. 172 00:09:50,120 --> 00:09:52,520 But you remember my first point was 173 00:09:52,520 --> 00:09:55,610 that when a matrix multiplies a vector, 174 00:09:55,610 --> 00:10:01,560 the result is a combination of the columns. 175 00:10:01,560 --> 00:10:05,250 And that's not always, because see, I'm looking at the picture 176 00:10:05,250 --> 00:10:07,040 not just by numbers. 177 00:10:07,040 --> 00:10:10,880 You know, with numbers I'm just doing this stuff. 178 00:10:10,880 --> 00:10:12,750 But now I'm stepping back a little bit 179 00:10:12,750 --> 00:10:14,830 and saying I'm combining. 180 00:10:14,830 --> 00:10:18,120 It's this vector times x_1. 181 00:10:18,120 --> 00:10:22,030 That vector times x_1 plus this vector times x_2 plus that 182 00:10:22,030 --> 00:10:25,790 one times x_3 added together gives me this. 183 00:10:25,790 --> 00:10:28,780 Saying nothing complicated here. 184 00:10:28,780 --> 00:10:36,960 It's just look at it by vectors, also. 185 00:10:36,960 --> 00:10:39,080 It's a little interesting, already. 186 00:10:39,080 --> 00:10:47,400 Here we multiplied these vectors by numbers. x_1, x_2, x_3. 187 00:10:47,400 --> 00:10:49,130 That was our thinking here. 188 00:10:49,130 --> 00:10:50,770 Now our thinking here is a little-- 189 00:10:50,770 --> 00:10:52,420 we've switched slightly. 190 00:10:52,420 --> 00:10:57,870 Now I'm multiplying the matrix times the numbers in x. 191 00:10:57,870 --> 00:11:01,220 Just a slight switch, multiply the matrix times the number. 192 00:11:01,220 --> 00:11:03,200 And I get some answer, b. 193 00:11:03,200 --> 00:11:07,880 Which is this, this is b. 194 00:11:07,880 --> 00:11:11,960 And of course, I can do a specific example like, 195 00:11:11,960 --> 00:11:19,850 suppose I take, well, I could take the squares to be in x. 196 00:11:19,850 --> 00:11:27,820 So suppose I take A times the first three squares, [1, 4, 9]. 197 00:11:27,820 --> 00:11:30,330 What answer would I get? 198 00:11:30,330 --> 00:11:35,130 Just to keep it clear that we're very specific here. 199 00:11:35,130 --> 00:11:38,650 So what would be the output be? 200 00:11:38,650 --> 00:11:42,260 I think of this as the input, the [1, 4, 9], the x's. 201 00:11:42,260 --> 00:11:48,140 Now the machine is multiply by A and here's the output. 202 00:11:48,140 --> 00:11:49,680 And what would be the output? 203 00:11:49,680 --> 00:11:51,270 What numbers am I going to get there? 204 00:11:51,270 --> 00:11:52,710 Yeah? 205 00:11:52,710 --> 00:11:57,700 One, three, something? [1, 3, 5]. 206 00:11:57,700 --> 00:12:01,020 Which is actually a little neat that you 207 00:12:01,020 --> 00:12:06,000 find the differences of the squares are the odd numbers. 208 00:12:06,000 --> 00:12:11,280 That appealed to me in school somehow. 209 00:12:11,280 --> 00:12:13,430 That was already a bad sign, right? 210 00:12:13,430 --> 00:12:18,790 This dumb kid notices that you take differences of squares 211 00:12:18,790 --> 00:12:21,780 and get odd numbers, whatever. 212 00:12:21,780 --> 00:12:25,170 So now is a big step. 213 00:12:25,170 --> 00:12:29,310 This was the forward direction, right? 214 00:12:29,310 --> 00:12:31,960 Input, and there's the output. 215 00:12:31,960 --> 00:12:36,460 But now the real reality-- That's easy and important, 216 00:12:36,460 --> 00:12:43,590 but the more deep problem is, what 217 00:12:43,590 --> 00:12:48,250 if I give you b and ask for x? 218 00:12:48,250 --> 00:12:52,450 So again, we're switching the direction here. 219 00:12:52,450 --> 00:12:55,990 We're solving an equation now, or three equations and three 220 00:12:55,990 --> 00:12:58,080 unknowns, Ax=b. 221 00:12:58,080 --> 00:13:02,960 So if I give you this b, can you get x? 222 00:13:02,960 --> 00:13:06,270 How do you solve three equations? 223 00:13:06,270 --> 00:13:07,960 We're looking backwards. 224 00:13:07,960 --> 00:13:14,760 Now that won't be too hard for this particular matrix 225 00:13:14,760 --> 00:13:18,350 that I chose; because it's triangular, 226 00:13:18,350 --> 00:13:23,180 we'll be able to go backwards. 227 00:13:23,180 --> 00:13:25,930 So let me do that. 228 00:13:25,930 --> 00:13:29,780 Let me take b to be-- It's a vector, 229 00:13:29,780 --> 00:13:31,630 it's got three components. 230 00:13:31,630 --> 00:13:36,190 And now I'm going to go backwards to find x. 231 00:13:36,190 --> 00:13:39,400 Or we will. 232 00:13:39,400 --> 00:13:41,950 So do you see the three equations? 233 00:13:41,950 --> 00:13:47,890 Here they are: x_1 is b_1, this is b_2, that difference is b_3. 234 00:13:47,890 --> 00:13:49,290 Those are my three equations. 235 00:13:49,290 --> 00:13:53,180 Three unknown x's, three known right-hand sides. 236 00:13:53,180 --> 00:13:57,230 Or I think of it as A times x, as a matrix 237 00:13:57,230 --> 00:14:00,500 times x giving a vector b. 238 00:14:00,500 --> 00:14:02,430 What's the answer? 239 00:14:02,430 --> 00:14:05,660 As I said, we're going to be able to do this. 240 00:14:05,660 --> 00:14:09,300 We're going to be able to solve this system easily because it's 241 00:14:09,300 --> 00:14:11,810 already triangular. 242 00:14:11,810 --> 00:14:14,290 And it's actually lower triangular 243 00:14:14,290 --> 00:14:17,890 so that means we'll start from the top. 244 00:14:17,890 --> 00:14:24,750 So the answers, the solution will be what? 245 00:14:24,750 --> 00:14:30,490 Let's make room for it. x_1, x_2, and x_3 I want to find. 246 00:14:30,490 --> 00:14:33,860 And what's the answer? 247 00:14:33,860 --> 00:14:37,450 Can we just go from top to bottom now? 248 00:14:37,450 --> 00:14:42,120 What's x_1? b_1, great. 249 00:14:42,120 --> 00:14:44,790 What's x_2? 250 00:14:44,790 --> 00:14:47,010 So x_2 - x_1. 251 00:14:47,010 --> 00:14:48,280 These are my equations. 252 00:14:48,280 --> 00:14:50,620 So what's x_2 - x_1? 253 00:14:50,620 --> 00:14:59,720 Well, it's b_2, so what is x_2? b_1 + b_2, right? 254 00:14:59,720 --> 00:15:03,030 And what's x_3? 255 00:15:03,030 --> 00:15:05,870 What do we need there for x_3? 256 00:15:05,870 --> 00:15:08,550 So I'm looking at the third equation. 257 00:15:08,550 --> 00:15:11,090 That'll determine x_3. 258 00:15:11,090 --> 00:15:13,670 When I see it this way, I see those ones 259 00:15:13,670 --> 00:15:16,190 and I see it multiplying x_3. 260 00:15:16,190 --> 00:15:20,750 And what do I get? 261 00:15:20,750 --> 00:15:25,010 Yeah, so x_3 minus this guy is b_3, 262 00:15:25,010 --> 00:15:28,040 so I have to add in another b_3, right? 263 00:15:28,040 --> 00:15:31,570 I'm doing sort of substitution down as I go. 264 00:15:31,570 --> 00:15:38,240 Once I learned that x_1 was b_1 I used it there to find x_2. 265 00:15:38,240 --> 00:15:40,700 And now I'll use x_2 to find x_3. 266 00:15:40,700 --> 00:15:48,520 And what do I get again? x_3 is, I'll put the x_2 over there. 267 00:15:48,520 --> 00:15:50,810 I think you've got it. b_1 + b_2 + b_3. 268 00:15:50,810 --> 00:15:55,410 269 00:15:55,410 --> 00:15:59,760 So that's the solution. 270 00:15:59,760 --> 00:16:02,570 Not difficult because the matrix was triangular. 271 00:16:02,570 --> 00:16:06,700 But let's think about that solution. 272 00:16:06,700 --> 00:16:13,520 That solution is a matrix times b. 273 00:16:13,520 --> 00:16:18,540 When you look at that-- So this is like a good early step 274 00:16:18,540 --> 00:16:19,700 in linear algebra. 275 00:16:19,700 --> 00:16:26,090 When I look at that I see a matrix multiplying b. 276 00:16:26,090 --> 00:16:29,000 You take that step up to seeing a matrix. 277 00:16:29,000 --> 00:16:30,560 And you can just read it off. 278 00:16:30,560 --> 00:16:34,930 So let me say, what's the matrix there that's multiplying 279 00:16:34,930 --> 00:16:41,550 b to give that answer? 280 00:16:41,550 --> 00:16:46,050 Remember the columns of this matrix-- well, 281 00:16:46,050 --> 00:16:47,960 I don't know how you want to read it off. 282 00:16:47,960 --> 00:16:52,810 But one way is, the think the columns of that matrix 283 00:16:52,810 --> 00:16:57,250 are multiplying b_1, b_2, and b_3 to give this. 284 00:16:57,250 --> 00:17:01,630 So what's the first column of the matrix? 285 00:17:01,630 --> 00:17:05,550 It's whatever I'm reading off, the coefficients, really, 286 00:17:05,550 --> 00:17:09,150 of b_1 here: [1, 1, 1]. 287 00:17:09,150 --> 00:17:13,101 And what's the second column of the matrix? [0, 1, 1]. 288 00:17:13,101 --> 00:17:13,600 Good. 289 00:17:13,600 --> 00:17:16,640 Zero b_2's, one, one. 290 00:17:16,640 --> 00:17:19,830 And the third is? [0, 0, 1]. 291 00:17:19,830 --> 00:17:23,200 Good. 292 00:17:23,200 --> 00:17:27,780 Now, so lots of things to comment here. 293 00:17:27,780 --> 00:17:30,700 Let me write up again here, this is x. 294 00:17:30,700 --> 00:17:34,390 That was the answer. 295 00:17:34,390 --> 00:17:38,630 It's a matrix times b. 296 00:17:38,630 --> 00:17:42,580 And what's the name of that matrix? 297 00:17:42,580 --> 00:17:44,470 It's the inverse matrix. 298 00:17:44,470 --> 00:17:48,120 If Ax gives b, the inverse matrix 299 00:17:48,120 --> 00:17:50,930 does it the other way around, x is A inverse b. 300 00:17:50,930 --> 00:17:53,550 Let me just put that over here. 301 00:17:53,550 --> 00:18:01,070 If Ax is b, then x should be A inverse b. 302 00:18:01,070 --> 00:18:04,680 So we had inverse, I wrote down inverse this morning 303 00:18:04,680 --> 00:18:10,150 but without saying the point, but so you see how that comes? 304 00:18:10,150 --> 00:18:11,690 I mean, if I want to go formally, 305 00:18:11,690 --> 00:18:15,100 I multiply both sides by A inverse. 306 00:18:15,100 --> 00:18:18,010 If there is an A inverse. 307 00:18:18,010 --> 00:18:20,180 That's a critical thing as we saw. 308 00:18:20,180 --> 00:18:21,740 Is the matrix invertible? 309 00:18:21,740 --> 00:18:24,430 The answer here is, yes, there is an inverse. 310 00:18:24,430 --> 00:18:26,410 And what does that really mean? 311 00:18:26,410 --> 00:18:32,140 The inverse is the thing that takes us from b back to x. 312 00:18:32,140 --> 00:18:34,820 Think of A as kind of a-- multiplying by A 313 00:18:34,820 --> 00:18:39,150 is kind of a mapping, mathematicians use the word, 314 00:18:39,150 --> 00:18:40,930 or transform. 315 00:18:40,930 --> 00:18:42,670 Transform would be good. 316 00:18:42,670 --> 00:18:45,850 Transform from x to b. 317 00:18:45,850 --> 00:18:49,100 And this is the inverse transform. 318 00:18:49,100 --> 00:18:52,490 So it doesn't happen to be the discrete Fourier 319 00:18:52,490 --> 00:18:55,940 transform or a wavelet transform, it's a-- well, 320 00:18:55,940 --> 00:18:57,830 actually we could give it a name. 321 00:18:57,830 --> 00:19:00,740 This is kind of a difference transform, right? 322 00:19:00,740 --> 00:19:03,570 That's what A did, took differences. 323 00:19:03,570 --> 00:19:06,780 So what does A inverse do? 324 00:19:06,780 --> 00:19:09,090 It takes sums. 325 00:19:09,090 --> 00:19:10,030 It takes sums. 326 00:19:10,030 --> 00:19:15,550 That's why you see 1, 1 and 1, 1, 1 along the rows 327 00:19:15,550 --> 00:19:18,160 because it's just adding, and you see it 328 00:19:18,160 --> 00:19:20,130 here in fully display. 329 00:19:20,130 --> 00:19:21,730 It's a sum matrix. 330 00:19:21,730 --> 00:19:25,170 I might as well call it S for sum. 331 00:19:25,170 --> 00:19:28,610 So that matrix, that sum matrix is the inverse 332 00:19:28,610 --> 00:19:33,750 of the different matrix. 333 00:19:33,750 --> 00:19:40,220 And maybe, since I hit on calculus earlier, 334 00:19:40,220 --> 00:19:43,590 you could say that calculus is all about one 335 00:19:43,590 --> 00:19:45,420 thing and its inverse. 336 00:19:45,420 --> 00:19:50,170 The derivative is A, and what's S? 337 00:19:50,170 --> 00:19:51,990 In calculus. 338 00:19:51,990 --> 00:19:53,510 The integral. 339 00:19:53,510 --> 00:19:57,880 The whole subject is about one operation, 340 00:19:57,880 --> 00:20:00,330 now admittedly it's not a matrix, 341 00:20:00,330 --> 00:20:04,810 it operates on functions instead of just little vectors, 342 00:20:04,810 --> 00:20:06,870 but that's the main point. 343 00:20:06,870 --> 00:20:09,390 The fundamental theorem of calculus 344 00:20:09,390 --> 00:20:14,380 is telling us that integration's the inverse of differentiation. 345 00:20:14,380 --> 00:20:20,390 So this is good and if I put in b = [1, 3, 5] 346 00:20:20,390 --> 00:20:24,660 for example just to put in some numbers, 347 00:20:24,660 --> 00:20:31,340 if I put in b = [1, 3, 5], what would the x that comes out be? 348 00:20:31,340 --> 00:20:33,170 [1, 4, 9]. 349 00:20:33,170 --> 00:20:33,870 Right? 350 00:20:33,870 --> 00:20:35,580 Because it takes us back. 351 00:20:35,580 --> 00:20:39,080 Here, previously we started, we took differences of [1, 4, 9], 352 00:20:39,080 --> 00:20:40,690 got [1, 3, 5]. 353 00:20:40,690 --> 00:20:46,300 Now if we take sums of [1, 3, 5], we get [1, 4, 9]. 354 00:20:46,300 --> 00:20:51,240 Now we have a system of linear equations. 355 00:20:51,240 --> 00:20:53,760 Now I want to step back and see what 356 00:20:53,760 --> 00:20:56,570 was good about this matrix. 357 00:20:56,570 --> 00:20:59,520 Somehow it has an inverse. 358 00:20:59,520 --> 00:21:03,040 Ax=b has a solution, in other words. 359 00:21:03,040 --> 00:21:05,510 And it has only one solution, right? 360 00:21:05,510 --> 00:21:07,190 Because we worked it out. 361 00:21:07,190 --> 00:21:08,120 We had no choice. 362 00:21:08,120 --> 00:21:10,340 That was it. 363 00:21:10,340 --> 00:21:12,300 So there's just one solution. 364 00:21:12,300 --> 00:21:14,830 There's always one and only one solution. 365 00:21:14,830 --> 00:21:18,430 It's like a perfect transform from the x's 366 00:21:18,430 --> 00:21:20,400 to the b's and back again. 367 00:21:20,400 --> 00:21:24,200 Yeah so that's what an invertible matrix is. 368 00:21:24,200 --> 00:21:29,810 It's a perfect map from one set of x's to the x's and you 369 00:21:29,810 --> 00:21:33,580 can get back again. 370 00:21:33,580 --> 00:21:36,120 Questions always. 371 00:21:36,120 --> 00:21:39,850 Now I think I'm ready for another example. 372 00:21:39,850 --> 00:21:41,140 There are only two examples. 373 00:21:41,140 --> 00:21:47,550 And actually these two examples are on the 18.06 web page. 374 00:21:47,550 --> 00:21:51,480 If some people asked after class how 375 00:21:51,480 --> 00:21:55,650 to get sort of a review of linear algebra, 376 00:21:55,650 --> 00:22:06,220 well the 18.06 website would be definitely a possibility. 377 00:22:06,220 --> 00:22:12,070 Well, I'll put down the OpenCourseWare website, mit.edu 378 00:22:12,070 --> 00:22:14,780 and then you would look at the linear algebra 379 00:22:14,780 --> 00:22:20,640 course or the math one. 380 00:22:20,640 --> 00:22:26,130 What is it? web.math.edu, is that it? 381 00:22:26,130 --> 00:22:34,930 No, maybe that's an MIT-- so is it math? 382 00:22:34,930 --> 00:22:39,190 I can't live without edu at the end, right? 383 00:22:39,190 --> 00:22:41,700 Is it just edu? 384 00:22:41,700 --> 00:22:49,690 Whatever! 385 00:22:49,690 --> 00:22:54,070 So that website has, well, all the old exams you could ever 386 00:22:54,070 --> 00:22:55,730 want if you wanted any. 387 00:22:55,730 --> 00:23:03,630 And it has this example and you click 388 00:23:03,630 --> 00:23:06,720 on Starting With Two Matrices. 389 00:23:06,720 --> 00:23:09,050 And this is one of them. 390 00:23:09,050 --> 00:23:11,240 Okay, ready for the other. 391 00:23:11,240 --> 00:23:14,100 So here comes the second matrix, second example 392 00:23:14,100 --> 00:23:16,560 that you can contrast. 393 00:23:16,560 --> 00:23:20,350 Second example is going to have the same u. 394 00:23:20,350 --> 00:23:24,996 Let me put-- our matrix, I'm going to call it, 395 00:23:24,996 --> 00:23:26,120 what am I going to call it? 396 00:23:26,120 --> 00:23:33,080 Maybe C. So it'll have the same u. 397 00:23:33,080 --> 00:23:39,580 And the same v. But I'm going to change w. 398 00:23:39,580 --> 00:23:41,430 And that's going to make all the difference. 399 00:23:41,430 --> 00:23:48,630 My w, I'm going to make that into w. 400 00:23:48,630 --> 00:23:52,700 So now I have three vectors. 401 00:23:52,700 --> 00:23:55,260 I can take their combinations. 402 00:23:55,260 --> 00:23:58,920 I can look at the equation Cx=b . 403 00:23:58,920 --> 00:24:00,710 I can try to solve it. 404 00:24:00,710 --> 00:24:06,020 All the normal stuff with those combinations 405 00:24:06,020 --> 00:24:09,200 of those three vectors. 406 00:24:09,200 --> 00:24:12,770 And we'll see a difference. 407 00:24:12,770 --> 00:24:15,880 So now, what happens if I do-- Could 408 00:24:15,880 --> 00:24:24,160 I even like do just a little erase to deal with C now? 409 00:24:24,160 --> 00:24:26,010 How does C differ? 410 00:24:26,010 --> 00:24:31,620 If I change this multiplication from A 411 00:24:31,620 --> 00:24:34,420 to C, to this new matrix. 412 00:24:34,420 --> 00:24:38,690 Then what we've done is to put in a minus one there, right? 413 00:24:38,690 --> 00:24:41,390 That's the only change we made. 414 00:24:41,390 --> 00:24:49,710 And what's the change in Cx? 415 00:24:49,710 --> 00:24:51,350 I've changed the first row, so I'm 416 00:24:51,350 --> 00:24:56,720 going to change the first row of the answer to what? x_1 - x_3. 417 00:24:56,720 --> 00:25:04,110 418 00:25:04,110 --> 00:25:06,270 You could say again, as I said this morning, 419 00:25:06,270 --> 00:25:09,050 you've sort of changed the boundary condition maybe. 420 00:25:09,050 --> 00:25:14,540 You've made this difference equation somehow circular. 421 00:25:14,540 --> 00:25:23,370 That's why I'm using that letter C. 422 00:25:23,370 --> 00:25:25,560 Is it different? 423 00:25:25,560 --> 00:25:26,920 Ah, yes! 424 00:25:26,920 --> 00:25:29,240 I didn't get it right here. 425 00:25:29,240 --> 00:25:33,820 Thank you, thank you very much. 426 00:25:33,820 --> 00:25:35,147 Absolutely. 427 00:25:35,147 --> 00:25:37,480 I mean that would have been another matrix that we could 428 00:25:37,480 --> 00:25:39,021 think about but it wouldn't have made 429 00:25:39,021 --> 00:25:42,560 the point I wanted, so thanks, that's absolutely great. 430 00:25:42,560 --> 00:25:48,260 So now it's correct here and this is correct 431 00:25:48,260 --> 00:25:53,970 and I can look at equations but can I solve them? 432 00:25:53,970 --> 00:25:56,320 Can I solve them? 433 00:25:56,320 --> 00:26:00,170 And you're guessing already, no we can't do it. 434 00:26:00,170 --> 00:26:02,380 Right? 435 00:26:02,380 --> 00:26:07,480 So now let me maybe go to a board, work 436 00:26:07,480 --> 00:26:11,440 below, because I'd hate to erase, that was so great, 437 00:26:11,440 --> 00:26:14,490 that being able to solve it in a nice clear solution 438 00:26:14,490 --> 00:26:17,010 and some matrix coming in. 439 00:26:17,010 --> 00:26:19,090 But now, how about this one? 440 00:26:19,090 --> 00:26:24,370 Okay. 441 00:26:24,370 --> 00:26:27,480 One comment I should have made here. 442 00:26:27,480 --> 00:26:30,640 Suppose the b's were zero. 443 00:26:30,640 --> 00:26:32,610 Suppose I was looking at, originally 444 00:26:32,610 --> 00:26:38,290 at A times x equal all zeroes, What's x? 445 00:26:38,290 --> 00:26:41,600 If all the b's were zero in this, 446 00:26:41,600 --> 00:26:46,220 this was the one that dealt with the matrix A. If all the b's 447 00:26:46,220 --> 00:26:50,280 are zero then the x's are zero. 448 00:26:50,280 --> 00:26:53,930 The only way to get zero right-hand sides, 449 00:26:53,930 --> 00:26:58,630 b's, was to have zero x's. 450 00:26:58,630 --> 00:27:01,670 Right? 451 00:27:01,670 --> 00:27:06,080 If you wanted to get zero out, you had to put zero in. 452 00:27:06,080 --> 00:27:08,360 Well, you can always put zero in and get zero out, 453 00:27:08,360 --> 00:27:12,250 but here you can put other vectors in and get zero out. 454 00:27:12,250 --> 00:27:17,320 So I want to say there's a solution with zeroes 455 00:27:17,320 --> 00:27:22,230 out, coming out of C, but some non-zeroes going in. 456 00:27:22,230 --> 00:27:25,630 And of course we know from this morning 457 00:27:25,630 --> 00:27:29,900 that that's a signal that it's a different sort of matrix, 458 00:27:29,900 --> 00:27:36,510 there won't be an inverse, we've got questions. 459 00:27:36,510 --> 00:27:39,260 Tell me all the solutions. 460 00:27:39,260 --> 00:27:42,780 All the solutions, so actually not just one, well you could 461 00:27:42,780 --> 00:27:44,177 tell me one, tell me one first. 462 00:27:44,177 --> 00:27:45,260 AUDIENCE: [UNINTELLIGIBLE] 463 00:27:45,260 --> 00:27:46,426 PROFESSOR STRANG: [1, 1, 1]. 464 00:27:46,426 --> 00:27:47,260 Okay. 465 00:27:47,260 --> 00:27:48,290 Now tell me all. 466 00:27:48,290 --> 00:27:50,754 AUDIENCE: C, C, C. 467 00:27:50,754 --> 00:27:51,920 PROFESSOR STRANG: [C, C, C]. 468 00:27:51,920 --> 00:27:52,450 Yeah. 469 00:27:52,450 --> 00:27:55,230 That whole line through [1, 1, 1]. 470 00:27:55,230 --> 00:27:57,910 And that would be normal. 471 00:27:57,910 --> 00:28:01,380 So this is a line of solutions. 472 00:28:01,380 --> 00:28:01,880 Right. 473 00:28:01,880 --> 00:28:03,110 A line of a solutions. 474 00:28:03,110 --> 00:28:08,190 I think of [1, 1, 1] as in some solution space, and then all 475 00:28:08,190 --> 00:28:08,710 multiples. 476 00:28:08,710 --> 00:28:10,550 That whole line. 477 00:28:10,550 --> 00:28:13,470 Later I would say it's a subspace. 478 00:28:13,470 --> 00:28:16,570 When I say what that word subspace means 479 00:28:16,570 --> 00:28:20,780 it's just this-- linear algebra's 480 00:28:20,780 --> 00:28:23,810 done its job beyond just [1, 1, 1]. 481 00:28:23,810 --> 00:28:25,030 Okay. 482 00:28:25,030 --> 00:28:33,510 So, again, it's this fact of-- if we only 483 00:28:33,510 --> 00:28:38,290 know the differences-- Yeah. 484 00:28:38,290 --> 00:28:42,040 You can see different ways that this has got problems. 485 00:28:42,040 --> 00:28:43,980 So that's C times x. 486 00:28:43,980 --> 00:28:48,520 Now one way to see a problem is to say 487 00:28:48,520 --> 00:28:52,020 we can get the answer of all zeroes by putting constants. 488 00:28:52,020 --> 00:28:55,480 All that's saying in words the differences 489 00:28:55,480 --> 00:28:58,720 of a constant factor are all zeroes, right? 490 00:28:58,720 --> 00:29:00,610 That's all that happened. 491 00:29:00,610 --> 00:29:06,410 Another way to see a problem if I had this system of equations, 492 00:29:06,410 --> 00:29:08,470 how would you see that there's a problem, 493 00:29:08,470 --> 00:29:10,960 and how would you see that there is sometimes an answer 494 00:29:10,960 --> 00:29:13,270 and even decide when? 495 00:29:13,270 --> 00:29:17,360 I don't know if you can take a quick look. 496 00:29:17,360 --> 00:29:20,990 If I had three equations, x_1-x_3 is b_1, 497 00:29:20,990 --> 00:29:23,150 this equals b_2, this equals b_3. 498 00:29:23,150 --> 00:29:27,330 499 00:29:27,330 --> 00:29:32,010 Do you see something that I can do to the left sides that's 500 00:29:32,010 --> 00:29:36,070 important somehow? 501 00:29:36,070 --> 00:29:39,510 Suppose I add those left-hand sides. 502 00:29:39,510 --> 00:29:41,040 What do I get? 503 00:29:41,040 --> 00:29:42,510 And I'm allowed to do that, right? 504 00:29:42,510 --> 00:29:45,510 If I've got three equations I'm allowed to add them, 505 00:29:45,510 --> 00:29:50,970 and I would get zero, if I add, I get zero equals -- 506 00:29:50,970 --> 00:29:53,370 I have to add the right-sides of course -- b_1+b_2+b_3. 507 00:29:53,370 --> 00:29:57,810 508 00:29:57,810 --> 00:30:00,890 I hesitate to say a fourth equation because it's not 509 00:30:00,890 --> 00:30:02,990 independent of those three, but it's 510 00:30:02,990 --> 00:30:04,930 a consequence of those three. 511 00:30:04,930 --> 00:30:11,470 So actually this is telling me when I could get an answer 512 00:30:11,470 --> 00:30:14,390 and when I couldn't. 513 00:30:14,390 --> 00:30:16,500 If I get zero on the left side I have 514 00:30:16,500 --> 00:30:19,540 to have zero on the right side or I'm lost. 515 00:30:19,540 --> 00:30:23,640 So I could actually solve this when b_1+b_2+b_3=0. 516 00:30:23,640 --> 00:30:30,920 517 00:30:30,920 --> 00:30:33,530 So I've taken a step there. 518 00:30:33,530 --> 00:30:37,210 I've said that okay, we're in trouble often, 519 00:30:37,210 --> 00:30:42,300 but in case the right-side adds up to zero then we're not. 520 00:30:42,300 --> 00:30:47,670 And if you'll allow me to jump to a mechanical meaning 521 00:30:47,670 --> 00:30:53,940 of this, if these were springs or something, masses, 522 00:30:53,940 --> 00:30:58,210 and these were forces on them -- so I'm solving 523 00:30:58,210 --> 00:31:02,150 for displacements of masses that we'll see very soon, 524 00:31:02,150 --> 00:31:07,790 and these are forces -- what that equation is saying is, 525 00:31:07,790 --> 00:31:12,200 because they're sorta cyclical, it's somehow saying that 526 00:31:12,200 --> 00:31:17,230 if the forces add up to zero, if the resulting force is zero, 527 00:31:17,230 --> 00:31:19,160 then you're okay. 528 00:31:19,160 --> 00:31:23,060 The springs and masses don't like take off, or start 529 00:31:23,060 --> 00:31:25,750 spinning or whatever. 530 00:31:25,750 --> 00:31:30,030 So there's a physical meaning for that condition 531 00:31:30,030 --> 00:31:35,730 that it's okay provided, if, the b's add up to zero. 532 00:31:35,730 --> 00:31:38,850 But of course, if the b's don't add up to zero we're lost. 533 00:31:38,850 --> 00:31:40,790 Right yeah. 534 00:31:40,790 --> 00:31:42,070 Okay. 535 00:31:42,070 --> 00:31:52,200 So Cx=b could be solved sometimes, but not always. 536 00:31:52,200 --> 00:31:55,520 The difficulty with C is showing up several ways. 537 00:31:55,520 --> 00:32:00,320 It's showing up in C times a vector x giving zero. 538 00:32:00,320 --> 00:32:02,830 That's bad news. 539 00:32:02,830 --> 00:32:05,430 Because no C inverse can bring you back. 540 00:32:05,430 --> 00:32:07,650 I mean it's like you can't come back from zero. 541 00:32:07,650 --> 00:32:11,600 Once you get to zero, C inverse can never bring you back 542 00:32:11,600 --> 00:32:14,010 to x, right? 543 00:32:14,010 --> 00:32:21,220 A took x into b up there, and then A inverse brought back x. 544 00:32:21,220 --> 00:32:23,150 But here there's no way to bring back 545 00:32:23,150 --> 00:32:25,620 that x because I can't multiply zero by anything 546 00:32:25,620 --> 00:32:27,380 and get back to x. 547 00:32:27,380 --> 00:32:30,150 So that's why I see it's got troubles here. 548 00:32:30,150 --> 00:32:33,410 Here I see it's got troubles because if I add the left sides 549 00:32:33,410 --> 00:32:35,210 I get zero. 550 00:32:35,210 --> 00:32:37,590 And therefore the right sides must add to zero. 551 00:32:37,590 --> 00:32:41,220 So you've got trouble several ways. 552 00:32:41,220 --> 00:32:45,290 Ah, let's see another way, let's see geometrically 553 00:32:45,290 --> 00:32:46,890 why were in trouble. 554 00:32:46,890 --> 00:32:53,930 Okay, so let me draw a picture to go with that picture. 555 00:32:53,930 --> 00:32:56,880 So there's three-dimensional space. 556 00:32:56,880 --> 00:33:00,940 I didn't change u, I didn't change v, 557 00:33:00,940 --> 00:33:06,450 but I changed w to minus one, what does that mean? 558 00:33:06,450 --> 00:33:09,840 Minus one sort of going this way maybe, zero, 559 00:33:09,840 --> 00:33:13,670 one is the z direction, somehow I changed it to there. 560 00:33:13,670 --> 00:33:17,630 So this is w star maybe, a different w. 561 00:33:17,630 --> 00:33:23,330 This is the w that gave me problems. 562 00:33:23,330 --> 00:33:26,160 What's the problem? 563 00:33:26,160 --> 00:33:37,170 How does the picture show the problem? 564 00:33:37,170 --> 00:33:41,030 What's the problem with those three vectors, those three 565 00:33:41,030 --> 00:33:45,730 columns of C? 566 00:33:45,730 --> 00:33:46,230 Yeah? 567 00:33:46,230 --> 00:33:47,380 AUDIENCE: [UNINTELLIGIBLE] 568 00:33:47,380 --> 00:33:49,580 PROFESSOR STRANG: There in the same plane. 569 00:33:49,580 --> 00:33:53,560 There in the same plane. w* gave us nothing new. 570 00:33:53,560 --> 00:33:56,690 We had the combinations of u and v made a plane, 571 00:33:56,690 --> 00:34:00,280 and w* happened to fall in that plane. 572 00:34:00,280 --> 00:34:06,360 So this is a plane here somehow, and goes 573 00:34:06,360 --> 00:34:09,340 through the origin of course. 574 00:34:09,340 --> 00:34:10,310 What is that plane? 575 00:34:10,310 --> 00:34:17,830 This is all combinations, all combinations of u, v, 576 00:34:17,830 --> 00:34:21,370 and the third guy, w*. 577 00:34:21,370 --> 00:34:22,040 Right. 578 00:34:22,040 --> 00:34:23,950 It's a plane, and I drew a triangle, 579 00:34:23,950 --> 00:34:28,200 but of course, I should draw the plane goes out to infinity. 580 00:34:28,200 --> 00:34:32,060 But the point is there are lots of b's, lots 581 00:34:32,060 --> 00:34:36,590 of right-hand sides not on that plane. 582 00:34:36,590 --> 00:34:37,190 Okay. 583 00:34:37,190 --> 00:34:43,830 Now if I drew all combinations of u, v, w, the original w, 584 00:34:43,830 --> 00:34:45,180 what have I got? 585 00:34:45,180 --> 00:34:48,750 So let me bring that picture back for a moment. 586 00:34:48,750 --> 00:34:51,020 If I took all combinations of those 587 00:34:51,020 --> 00:34:55,560 does w lie in the plane of u and v? 588 00:34:55,560 --> 00:34:56,580 No, right? 589 00:34:56,580 --> 00:34:58,760 I would call it independent. 590 00:34:58,760 --> 00:35:00,900 These three vectors are independent. 591 00:35:00,900 --> 00:35:05,370 These three, u, v, and w* I would call dependent. 592 00:35:05,370 --> 00:35:09,670 Because the third guy was a combination of the first two. 593 00:35:09,670 --> 00:35:13,310 Okay, so tell me what do I get now? 594 00:35:13,310 --> 00:35:16,610 So now you're really up to 3-D. What 595 00:35:16,610 --> 00:35:20,580 do you get if you take all combinations of u, v, and w? 596 00:35:20,580 --> 00:35:23,640 AUDIENCE: [INAUDIBLE]. 597 00:35:23,640 --> 00:35:25,500 PROFESSOR STRANG: Say it again. 598 00:35:25,500 --> 00:35:29,130 The whole space. 599 00:35:29,130 --> 00:35:31,495 If taking all combinations of u, v, w 600 00:35:31,495 --> 00:35:33,390 will give you the whole space. 601 00:35:33,390 --> 00:35:34,550 Why is that? 602 00:35:34,550 --> 00:35:38,620 Well we just showed-- when it was A we 603 00:35:38,620 --> 00:35:43,500 showed that we could get every b. 604 00:35:43,500 --> 00:35:48,330 We wanted the combination that gave b and we found it. 605 00:35:48,330 --> 00:35:53,990 So in the beginning when we were working with u, v, w, 606 00:35:53,990 --> 00:36:01,330 we found -- and this was short hand here -- 607 00:36:01,330 --> 00:36:04,200 this said find a combination to give b, 608 00:36:04,200 --> 00:36:06,830 and this says that combination will work. 609 00:36:06,830 --> 00:36:09,310 And we wrote out what x was. 610 00:36:09,310 --> 00:36:13,170 Now what's the difference-- Okay, here. 611 00:36:13,170 --> 00:36:21,080 So those were dependent, sorry, those were independent. 612 00:36:21,080 --> 00:36:24,580 I would even call those three vectors a basis 613 00:36:24,580 --> 00:36:25,860 for three-dimensional space. 614 00:36:25,860 --> 00:36:28,510 That word "basis" is a big deal. 615 00:36:28,510 --> 00:36:31,065 So a basis for five-dimensional space 616 00:36:31,065 --> 00:36:35,610 is five vectors that are independent. 617 00:36:35,610 --> 00:36:37,190 That's one way to say it. 618 00:36:37,190 --> 00:36:39,690 The second way to say it would be their combinations 619 00:36:39,690 --> 00:36:42,390 give the whole five-dimensional space. 620 00:36:42,390 --> 00:36:45,590 A third way to say it-- See if you can finish this sentence. 621 00:36:45,590 --> 00:36:48,140 This is for the independent, the good guys. 622 00:36:48,140 --> 00:36:54,000 If I put those five vectors into a five by five matrix, 623 00:36:54,000 --> 00:37:01,530 that matrix will be... invertible. 624 00:37:01,530 --> 00:37:04,060 That matrix will be invertible. 625 00:37:04,060 --> 00:37:06,650 So an invertible matrix is one with a basis 626 00:37:06,650 --> 00:37:09,240 sitting in its columns. 627 00:37:09,240 --> 00:37:12,810 It's a transform that has an inverse transform. 628 00:37:12,810 --> 00:37:16,300 This matrix is not invertible, those three vectors 629 00:37:16,300 --> 00:37:17,970 are not a basis. 630 00:37:17,970 --> 00:37:21,980 Their combinations are only in a plane. 631 00:37:21,980 --> 00:37:24,390 By the way, a plane as a subspace. 632 00:37:24,390 --> 00:37:27,490 A plane would be a typical subspace. 633 00:37:27,490 --> 00:37:29,030 It's like fill it out. 634 00:37:29,030 --> 00:37:31,900 You took all the combinations, you did your job, 635 00:37:31,900 --> 00:37:37,460 but in that case the whole space would count as a subspace too. 636 00:37:37,460 --> 00:37:39,540 That's the way you get subspaces, 637 00:37:39,540 --> 00:37:42,010 by taking all combinations. 638 00:37:42,010 --> 00:37:46,180 Okay, now I'm even going to push you one more step and then 639 00:37:46,180 --> 00:37:50,900 this example is complete. 640 00:37:50,900 --> 00:37:56,250 Can you tell me what vectors do you get? 641 00:37:56,250 --> 00:37:58,190 All combinations of u, v, w. 642 00:37:58,190 --> 00:37:59,710 Let me try to write something. 643 00:37:59,710 --> 00:38:08,190 This gives only a plane. 644 00:38:08,190 --> 00:38:10,470 Because we've got two independent vectors but not 645 00:38:10,470 --> 00:38:12,180 the third. 646 00:38:12,180 --> 00:38:15,370 Okay. 647 00:38:15,370 --> 00:38:17,570 I don't know if I should even ask. 648 00:38:17,570 --> 00:38:20,960 Do we know an equation for that plane? 649 00:38:20,960 --> 00:38:25,880 Well I think we do if we think about it correctly. 650 00:38:25,880 --> 00:38:33,640 All combinations of u, v, w* is the same as saying all vectors 651 00:38:33,640 --> 00:38:41,610 C times x, right? 652 00:38:41,610 --> 00:38:48,780 Do you agree that those two are exactly the same thing? 653 00:38:48,780 --> 00:38:51,940 This is the key, because we're moving up 654 00:38:51,940 --> 00:38:56,670 to vectors, combinations, and now comes subspaces. 655 00:38:56,670 --> 00:38:59,590 If I take all combinations of u, v, w*, 656 00:38:59,590 --> 00:39:02,610 I say that that's the same as all vectors C times x, 657 00:39:02,610 --> 00:39:07,520 why's that? 658 00:39:07,520 --> 00:39:12,670 It's what I said in the very first sentence at 4 o'clock. 659 00:39:12,670 --> 00:39:17,210 The combinations of u, v, w*, how do I produce them? 660 00:39:17,210 --> 00:39:21,260 I create the matrix with those columns. 661 00:39:21,260 --> 00:39:28,150 I multiply them by x's, and I get all the combinations. 662 00:39:28,150 --> 00:39:31,510 And this is just C times x. 663 00:39:31,510 --> 00:39:35,870 So what I've said there is just another way 664 00:39:35,870 --> 00:39:38,670 of saying how does matrix multiplication work. 665 00:39:38,670 --> 00:39:46,440 Put the guys in its columns and multiply by a vector. 666 00:39:46,440 --> 00:39:49,190 So we're getting all vectors C times x, 667 00:39:49,190 --> 00:39:55,870 and now I was going to stretch it that little bit further. 668 00:39:55,870 --> 00:39:58,120 Can we describe what vectors we get? 669 00:39:58,120 --> 00:40:02,220 So that's my question. 670 00:40:02,220 --> 00:40:09,550 What b's -- so this is b = [b 1, b 2, b 3] -- do we get? 671 00:40:09,550 --> 00:40:14,912 We don't get them all. 672 00:40:14,912 --> 00:40:16,120 Right, we don't get them all. 673 00:40:16,120 --> 00:40:21,050 That's the trouble with C. We only get a plane of them. 674 00:40:21,050 --> 00:40:25,440 And now can you tell me which b's 675 00:40:25,440 --> 00:40:33,480 we do get when we look at all combinations of these three 676 00:40:33,480 --> 00:40:36,910 dependent vectors. 677 00:40:36,910 --> 00:40:39,850 Well we've done a lot today. 678 00:40:39,850 --> 00:40:42,850 Let me just tell you the answer because it's here. 679 00:40:42,850 --> 00:40:45,560 The b's have to add to zero. 680 00:40:45,560 --> 00:40:49,400 That's the equation that the b's have to satisfy. 681 00:40:49,400 --> 00:40:54,820 Because when we wrote out Cx we noticed that the components 682 00:40:54,820 --> 00:40:58,550 always added to zero. 683 00:40:58,550 --> 00:41:00,270 Which b's do we get? 684 00:41:00,270 --> 00:41:05,260 We get the ones where the components add to zero. 685 00:41:05,260 --> 00:41:10,170 In other words that's the equation of the plane, 686 00:41:10,170 --> 00:41:11,000 you could say. 687 00:41:11,000 --> 00:41:11,500 Yeah. 688 00:41:11,500 --> 00:41:13,780 Actually that's a good way to look at it. 689 00:41:13,780 --> 00:41:19,720 All these vectors are on the plane. 690 00:41:19,720 --> 00:41:23,520 Do the components of u add to zero? 691 00:41:23,520 --> 00:41:25,190 Look at u. 692 00:41:25,190 --> 00:41:26,750 Yes. 693 00:41:26,750 --> 00:41:30,930 Do the components of v add to zero? 694 00:41:30,930 --> 00:41:31,510 Yes. 695 00:41:31,510 --> 00:41:32,580 Add them up. 696 00:41:32,580 --> 00:41:37,010 Do the components of w*, now that you've fixed it correctly, 697 00:41:37,010 --> 00:41:38,001 do they add to zero? 698 00:41:38,001 --> 00:41:38,500 Yes. 699 00:41:38,500 --> 00:41:40,470 So all the combinations will add to zero. 700 00:41:40,470 --> 00:41:42,800 That's the plane. 701 00:41:42,800 --> 00:41:44,370 That's the plane. 702 00:41:44,370 --> 00:41:47,880 You see there are so many different ways to see, 703 00:41:47,880 --> 00:41:50,610 and none of this is difficult, but it's 704 00:41:50,610 --> 00:41:55,100 coming fast because we're seeing the same thing 705 00:41:55,100 --> 00:41:56,310 in different languages. 706 00:41:56,310 --> 00:41:59,620 We're seeing it geometrically in a picture of a plane. 707 00:41:59,620 --> 00:42:02,190 We're seeing it as a combination of vectors. 708 00:42:02,190 --> 00:42:05,310 We're seeing it as a multiplication by a matrix. 709 00:42:05,310 --> 00:42:11,280 And we saw it sort of here by operation, 710 00:42:11,280 --> 00:42:16,140 operating and simplifying, and getting the key fact out 711 00:42:16,140 --> 00:42:21,000 of the equations. 712 00:42:21,000 --> 00:42:22,880 Well, okay. 713 00:42:22,880 --> 00:42:28,580 I wanted to give you this example, the two examples, 714 00:42:28,580 --> 00:42:31,380 because they bring out so many of the key ideas. 715 00:42:31,380 --> 00:42:34,990 The key idea of a subspace. 716 00:42:34,990 --> 00:42:38,670 Shall I just say a little about what that word means? 717 00:42:38,670 --> 00:42:41,430 A subspace. 718 00:42:41,430 --> 00:42:43,560 What's a subspace? 719 00:42:43,560 --> 00:42:48,360 Well, what's a vector space first of all? 720 00:42:48,360 --> 00:42:51,570 A vector space is a bunch of vectors. 721 00:42:51,570 --> 00:42:54,770 And the rule is you have to be able to take 722 00:42:54,770 --> 00:42:56,130 their combinations. 723 00:42:56,130 --> 00:42:57,760 That what linear algebra does. 724 00:42:57,760 --> 00:42:59,370 Takes combinations. 725 00:42:59,370 --> 00:43:05,810 So a vector space is one where you take all combinations. 726 00:43:05,810 --> 00:43:09,690 So if I only took just this triangle that would not 727 00:43:09,690 --> 00:43:14,380 be a subspace because one combination would be 2u 728 00:43:14,380 --> 00:43:16,190 and it would be out of the triangle. 729 00:43:16,190 --> 00:43:22,210 So a subspace, just think of it as a plane, 730 00:43:22,210 --> 00:43:25,370 but then of course it could be in higher dimensions. 731 00:43:25,370 --> 00:43:28,010 You know it could be a 7-dimensional subspace 732 00:43:28,010 --> 00:43:30,820 inside a 15-dimensional space. 733 00:43:30,820 --> 00:43:37,780 And I don't know if you're good at visualizing that, I'm not. 734 00:43:37,780 --> 00:43:38,730 Never mind. 735 00:43:38,730 --> 00:43:41,730 You you've got seven vectors, you think okay, 736 00:43:41,730 --> 00:43:44,800 their combinations give us seven-dimensional subspace. 737 00:43:44,800 --> 00:43:47,250 Each factor has 15 components. 738 00:43:47,250 --> 00:43:48,300 No problem. 739 00:43:48,300 --> 00:43:50,110 I mean no problem for MATLAB certainly. 740 00:43:50,110 --> 00:43:53,080 It's got what, a matrix with a 105 entries. 741 00:43:53,080 --> 00:43:55,770 It deals with that instantly. 742 00:43:55,770 --> 00:44:02,480 Okay, so a subspace is like a vector space inside a bigger 743 00:44:02,480 --> 00:44:03,200 one. 744 00:44:03,200 --> 00:44:06,460 That's why the prefix "sub-" is there. 745 00:44:06,460 --> 00:44:07,160 Right? 746 00:44:07,160 --> 00:44:11,720 And mathematics always counts the biggest possibility too, 747 00:44:11,720 --> 00:44:13,640 which would be the whole space. 748 00:44:13,640 --> 00:44:15,680 And what's the smallest? 749 00:44:15,680 --> 00:44:19,160 So what's the smallest subspace of R^3? 750 00:44:19,160 --> 00:44:21,880 So I have 3-dimensional space-- you can tell me all 751 00:44:21,880 --> 00:44:23,880 the subspaces of R^3. 752 00:44:23,880 --> 00:44:25,980 So there is one, a plane. 753 00:44:25,980 --> 00:44:28,180 Yeah, tell me all the subspaces of R^3. 754 00:44:28,180 --> 00:44:31,450 And then you'll have that word kind of down. 755 00:44:31,450 --> 00:44:33,050 AUDIENCE: [UNINTELLIGIBLE] 756 00:44:33,050 --> 00:44:34,660 PROFESSOR STRANG: A line. 757 00:44:34,660 --> 00:44:37,420 So planes and lines, those you could say, 758 00:44:37,420 --> 00:44:39,490 the real, the proper subspaces. 759 00:44:39,490 --> 00:44:41,330 The best, the right ones. 760 00:44:41,330 --> 00:44:45,230 But there are a couple more possibilities which are? 761 00:44:45,230 --> 00:44:46,337 AUDIENCE: [UNINTELLIGIBLE] 762 00:44:46,337 --> 00:44:47,420 PROFESSOR STRANG: A point. 763 00:44:47,420 --> 00:44:49,140 Which point? 764 00:44:49,140 --> 00:44:50,010 The origin. 765 00:44:50,010 --> 00:44:50,980 Only the origin. 766 00:44:50,980 --> 00:44:56,020 Because if you tried to say that point was a subspace, no way. 767 00:44:56,020 --> 00:44:56,910 Why not? 768 00:44:56,910 --> 00:44:58,520 Because I wouldn't be able to multiply 769 00:44:58,520 --> 00:45:03,870 that vector by five and I would be away from the point. 770 00:45:03,870 --> 00:45:08,790 But the zero subspace, the really small subspace 771 00:45:08,790 --> 00:45:12,670 that just has the zero vector, it's got one vector in it. 772 00:45:12,670 --> 00:45:13,330 Not empty. 773 00:45:13,330 --> 00:45:15,520 It's got that one point but that's all. 774 00:45:15,520 --> 00:45:20,090 Okay, so planes, lines, the origin, 775 00:45:20,090 --> 00:45:25,030 and then the other possibility for a subspaces is? 776 00:45:25,030 --> 00:45:26,000 The whole space. 777 00:45:26,000 --> 00:45:26,520 Right. 778 00:45:26,520 --> 00:45:27,080 Right. 779 00:45:27,080 --> 00:45:30,390 So the dimensions could be three for the whole space, 780 00:45:30,390 --> 00:45:35,420 two for a plane, one for a line, zero for a point. 781 00:45:35,420 --> 00:45:41,300 It just kicks together. 782 00:45:41,300 --> 00:45:42,300 How are we for time? 783 00:45:42,300 --> 00:45:48,120 Maybe I went more than half, but now is a chance to just ask me, 784 00:45:48,120 --> 00:45:53,160 if you want to, like anything about the course. 785 00:45:53,160 --> 00:45:54,540 Is at all linear algebra? 786 00:45:54,540 --> 00:45:55,380 No. 787 00:45:55,380 --> 00:46:02,520 But I think I can't do anything more helpful to you 788 00:46:02,520 --> 00:46:06,230 then to for you to begin to see-- 789 00:46:06,230 --> 00:46:09,960 when you look at a matrix, begin to see what is it doing. 790 00:46:09,960 --> 00:46:11,640 What is it about. 791 00:46:11,640 --> 00:46:14,910 Right, and of course matrices can be rectangular. 792 00:46:14,910 --> 00:46:17,970 So I'll give you a hint about what's 793 00:46:17,970 --> 00:46:20,780 coming in the course itself. 794 00:46:20,780 --> 00:46:26,340 We'll have rectangular matrices A, okay. 795 00:46:26,340 --> 00:46:28,310 They're not invertible. 796 00:46:28,310 --> 00:46:30,760 They're taking seven-dimensional space 797 00:46:30,760 --> 00:46:33,720 to three-dimensional space or something. 798 00:46:33,720 --> 00:46:35,750 You can't invert that. 799 00:46:35,750 --> 00:46:40,070 What comes up every time-- I sort of got the idea finally. 800 00:46:40,070 --> 00:46:45,510 Every time I see a rectangular matrix, maybe seven by three, 801 00:46:45,510 --> 00:46:48,890 that would be seven rows by three columns. 802 00:46:48,890 --> 00:46:53,170 Then what comes up with a rectangular matrix 803 00:46:53,170 --> 00:46:57,820 A is sooner or later A transpose sticks its nose in 804 00:46:57,820 --> 00:47:07,970 and multiplies that A. And we couldn't do it for our A here. 805 00:47:07,970 --> 00:47:10,730 Actually if I did it for that original matrix A 806 00:47:10,730 --> 00:47:14,770 I would get something you'd recognize. 807 00:47:14,770 --> 00:47:18,790 What I want to say is that the course focuses 808 00:47:18,790 --> 00:47:23,860 on A transpose A. I'll just say now that that matrix always 809 00:47:23,860 --> 00:47:27,390 comes out square, because this would be three times seven 810 00:47:27,390 --> 00:47:31,380 times seven times three, so this would be three by three, 811 00:47:31,380 --> 00:47:34,300 and it always comes out symmetric. 812 00:47:34,300 --> 00:47:36,300 That's the nice thing. 813 00:47:36,300 --> 00:47:37,190 And even more. 814 00:47:37,190 --> 00:47:39,760 We'll see more. 815 00:47:39,760 --> 00:47:41,590 That's like a hint. 816 00:47:41,590 --> 00:47:48,320 Watch for A transpose A. And watch for it 817 00:47:48,320 --> 00:47:52,740 in applications of all kinds. 818 00:47:52,740 --> 00:47:56,740 In networks an A will be associated 819 00:47:56,740 --> 00:47:59,270 with Kirchhoff's voltage law, and A transpose 820 00:47:59,270 --> 00:48:00,530 with Kirchhoff's current law. 821 00:48:00,530 --> 00:48:04,720 They just teamed up together. 822 00:48:04,720 --> 00:48:06,780 We'll see more. 823 00:48:06,780 --> 00:48:10,230 Alright now let me give you a chance to ask any question. 824 00:48:10,230 --> 00:48:14,230 Whatever. 825 00:48:14,230 --> 00:48:15,370 Homework. 826 00:48:15,370 --> 00:48:17,460 Did I mention homework? 827 00:48:17,460 --> 00:48:21,450 You may have said that's a crazy homework 828 00:48:21,450 --> 00:48:24,730 to say three problems 1.1. 829 00:48:24,730 --> 00:48:30,200 I've never done this before so essentially you 830 00:48:30,200 --> 00:48:34,380 can get away with anything this week, 831 00:48:34,380 --> 00:48:37,780 and indefinitely actually. 832 00:48:37,780 --> 00:48:42,300 How many are-- Is this the first day of MIT classes? 833 00:48:42,300 --> 00:48:43,000 Oh wow. 834 00:48:43,000 --> 00:48:43,840 Okay. 835 00:48:43,840 --> 00:48:46,380 Well, welcome to MIT. 836 00:48:46,380 --> 00:48:49,930 I hope you like it. 837 00:48:49,930 --> 00:48:56,050 It's not so high pressure or whatever 838 00:48:56,050 --> 00:48:57,560 is associated with MIT. 839 00:48:57,560 --> 00:49:01,450 It's kind of tolerant. 840 00:49:01,450 --> 00:49:04,160 If my advisees ask for something I always say yes. 841 00:49:04,160 --> 00:49:05,610 It's easier that way. 842 00:49:05,610 --> 00:49:11,030 AUDIENCE: [LAUGHTER]. 843 00:49:11,030 --> 00:49:15,400 PROFESSOR STRANG: And let me just again-- 844 00:49:15,400 --> 00:49:19,000 and I'll say it often and in private. 845 00:49:19,000 --> 00:49:21,170 This is like a grown-up course. 846 00:49:21,170 --> 00:49:23,430 I'm figuring you're here to learn, 847 00:49:23,430 --> 00:49:26,090 so it's not my job to force it. 848 00:49:26,090 --> 00:49:31,570 My job is to help it, and hope this is some help.