1 00:00:09,760 --> 00:00:13,450 OK, here is lecture ten in linear algebra. 2 00:00:13,450 --> 00:00:17,400 Two important things to do in this lecture. 3 00:00:17,400 --> 00:00:21,840 One is to correct an error from lecture nine. 4 00:00:21,840 --> 00:00:27,250 So the blackboard with that awful error is still with us. 5 00:00:27,250 --> 00:00:29,580 And the second, the big thing to do 6 00:00:29,580 --> 00:00:34,200 is to tell you about the four subspaces that 7 00:00:34,200 --> 00:00:35,930 come with a matrix. 8 00:00:35,930 --> 00:00:39,030 We've seen two subspaces, the column space 9 00:00:39,030 --> 00:00:39,900 and the null space. 10 00:00:39,900 --> 00:00:42,490 There's two to go. 11 00:00:42,490 --> 00:00:45,760 First of all, and this is a great way to 12 00:00:45,760 --> 00:00:50,690 OK. recap and correct the previous lecture -- 13 00:00:50,690 --> 00:00:54,190 so you remember I was just doing R^3. 14 00:00:54,190 --> 00:00:58,620 I couldn't have taken a simpler example than R^3. 15 00:00:58,620 --> 00:01:02,230 And I wrote down the standard basis. 16 00:01:02,230 --> 00:01:03,945 That's the standard basis. 17 00:01:06,780 --> 00:01:11,750 The basis -- the obvious basis for the whole three dimensional 18 00:01:11,750 --> 00:01:12,820 space. 19 00:01:12,820 --> 00:01:16,500 And then I wanted to make the point 20 00:01:16,500 --> 00:01:22,940 that there was nothing special, nothing about that basis 21 00:01:22,940 --> 00:01:25,130 that another basis couldn't have. 22 00:01:25,130 --> 00:01:26,970 It could have linear independence, 23 00:01:26,970 --> 00:01:28,530 it could span a space. 24 00:01:28,530 --> 00:01:30,350 There's lots of other bases. 25 00:01:30,350 --> 00:01:34,280 So I started with these vectors, one one two and two two five, 26 00:01:34,280 --> 00:01:36,710 and those were independent. 27 00:01:36,710 --> 00:01:39,200 And then I said three three seven 28 00:01:39,200 --> 00:01:42,990 wouldn't do, because three three seven is the sum of those. 29 00:01:42,990 --> 00:01:47,920 So in my innocence, I put in three three eight. 30 00:01:47,920 --> 00:01:53,150 I figured probably if three three seven is on the plane, 31 00:01:53,150 --> 00:01:56,210 is -- which I know, it's in the plane with these two, 32 00:01:56,210 --> 00:01:59,100 then probably three three eight sticks a little bit out 33 00:01:59,100 --> 00:02:02,010 of the plane and it's independent and it gives 34 00:02:02,010 --> 00:02:02,530 a basis. 35 00:02:02,530 --> 00:02:09,810 But after class, to my sorrow, a student tells me, 36 00:02:09,810 --> 00:02:12,620 "Wait a minute, that ba- that third vector, three three 37 00:02:12,620 --> 00:02:14,690 eight, is not independent." 38 00:02:14,690 --> 00:02:17,620 And why did she say that? 39 00:02:17,620 --> 00:02:22,560 She didn't actually take the time, didn't have to, 40 00:02:22,560 --> 00:02:27,890 to find w- w- what combination of this one and this one 41 00:02:27,890 --> 00:02:30,040 gives three three eight. 42 00:02:30,040 --> 00:02:33,170 She did something else. 43 00:02:33,170 --> 00:02:35,090 In other words, she looked ahead, 44 00:02:35,090 --> 00:02:41,080 because she said, wait a minute, if I look at that matrix, 45 00:02:41,080 --> 00:02:43,430 it's not invertible. 46 00:02:43,430 --> 00:02:47,530 That third column can't be independent of the first two, 47 00:02:47,530 --> 00:02:49,270 because when I look at that matrix, 48 00:02:49,270 --> 00:02:53,110 it's got two identical rows. 49 00:02:53,110 --> 00:02:56,490 I have a square matrix. 50 00:02:56,490 --> 00:03:01,240 Its rows are obviously dependent. 51 00:03:01,240 --> 00:03:06,670 And that makes the columns dependent. 52 00:03:06,670 --> 00:03:08,620 So there's my error. 53 00:03:08,620 --> 00:03:13,800 When I look at the matrix A that has those three columns, 54 00:03:13,800 --> 00:03:16,090 those three columns can't be independent 55 00:03:16,090 --> 00:03:18,170 because that matrix is not invertible 56 00:03:18,170 --> 00:03:21,920 because it's got two equal rows. 57 00:03:21,920 --> 00:03:27,260 And today's lecture will reach the conclusion, 58 00:03:27,260 --> 00:03:33,110 the great conclusion, that connects the column 59 00:03:33,110 --> 00:03:35,305 space with the row space. 60 00:03:38,030 --> 00:03:43,020 So those are -- the row space is now going to be another one 61 00:03:43,020 --> 00:03:45,510 of my fundamental subspaces. 62 00:03:45,510 --> 00:03:49,430 The row space of this matrix, or of this one -- well, 63 00:03:49,430 --> 00:03:54,880 the row space of this one is OK, but the row space of this one, 64 00:03:54,880 --> 00:03:58,240 I'm looking at the rows of the matrix -- oh, anyway, 65 00:03:58,240 --> 00:04:02,620 I'll have two equal rows and the row space will be only two 66 00:04:02,620 --> 00:04:04,060 dimensional. 67 00:04:04,060 --> 00:04:08,320 The rank of the matrix with these columns will only be two. 68 00:04:08,320 --> 00:04:13,730 So only two of those columns, columns can be independent too. 69 00:04:13,730 --> 00:04:17,730 The rows tell me something about the columns, in other words, 70 00:04:17,730 --> 00:04:20,560 something that I should have noticed and I didn't. 71 00:04:20,560 --> 00:04:21,500 OK. 72 00:04:21,500 --> 00:04:27,470 So now let me pin down these four fundamental subspaces. 73 00:04:27,470 --> 00:04:30,500 So here are the four fundamental subspaces. 74 00:04:36,510 --> 00:04:40,470 This is really the heart of this approach to linear algebra, 75 00:04:40,470 --> 00:04:44,520 to see these four subspaces, how they're related. 76 00:04:44,520 --> 00:04:45,830 So what are they? 77 00:04:45,830 --> 00:04:53,540 The column space, C of A. 78 00:04:53,540 --> 00:05:00,420 The null space, N of A. 79 00:05:00,420 --> 00:05:05,190 And now comes the row space, something new. 80 00:05:05,190 --> 00:05:08,700 The row space, what's in that? 81 00:05:08,700 --> 00:05:12,190 It's all combinations of the rows. 82 00:05:12,190 --> 00:05:13,620 That's natural. 83 00:05:13,620 --> 00:05:16,440 We want a space, so we have to take all combinations, 84 00:05:16,440 --> 00:05:18,240 and we start with the rows. 85 00:05:18,240 --> 00:05:22,450 So the rows span the row space. 86 00:05:22,450 --> 00:05:26,200 Are the rows a basis for the row space? 87 00:05:26,200 --> 00:05:29,140 Maybe so, maybe no. 88 00:05:29,140 --> 00:05:33,080 The rows are a basis for the row space when they're independent, 89 00:05:33,080 --> 00:05:39,250 but if they're dependent, as in this example, 90 00:05:39,250 --> 00:05:43,500 my error from last time, they're not -- 91 00:05:43,500 --> 00:05:45,440 those three rows are not a basis. 92 00:05:45,440 --> 00:05:48,280 The row space wouldn't -- would only be two dimensional. 93 00:05:48,280 --> 00:05:50,280 I only need two rows for a basis. 94 00:05:50,280 --> 00:05:52,990 So the row space, now what's in it? 95 00:05:52,990 --> 00:05:55,610 It's all combinations of the rows of A. 96 00:05:55,610 --> 00:06:01,330 All combinations of the rows of A. 97 00:06:01,330 --> 00:06:04,610 But I don't like working with row vectors. 98 00:06:04,610 --> 00:06:06,680 All my vectors have been column vectors. 99 00:06:06,680 --> 00:06:09,710 I'd like to stay with column vectors. 100 00:06:09,710 --> 00:06:14,530 How can I get to column vectors out of these rows? 101 00:06:14,530 --> 00:06:16,800 I transpose the matrix. 102 00:06:16,800 --> 00:06:20,020 So if that's OK with you, I'm going to transpose the 103 00:06:20,020 --> 00:06:20,590 matrix. 104 00:06:20,590 --> 00:06:24,450 I'm, I'm going to say all combinations 105 00:06:24,450 --> 00:06:31,875 of the columns of A transpose. 106 00:06:34,990 --> 00:06:40,750 And that allows me to use the convenient notation, the column 107 00:06:40,750 --> 00:06:42,755 space of A transpose. 108 00:06:47,310 --> 00:06:50,320 Nothing, no mathematics went on there. 109 00:06:50,320 --> 00:06:55,130 We just got some vectors that were lying down to stand up. 110 00:06:55,130 --> 00:06:59,730 But it means that we can use this column 111 00:06:59,730 --> 00:07:04,120 space of A transpose, that's telling me in a nice matrix 112 00:07:04,120 --> 00:07:08,950 notation what the row space is. 113 00:07:08,950 --> 00:07:09,470 OK. 114 00:07:09,470 --> 00:07:15,900 And finally is another null space. 115 00:07:15,900 --> 00:07:19,900 The fourth fundamental space will be 116 00:07:19,900 --> 00:07:22,120 the null space of A transpose. 117 00:07:22,120 --> 00:07:30,140 The fourth guy is the null space of A transpose. 118 00:07:30,140 --> 00:07:33,395 And of course my notation is N of A transpose. 119 00:07:36,230 --> 00:07:40,010 That's the null space of A transpose. 120 00:07:40,010 --> 00:07:46,250 Eh, we don't have a perfect name for this space as a -- 121 00:07:46,250 --> 00:07:51,770 connecting with A, but our usual name is the left null space, 122 00:07:51,770 --> 00:07:54,120 and I'll show you why in a moment. 123 00:07:54,120 --> 00:07:57,480 So often I call this the -- 124 00:07:57,480 --> 00:08:10,570 just to write that word -- the left null space of A. 125 00:08:10,570 --> 00:08:13,420 So just the way we have the row space of A 126 00:08:13,420 --> 00:08:16,810 and we switch it to the column space of A transpose, 127 00:08:16,810 --> 00:08:21,590 so we have this space of guys l- that I 128 00:08:21,590 --> 00:08:25,040 call the left null space of A, but the good notation 129 00:08:25,040 --> 00:08:27,910 is it's the null space of A transpose. 130 00:08:27,910 --> 00:08:28,410 OK. 131 00:08:28,410 --> 00:08:29,770 Those are four spaces. 132 00:08:32,980 --> 00:08:35,280 Where are those spaces? 133 00:08:35,280 --> 00:08:40,059 What, what big space are they in for -- when A is m by n? 134 00:08:47,720 --> 00:08:52,370 In that case, the null space of A, 135 00:08:52,370 --> 00:08:54,100 what's in the null space of A? 136 00:08:54,100 --> 00:08:58,850 Vectors with n components, solutions to A x equals zero. 137 00:08:58,850 --> 00:09:01,663 So the null space of A is in R^n. 138 00:09:04,800 --> 00:09:06,480 What's in the column space of A? 139 00:09:06,480 --> 00:09:08,220 Well, columns. 140 00:09:08,220 --> 00:09:11,320 How many components dothose columns have? 141 00:09:11,320 --> 00:09:12,350 m. 142 00:09:12,350 --> 00:09:15,840 So this column space is in R^m. 143 00:09:19,680 --> 00:09:22,010 What about the column space of A transpose, 144 00:09:22,010 --> 00:09:26,960 which are just a disguised way of saying the rows of A? 145 00:09:26,960 --> 00:09:31,040 The rows of A, in this three by six matrix, 146 00:09:31,040 --> 00:09:36,680 have six components, n components. 147 00:09:36,680 --> 00:09:38,420 The column space is in R^n. 148 00:09:42,790 --> 00:09:45,950 And the null space of A transpose, 149 00:09:45,950 --> 00:09:51,800 I see that this fourth space is already getting second, 150 00:09:51,800 --> 00:09:55,640 you know, second class citizen treatment 151 00:09:55,640 --> 00:09:57,550 and it doesn't deserve it. 152 00:09:57,550 --> 00:10:00,830 It's, it should be there, it is there, 153 00:10:00,830 --> 00:10:03,430 and shouldn't be squeezed. 154 00:10:03,430 --> 00:10:06,650 The null space of A transpose -- 155 00:10:06,650 --> 00:10:10,520 well, if the null space of A had vectors with n components, 156 00:10:10,520 --> 00:10:14,970 the null space of A transpose will be in R^m. 157 00:10:14,970 --> 00:10:19,260 I want to draw a picture of the four spaces. 158 00:10:19,260 --> 00:10:19,760 OK. 159 00:10:24,580 --> 00:10:25,130 OK. 160 00:10:25,130 --> 00:10:26,380 Here are the four spaces. 161 00:10:32,450 --> 00:10:38,580 OK, Let me put n dimensional space over on this side. 162 00:10:38,580 --> 00:10:42,760 Then which were the subspaces in R^n? 163 00:10:42,760 --> 00:10:47,490 The null space was and the row space was. 164 00:10:47,490 --> 00:10:53,471 So here we have the -- can I make that picture of the row 165 00:10:53,471 --> 00:10:53,970 space? 166 00:10:57,180 --> 00:11:00,880 And can I make this kind of picture of the null space? 167 00:11:06,140 --> 00:11:07,880 That's just meant to be a sketch, 168 00:11:07,880 --> 00:11:13,800 to remind you that they're in this -- which you know, how -- 169 00:11:13,800 --> 00:11:16,420 what type of vectors are in it? 170 00:11:16,420 --> 00:11:18,560 Vectors with n components. 171 00:11:18,560 --> 00:11:25,560 Over here, inside, consisting of vectors with m components, 172 00:11:25,560 --> 00:11:33,890 is the column space and what I'm calling 173 00:11:33,890 --> 00:11:39,600 the null space of A transpose. 174 00:11:39,600 --> 00:11:42,610 Those are the ones with m components. 175 00:11:42,610 --> 00:11:44,820 OK. 176 00:11:44,820 --> 00:11:48,760 To understand these spaces is our, is our job now. 177 00:11:48,760 --> 00:11:50,880 Because by understanding those spaces, 178 00:11:50,880 --> 00:11:56,730 we know everything about this half of linear algebra. 179 00:11:56,730 --> 00:11:59,640 What do I mean by understanding those spaces? 180 00:11:59,640 --> 00:12:03,250 I would like to know a basis for those spaces. 181 00:12:07,640 --> 00:12:10,090 For each one of those spaces, how would I create -- 182 00:12:10,090 --> 00:12:11,440 construct a basis? 183 00:12:11,440 --> 00:12:15,260 What systematic way would produce a basis? 184 00:12:15,260 --> 00:12:17,180 And what's their dimension? 185 00:12:24,611 --> 00:12:25,110 OK. 186 00:12:28,380 --> 00:12:30,070 So for each of the four spaces, I 187 00:12:30,070 --> 00:12:32,500 have to answer those questions. 188 00:12:32,500 --> 00:12:34,780 How do I produce a basis? 189 00:12:34,780 --> 00:12:38,620 And then -- which has a somewhat long answer. 190 00:12:38,620 --> 00:12:42,210 And what's the dimension, which is just a number, 191 00:12:42,210 --> 00:12:44,030 so it has a real short answer. 192 00:12:44,030 --> 00:12:46,250 Can I give you the short answer first? 193 00:12:46,250 --> 00:12:50,930 I shouldn't do it, but here it is. 194 00:12:50,930 --> 00:12:56,206 I can tell you the dimension of the column space. 195 00:12:56,206 --> 00:12:57,330 Let me start with this guy. 196 00:12:57,330 --> 00:12:58,590 What's its dimension? 197 00:12:58,590 --> 00:13:01,360 I have an m by n matrix. 198 00:13:01,360 --> 00:13:09,041 The dimension of the column space is the rank, 199 00:13:09,041 --> 00:13:09,540 r. 200 00:13:13,600 --> 00:13:17,210 We actually got to that at the end of the last lecture, 201 00:13:17,210 --> 00:13:19,570 but only for an example. 202 00:13:19,570 --> 00:13:24,520 So I really have to say, OK, what's going on there. 203 00:13:24,520 --> 00:13:28,340 I should produce a basis and then I 204 00:13:28,340 --> 00:13:31,260 just look to see how many vectors I needed in that basis, 205 00:13:31,260 --> 00:13:34,360 and the answer will be r. 206 00:13:34,360 --> 00:13:38,980 Actually, I'll do that, before I get on to the others. 207 00:13:38,980 --> 00:13:41,050 What's a basis for the columns space? 208 00:13:43,770 --> 00:13:46,980 We've done all the work of row reduction, 209 00:13:46,980 --> 00:13:50,810 identifying the pivot columns, the ones that 210 00:13:50,810 --> 00:13:53,940 have pivots, the ones that end up with pivots. 211 00:13:53,940 --> 00:13:57,500 But now I -- the pivot columns I'm interested in are columns 212 00:13:57,500 --> 00:14:00,530 of A, the original A. 213 00:14:00,530 --> 00:14:03,610 And those pivot columns, there are r of them. 214 00:14:03,610 --> 00:14:05,840 The rank r counts those. 215 00:14:05,840 --> 00:14:07,550 Those are a basis. 216 00:14:07,550 --> 00:14:12,520 So if I answer this question for the column space, 217 00:14:12,520 --> 00:14:18,100 the answer will be a basis is the pivot columns 218 00:14:18,100 --> 00:14:23,790 and the dimension is the rank r, and there are r pivot columns 219 00:14:23,790 --> 00:14:26,360 and everything great. 220 00:14:26,360 --> 00:14:26,950 OK. 221 00:14:26,950 --> 00:14:29,920 So that space we pretty well understand. 222 00:14:32,560 --> 00:14:36,090 I probably have a little going back to see that -- 223 00:14:36,090 --> 00:14:39,750 to prove that this is a right answer, 224 00:14:39,750 --> 00:14:42,460 but you know it's the right answer. 225 00:14:42,460 --> 00:14:45,255 Now let me look at the row space. 226 00:14:49,340 --> 00:14:51,830 OK. 227 00:14:51,830 --> 00:14:55,190 Shall I tell you the dimension of the row space? 228 00:14:55,190 --> 00:14:55,830 Yes. 229 00:14:55,830 --> 00:14:58,180 Before we do even an example, let me tell you 230 00:14:58,180 --> 00:15:00,220 the dimension of the row space. 231 00:15:00,220 --> 00:15:03,650 Its dimension is also r. 232 00:15:03,650 --> 00:15:06,830 The row space and the column space have the same dimension. 233 00:15:06,830 --> 00:15:08,440 That's a wonderful fact. 234 00:15:08,440 --> 00:15:12,950 The dimension of the column space of A transpose -- 235 00:15:12,950 --> 00:15:17,170 that's the row space -- is r. 236 00:15:17,170 --> 00:15:19,430 That, that space is r dimensional. 237 00:15:22,100 --> 00:15:24,790 Snd so is this one. 238 00:15:24,790 --> 00:15:27,570 OK. 239 00:15:27,570 --> 00:15:35,430 That's the sort of insight that got used in this example. 240 00:15:35,430 --> 00:15:40,990 If those -- are the three columns of a matrix -- 241 00:15:40,990 --> 00:15:44,720 let me make them the three columns of a matrix by just 242 00:15:44,720 --> 00:15:47,000 erasing some brackets. 243 00:15:47,000 --> 00:15:51,320 OK, those are the three columns of a matrix. 244 00:15:51,320 --> 00:15:54,130 The rank of that matrix, if I look at the columns, 245 00:15:54,130 --> 00:15:57,560 it wasn't obvious to me anyway. 246 00:15:57,560 --> 00:16:01,250 But if I look at the rows, now it's obvious. 247 00:16:01,250 --> 00:16:03,860 The row space of that matrix obviously 248 00:16:03,860 --> 00:16:07,380 is two dimensional, because I see a basis for the row 249 00:16:07,380 --> 00:16:10,650 space, this row and that row. 250 00:16:10,650 --> 00:16:12,370 And of course, strictly speaking, 251 00:16:12,370 --> 00:16:16,290 I'm supposed to transpose those guys, make them stand up. 252 00:16:16,290 --> 00:16:19,270 But the rank is two, and therefore the column space 253 00:16:19,270 --> 00:16:21,940 is two dimensional by this wonderful fact 254 00:16:21,940 --> 00:16:25,100 that the row space and column space have the same dimension. 255 00:16:25,100 --> 00:16:29,130 And therefore there are only two pivot columns, not three, 256 00:16:29,130 --> 00:16:34,310 and, those, the three columns are dependent. 257 00:16:34,310 --> 00:16:35,430 OK. 258 00:16:35,430 --> 00:16:45,470 Now let me bury that error and talk about the row space. 259 00:16:45,470 --> 00:16:48,690 Well, I'm going to give you the dimensions of all the spaces. 260 00:16:48,690 --> 00:16:52,480 Because that's such a nice answer. 261 00:16:52,480 --> 00:16:53,100 OK. 262 00:16:53,100 --> 00:16:56,460 So let me come back here. 263 00:16:56,460 --> 00:16:59,640 So we have this great fact to establish, 264 00:16:59,640 --> 00:17:06,790 that the row space, its dimension is also the rank. 265 00:17:06,790 --> 00:17:07,910 What about the null space? 266 00:17:07,910 --> 00:17:10,020 OK. 267 00:17:10,020 --> 00:17:13,630 What's a basis for the null space? 268 00:17:13,630 --> 00:17:15,970 What's the dimension of the null space? 269 00:17:15,970 --> 00:17:20,274 Let me, I'll put that answer up here for the null space. 270 00:17:24,200 --> 00:17:27,849 Well, how have we constructed the null space? 271 00:17:27,849 --> 00:17:32,260 We took the matrix A, we did those row operations 272 00:17:32,260 --> 00:17:36,170 to get it into a form U or, or even further. 273 00:17:36,170 --> 00:17:39,730 We got it into the reduced form R. 274 00:17:39,730 --> 00:17:43,490 And then we read off special solutions. 275 00:17:43,490 --> 00:17:44,580 Special solutions. 276 00:17:44,580 --> 00:17:48,110 And every special solution came from a free variable. 277 00:17:48,110 --> 00:17:50,720 And those special solutions are in the null space, 278 00:17:50,720 --> 00:17:54,360 and the great thing is they're a basis for it. 279 00:17:54,360 --> 00:17:59,445 So for the null space, a basis will be the special solutions. 280 00:18:03,280 --> 00:18:07,250 And there's one for every free variable, right? 281 00:18:07,250 --> 00:18:11,190 For each free variable, we give that variable the value one, 282 00:18:11,190 --> 00:18:13,350 the other free variables zero. 283 00:18:13,350 --> 00:18:17,840 We get the pivot variables, we get a vector in the -- 284 00:18:17,840 --> 00:18:20,680 we get a special solution. 285 00:18:20,680 --> 00:18:24,140 So we get altogether n-r of them, 286 00:18:24,140 --> 00:18:30,340 because that's the number of free variables. 287 00:18:30,340 --> 00:18:32,890 If we have r -- 288 00:18:32,890 --> 00:18:38,490 this is the dimension is r, is the number of pivot variables. 289 00:18:38,490 --> 00:18:40,810 This is the number of free variables. 290 00:18:40,810 --> 00:18:44,180 So the beauty is that those special solutions do form 291 00:18:44,180 --> 00:18:51,380 a basis and tell us immediately that the dimension of the null 292 00:18:51,380 --> 00:18:55,570 space is n -- 293 00:18:55,570 --> 00:19:00,590 I better write this well, because it's so nice -- n-r. 294 00:19:00,590 --> 00:19:04,910 And do you see the nice thing? 295 00:19:04,910 --> 00:19:08,320 That the two dimensions in this n dimensional space, 296 00:19:08,320 --> 00:19:12,300 one subspace is r dimensional -- 297 00:19:12,300 --> 00:19:15,400 to be proved, that's the row space. 298 00:19:15,400 --> 00:19:18,680 The other subspace is n-r dimensional, 299 00:19:18,680 --> 00:19:21,150 that's the null space. 300 00:19:21,150 --> 00:19:25,550 And the two dimensions like together give n. 301 00:19:25,550 --> 00:19:28,870 The sum of r and n-R is n. 302 00:19:28,870 --> 00:19:31,270 And that's just great. 303 00:19:31,270 --> 00:19:35,720 It's really copying the fact that we have n variables, 304 00:19:35,720 --> 00:19:39,750 r of them are pivot variables and n-r are free variables, 305 00:19:39,750 --> 00:19:41,380 and n altogether. 306 00:19:41,380 --> 00:19:41,880 OK. 307 00:19:41,880 --> 00:19:47,090 And now what's the dimension of this poor misbegotten fourth 308 00:19:47,090 --> 00:19:48,500 subspace? 309 00:19:48,500 --> 00:19:53,700 It's got to be m-r. 310 00:19:53,700 --> 00:20:00,110 The dimension of this left null space, left out practically, 311 00:20:00,110 --> 00:20:01,540 is m-r. 312 00:20:04,120 --> 00:20:08,690 Well, that's really just saying that this -- again, 313 00:20:08,690 --> 00:20:15,140 the sum of that plus that is m, and m is correct, 314 00:20:15,140 --> 00:20:21,180 it's the number of columns in A transpose. 315 00:20:21,180 --> 00:20:25,810 A transpose is just as good a matrix as A. 316 00:20:25,810 --> 00:20:29,590 It just happens to be n by m. 317 00:20:29,590 --> 00:20:38,360 It happens to have m columns, so it will have m variables 318 00:20:38,360 --> 00:20:41,990 when I go to A x equals 0 and m of them, 319 00:20:41,990 --> 00:20:46,950 and r of them will be pivot variables and m-r will 320 00:20:46,950 --> 00:20:49,920 be free variables. 321 00:20:49,920 --> 00:20:52,650 A transpose is as good a matrix as A. 322 00:20:52,650 --> 00:20:57,410 It follows the same rule that the this plus the dimension -- 323 00:20:57,410 --> 00:21:00,830 this dimension plus this dimension adds up to the number 324 00:21:00,830 --> 00:21:03,130 of columns. 325 00:21:03,130 --> 00:21:06,770 And over here, A transpose has m columns. 326 00:21:06,770 --> 00:21:09,290 OK. 327 00:21:09,290 --> 00:21:09,790 OK. 328 00:21:09,790 --> 00:21:13,580 So I gave you the easy answer, the dimensions. 329 00:21:13,580 --> 00:21:21,160 Now can I go back to check on a basis? 330 00:21:21,160 --> 00:21:25,540 We would like to think that -- say the row space, 331 00:21:25,540 --> 00:21:29,280 because we've got a basis for the column space. 332 00:21:29,280 --> 00:21:33,570 The pivot columns give a basis for the column space. 333 00:21:33,570 --> 00:21:36,840 Now I'm asking you to look at the row space. 334 00:21:36,840 --> 00:21:40,790 And I -- you could say, OK, I can produce a basis for the row 335 00:21:40,790 --> 00:21:45,540 space by transposing my matrix, making those columns, 336 00:21:45,540 --> 00:21:48,670 then doing elimination, row reduction, 337 00:21:48,670 --> 00:21:54,690 and checking out the pivot columns in this transposed 338 00:21:54,690 --> 00:21:55,350 matrix. 339 00:21:55,350 --> 00:21:57,760 But that means you had to do all that row 340 00:21:57,760 --> 00:22:00,750 reduction on A transpose. 341 00:22:00,750 --> 00:22:05,590 It ought to be possible, if we take a matrix A -- 342 00:22:05,590 --> 00:22:08,620 let me take the matrix -- maybe we had this matrix in the last 343 00:22:08,620 --> 00:22:09,120 lecture. 344 00:22:09,120 --> 00:22:15,530 1 1 1, 2 1 2, 3 2 3, 1 1 1. 345 00:22:21,950 --> 00:22:22,670 OK. 346 00:22:22,670 --> 00:22:24,770 That, that matrix was so easy. 347 00:22:24,770 --> 00:22:29,100 We spotted its pivot columns, one and two, without actually 348 00:22:29,100 --> 00:22:30,930 doing row reduction. 349 00:22:30,930 --> 00:22:35,070 But now let's do the job properly. 350 00:22:35,070 --> 00:22:38,980 So I subtract this away from this to produce a zero. 351 00:22:38,980 --> 00:22:42,810 So one 2 3 1 is fine. 352 00:22:42,810 --> 00:22:47,419 Subtracting that away leaves me minus 1 -1 0, right? 353 00:22:47,419 --> 00:22:49,960 And subtracting that from the last row, oh, well that's easy. 354 00:22:53,140 --> 00:22:53,690 OK? 355 00:22:53,690 --> 00:22:56,680 I'm doing row reduction. 356 00:22:56,680 --> 00:23:00,700 Now I've -- the first column is all set. 357 00:23:00,700 --> 00:23:04,230 The second column I now see the pivot. 358 00:23:04,230 --> 00:23:07,800 And I can clean up, if I -- 359 00:23:07,800 --> 00:23:08,480 actually, 360 00:23:08,480 --> 00:23:09,860 OK. 361 00:23:09,860 --> 00:23:13,070 Why don't I make the pivot into a 1. 362 00:23:13,070 --> 00:23:18,805 I'll multiply that row through by by -1, and then I have 1 1. 363 00:23:22,200 --> 00:23:24,370 That was an elementary operation I'm allowed, 364 00:23:24,370 --> 00:23:26,790 multiply a row by a number. 365 00:23:26,790 --> 00:23:28,150 And now I'll do elimination. 366 00:23:28,150 --> 00:23:31,300 Two of those away from that will knock this guy out 367 00:23:31,300 --> 00:23:33,120 and make this into a 1. 368 00:23:33,120 --> 00:23:36,560 So that's now a 0 and that's a 369 00:23:36,560 --> 00:23:37,620 OK. 370 00:23:37,620 --> 00:23:39,230 Done. 371 00:23:39,230 --> 00:23:42,470 That's R. 372 00:23:42,470 --> 00:23:45,930 I'm seeing the identity matrix here. 373 00:23:45,930 --> 00:23:48,530 I'm seeing zeros below. 374 00:23:48,530 --> 00:23:49,990 And I'm seeing F there. 375 00:23:53,151 --> 00:23:53,650 OK. 376 00:23:56,340 --> 00:24:00,110 What about its row space? 377 00:24:00,110 --> 00:24:02,650 What happened to its row space -- well, what happened -- 378 00:24:02,650 --> 00:24:04,610 let me first ask, just because this is, is -- 379 00:24:04,610 --> 00:24:06,780 sometimes something does happen. 380 00:24:06,780 --> 00:24:09,010 Its column space changed. 381 00:24:09,010 --> 00:24:18,820 The column space of R is not the column space of A, right? 382 00:24:18,820 --> 00:24:22,000 Because 1 1 1 is certainly in the column space of A 383 00:24:22,000 --> 00:24:26,460 and certainly not in the column space of R. 384 00:24:26,460 --> 00:24:29,770 I did row operations. 385 00:24:29,770 --> 00:24:33,750 Those row operations preserve the row space. 386 00:24:33,750 --> 00:24:36,640 So the row, so the column spaces are different. 387 00:24:36,640 --> 00:24:39,550 Different column spaces, different column spaces. 388 00:24:45,950 --> 00:24:50,480 But I believe that they have the same row space. 389 00:24:55,080 --> 00:24:55,880 Same row space. 390 00:25:00,030 --> 00:25:04,040 I believe that the row space of that matrix and the row space 391 00:25:04,040 --> 00:25:06,210 of this matrix are identical. 392 00:25:06,210 --> 00:25:09,640 They have exactly the same vectors in them. 393 00:25:09,640 --> 00:25:14,400 Those vectors are vectors with four components, right? 394 00:25:14,400 --> 00:25:17,940 They're all combinations of those rows. 395 00:25:17,940 --> 00:25:19,800 Or I believe you get the same thing 396 00:25:19,800 --> 00:25:22,030 by taking all combinations of these rows. 397 00:25:24,690 --> 00:25:29,960 And if true, what's a basis? 398 00:25:29,960 --> 00:25:32,430 What's a basis for the row space of R, 399 00:25:32,430 --> 00:25:42,450 and it'll be a basis for the row space of the original A, 400 00:25:42,450 --> 00:25:45,120 but it's obviously a basis for the row space of R. 401 00:25:45,120 --> 00:25:47,840 What's a basis for the row space of that matrix? 402 00:25:47,840 --> 00:25:48,820 The first two rows. 403 00:25:51,930 --> 00:25:57,550 So a basis for the row -- so a basis is, 404 00:25:57,550 --> 00:26:15,500 for the row space of A or of R, is, is the first R rows of R. 405 00:26:15,500 --> 00:26:18,050 Not of A. 406 00:26:18,050 --> 00:26:21,960 Sometimes it's true for A, but not necessarily. 407 00:26:21,960 --> 00:26:29,280 But R, we definitely have a matrix here whose row space we 408 00:26:29,280 --> 00:26:32,060 can, we can identify. 409 00:26:32,060 --> 00:26:36,790 The row space is spanned by the three rows, 410 00:26:36,790 --> 00:26:40,580 but if we want a basis we want independence. 411 00:26:40,580 --> 00:26:43,380 So out goes row three. 412 00:26:43,380 --> 00:26:46,820 The row space is also spanned by the first two rows. 413 00:26:46,820 --> 00:26:48,920 This guy didn't contribute anything. 414 00:26:48,920 --> 00:26:52,706 And of course over here this 1 2 3 1 in the bottom 415 00:26:52,706 --> 00:26:53,830 didn't contribute anything. 416 00:26:53,830 --> 00:26:56,510 We had it already. 417 00:26:56,510 --> 00:26:58,260 So this, here is a basis. 418 00:26:58,260 --> 00:27:01,260 1 0 1 1 and 0 1 1 0. 419 00:27:04,710 --> 00:27:06,860 I believe those are in the row space. 420 00:27:06,860 --> 00:27:08,090 I know they're independent. 421 00:27:08,090 --> 00:27:10,680 Why are they in the row space? 422 00:27:10,680 --> 00:27:13,720 Why are those two vectors in the row space? 423 00:27:13,720 --> 00:27:17,440 Because all those operations we did, 424 00:27:17,440 --> 00:27:22,680 which started with these rows and took combinations of them 425 00:27:22,680 --> 00:27:23,900 -- 426 00:27:23,900 --> 00:27:28,910 I took this row minus this row, that gave me something 427 00:27:28,910 --> 00:27:30,780 that's still in the row space. 428 00:27:30,780 --> 00:27:32,400 That's the point. 429 00:27:32,400 --> 00:27:36,760 When I took a row minus a multiple of another row, 430 00:27:36,760 --> 00:27:38,380 I'm staying in the row space. 431 00:27:38,380 --> 00:27:41,330 The row space is not changing. 432 00:27:41,330 --> 00:27:43,240 My little basis for it is changing, 433 00:27:43,240 --> 00:27:46,680 and I've ended up with, sort of the best basis. 434 00:27:49,380 --> 00:27:53,240 If the columns of the identity matrix are the best basis 435 00:27:53,240 --> 00:28:00,490 for R^3 or R^n, the rows of this matrix are the best basis 436 00:28:00,490 --> 00:28:02,510 for the row space. 437 00:28:02,510 --> 00:28:06,250 Best in the sense of being as clean as I can make it. 438 00:28:06,250 --> 00:28:09,310 Starting off with the identity and then finishing up 439 00:28:09,310 --> 00:28:11,420 with whatever has to be in there. 440 00:28:11,420 --> 00:28:12,730 OK. 441 00:28:12,730 --> 00:28:16,720 Do you see then that the dimension is r? 442 00:28:16,720 --> 00:28:23,330 For sure, because we've got r pivots, r non-zero rows. 443 00:28:23,330 --> 00:28:26,420 We've got the right number of vectors, r. 444 00:28:26,420 --> 00:28:30,080 They're in the row space, they're independent. 445 00:28:30,080 --> 00:28:31,600 That's it. 446 00:28:31,600 --> 00:28:34,160 They are a basis for the row space. 447 00:28:34,160 --> 00:28:36,220 And we can even pin that down further. 448 00:28:36,220 --> 00:28:41,340 How do I know that every row of A is a combination? 449 00:28:41,340 --> 00:28:45,110 How do I know they span the row space? 450 00:28:45,110 --> 00:28:48,110 Well, somebody says, I've got the right number of them, 451 00:28:48,110 --> 00:28:48,740 so they must. 452 00:28:48,740 --> 00:28:49,870 But -- and that's true. 453 00:28:49,870 --> 00:28:54,400 But let me just say, how do I know that this row is 454 00:28:54,400 --> 00:28:57,300 a combination of these? 455 00:28:57,300 --> 00:29:01,870 By just reversing the steps of row reduction. 456 00:29:01,870 --> 00:29:07,760 If I just reverse the steps and go from A -- from R back to A, 457 00:29:07,760 --> 00:29:10,010 then what do I, what I doing? 458 00:29:10,010 --> 00:29:12,030 I'm starting with these rows, I'm 459 00:29:12,030 --> 00:29:15,730 taking combinations of them. 460 00:29:15,730 --> 00:29:19,670 After a couple of steps, undoing the subtractions 461 00:29:19,670 --> 00:29:22,710 that I did before, I'm back to these rows. 462 00:29:22,710 --> 00:29:25,500 So these rows are combinations of those rows. 463 00:29:25,500 --> 00:29:28,020 Those rows are combinations of those rows. 464 00:29:28,020 --> 00:29:31,740 The two row spaces are the same. 465 00:29:31,740 --> 00:29:34,830 The bases are the same. 466 00:29:34,830 --> 00:29:38,540 And the natural basis is this guy. 467 00:29:38,540 --> 00:29:41,530 Is that all right for the row space? 468 00:29:41,530 --> 00:29:45,030 The row space is sitting there in R 469 00:29:45,030 --> 00:29:47,750 in its cleanest possible form. 470 00:29:47,750 --> 00:29:48,800 OK. 471 00:29:48,800 --> 00:29:56,875 Now what about the fourth guy, the null space of A transpose? 472 00:29:59,630 --> 00:30:03,440 First of all, why do I call that the left null space? 473 00:30:03,440 --> 00:30:11,610 So let me save that and bring that down. 474 00:30:11,610 --> 00:30:14,050 OK. 475 00:30:14,050 --> 00:30:20,940 So the fourth space is the null space of A transpose. 476 00:30:23,700 --> 00:30:27,200 So it has in it vectors, let me call them y, 477 00:30:27,200 --> 00:30:30,470 so that A transpose y equals 0. 478 00:30:30,470 --> 00:30:35,360 If A transpose y equals 0, then y 479 00:30:35,360 --> 00:30:39,890 is in the null space of A transpose, of course. 480 00:30:39,890 --> 00:30:47,770 So this is a matrix times a column equaling zero. 481 00:30:50,480 --> 00:30:56,550 And now, because I want y to sit on the left 482 00:30:56,550 --> 00:31:00,200 and I want A instead of A transpose, 483 00:31:00,200 --> 00:31:03,380 I'll just transpose that equation. 484 00:31:03,380 --> 00:31:06,180 Can I just transpose that? 485 00:31:06,180 --> 00:31:10,705 On the right, it makes the zero vector lie down. 486 00:31:14,590 --> 00:31:21,510 And on the left, it's a product, A, A transpose times y. 487 00:31:21,510 --> 00:31:24,510 If I take the transpose, then they come in opposite order, 488 00:31:24,510 --> 00:31:25,540 right? 489 00:31:25,540 --> 00:31:30,156 So that's y transpose times A transpose transpose. 490 00:31:33,210 --> 00:31:35,540 But nobody's going to leave it like that. 491 00:31:35,540 --> 00:31:39,620 That's -- A transpose transpose is just A, of course. 492 00:31:39,620 --> 00:31:43,360 When I transposed A transpose I got back to A. 493 00:31:43,360 --> 00:31:45,870 Now do you see what I have now? 494 00:31:45,870 --> 00:31:51,820 I have a row vector, y transpose, 495 00:31:51,820 --> 00:31:58,540 multiplying A, and multiplying from the left. 496 00:31:58,540 --> 00:32:02,180 That's why I call it the left null space. 497 00:32:02,180 --> 00:32:05,590 But by making it -- putting it on the left, 498 00:32:05,590 --> 00:32:09,720 I had to make it into a row instead of a column vector, 499 00:32:09,720 --> 00:32:15,250 and so my convention is I usually don't do that. 500 00:32:15,250 --> 00:32:18,960 I usually stay with A transpose y equals 0. 501 00:32:18,960 --> 00:32:20,290 OK. 502 00:32:20,290 --> 00:32:27,990 And you might ask, how do we get a basis -- or I might ask, 503 00:32:27,990 --> 00:32:31,470 how do we get a basis for this fourth space, 504 00:32:31,470 --> 00:32:32,610 this left null space? 505 00:32:36,150 --> 00:32:36,720 OK. 506 00:32:36,720 --> 00:32:40,050 I'll do it in the example. 507 00:32:40,050 --> 00:32:43,350 As always -- not that one. 508 00:32:49,880 --> 00:32:53,310 The left null space is not jumping out at me here. 509 00:32:57,060 --> 00:33:00,220 I know which are the free variables -- 510 00:33:00,220 --> 00:33:03,890 the special solutions, but those are special solutions to A x 511 00:33:03,890 --> 00:33:06,230 equals zero, and now I'm looking at A transpose, 512 00:33:06,230 --> 00:33:08,420 and I'm not seeing it here. 513 00:33:08,420 --> 00:33:12,760 So -- but somehow you feel that the work that you did which 514 00:33:12,760 --> 00:33:19,310 simplified A to R should have revealed the left null space 515 00:33:19,310 --> 00:33:20,760 too. 516 00:33:20,760 --> 00:33:25,970 And it's slightly less immediate, but it's there. 517 00:33:25,970 --> 00:33:31,010 So from A to R, I took some steps, 518 00:33:31,010 --> 00:33:34,880 and I guess I'm interested in what were those steps, 519 00:33:34,880 --> 00:33:36,600 or what were all of them together. 520 00:33:36,600 --> 00:33:43,160 I don't -- I'm not interested in what particular ones they were. 521 00:33:43,160 --> 00:33:45,910 I'm interested in what was the whole matrix that 522 00:33:45,910 --> 00:33:51,620 took me from A to R. 523 00:33:51,620 --> 00:33:52,880 How would you find that? 524 00:33:55,640 --> 00:33:58,970 Do you remember Gauss-Jordan, where you 525 00:33:58,970 --> 00:34:02,260 tack on the identity matrix? 526 00:34:02,260 --> 00:34:03,810 Let's do that again. 527 00:34:03,810 --> 00:34:06,640 So I, I'll do it above, here. 528 00:34:06,640 --> 00:34:13,620 So this is now, this is now the idea of -- 529 00:34:13,620 --> 00:34:17,570 I take the matrix A, which is m by n. 530 00:34:20,389 --> 00:34:22,639 In Gauss-Jordan, when we saw him before -- 531 00:34:22,639 --> 00:34:25,150 A was a square invertible matrix and we 532 00:34:25,150 --> 00:34:27,900 were finding its inverse. 533 00:34:27,900 --> 00:34:29,409 Now the matrix isn't square. 534 00:34:29,409 --> 00:34:32,360 It's probably rectangular. 535 00:34:32,360 --> 00:34:36,989 But I'll still tack on the identity matrix, and of course 536 00:34:36,989 --> 00:34:42,179 since these have length m it better be m by m. 537 00:34:42,179 --> 00:34:49,374 And now I'll do the reduced row echelon form of this matrix. 538 00:34:52,480 --> 00:34:56,120 And what do I get? 539 00:35:01,640 --> 00:35:04,950 The reduced row echelon form starts with these columns, 540 00:35:04,950 --> 00:35:11,680 starts with the first columns, works like mad, and produces R. 541 00:35:11,680 --> 00:35:13,920 Of course, still that same size, m by n. 542 00:35:13,920 --> 00:35:15,700 And we did it before. 543 00:35:15,700 --> 00:35:19,140 And then whatever it did to get R, 544 00:35:19,140 --> 00:35:22,520 something else is going to show up here. 545 00:35:22,520 --> 00:35:26,690 Let me call it E, m by m. 546 00:35:26,690 --> 00:35:30,170 It's whatever -- do you see that E is just going to contain 547 00:35:30,170 --> 00:35:32,640 a record of what we did? 548 00:35:32,640 --> 00:35:38,780 We did whatever it took to get A to become R. 549 00:35:38,780 --> 00:35:40,930 And at the same time, we were doing it 550 00:35:40,930 --> 00:35:44,410 to the identity matrix. 551 00:35:44,410 --> 00:35:48,590 So we started with the identity matrix, we buzzed along. 552 00:35:48,590 --> 00:35:51,190 So we took some -- 553 00:35:51,190 --> 00:35:55,860 all this row reduction amounted to multiplying on the left 554 00:35:55,860 --> 00:36:00,040 by some matrix, some series of elementary matrices 555 00:36:00,040 --> 00:36:05,780 that altogether gave us one matrix, and that matrix is E. 556 00:36:05,780 --> 00:36:11,450 So all this row reduction stuff amounted to multiplying by E. 557 00:36:11,450 --> 00:36:13,120 How do I know that? 558 00:36:13,120 --> 00:36:16,680 It certainly amounted to multiply it by something. 559 00:36:16,680 --> 00:36:21,660 And that something took I to E, so that something was E. 560 00:36:21,660 --> 00:36:29,790 So now look at the first part, E A is R. 561 00:36:29,790 --> 00:36:31,520 No big deal. 562 00:36:31,520 --> 00:36:38,710 All I've said is that the row reduction steps that we all 563 00:36:38,710 --> 00:36:45,680 know -- well, taking A to R, are in a, in some matrix, 564 00:36:45,680 --> 00:36:49,340 and I can find out what that matrix is by just tacking I 565 00:36:49,340 --> 00:36:51,810 on and seeing what comes out. 566 00:36:51,810 --> 00:36:54,570 What comes out is E. 567 00:36:54,570 --> 00:36:58,400 Let's just review the invertible square case. 568 00:36:58,400 --> 00:37:00,860 What happened then? 569 00:37:00,860 --> 00:37:04,350 Because I was interested in it in chapter two also. 570 00:37:04,350 --> 00:37:08,260 When A was square and invertible, I took A I, 571 00:37:08,260 --> 00:37:10,480 I did row, row elimination. 572 00:37:10,480 --> 00:37:12,210 And what was the R that came out? 573 00:37:12,210 --> 00:37:14,770 It was I. 574 00:37:14,770 --> 00:37:24,530 So in chapter two, in chapter two, R was I. 575 00:37:24,530 --> 00:37:27,310 The row the, the reduced row echelon 576 00:37:27,310 --> 00:37:31,730 form of a nice invertible square matrix is the identity. 577 00:37:31,730 --> 00:37:41,510 So if R was I in that case, then E was -- then E was A inverse, 578 00:37:41,510 --> 00:37:44,430 because E A is I. 579 00:37:44,430 --> 00:37:45,200 Good. 580 00:37:45,200 --> 00:37:48,540 That's, that was good and easy. 581 00:37:48,540 --> 00:37:52,990 Now what I'm saying is that there still is an E. 582 00:37:52,990 --> 00:37:55,740 It's not A inverse any more, because A is rectangular. 583 00:37:55,740 --> 00:37:57,730 It hasn't got an inverse. 584 00:37:57,730 --> 00:38:05,010 But there is still some matrix E that connected this to this -- 585 00:38:05,010 --> 00:38:09,260 oh, I should have figured out in advanced what it was. 586 00:38:09,260 --> 00:38:11,810 Shoot. 587 00:38:11,810 --> 00:38:12,800 I didn't -- 588 00:38:12,800 --> 00:38:16,620 I did those steps and sort of erased as I went -- 589 00:38:16,620 --> 00:38:20,370 and, I should have done them to the identity too. 590 00:38:20,370 --> 00:38:22,500 Can I do that? 591 00:38:22,500 --> 00:38:23,460 Can I do that? 592 00:38:23,460 --> 00:38:26,140 I'll keep the identity matrix, like I'm supposed to do, 593 00:38:26,140 --> 00:38:29,750 and I'll do the same operations on it, and see what I end up 594 00:38:29,750 --> 00:38:30,340 with. 595 00:38:30,340 --> 00:38:31,300 OK. 596 00:38:31,300 --> 00:38:32,810 So I'm starting with the identity -- 597 00:38:32,810 --> 00:38:40,511 which I'll write in light, light enough, but -- 598 00:38:40,511 --> 00:38:41,010 OK. 599 00:38:41,010 --> 00:38:42,520 What did I do? 600 00:38:42,520 --> 00:38:45,950 I subtracted that row from that one and that row from that one. 601 00:38:45,950 --> 00:38:47,950 OK, I'll do that to the identity. 602 00:38:47,950 --> 00:38:52,990 So I subtract that first row from row two and row three. 603 00:38:52,990 --> 00:38:55,310 Good. 604 00:38:55,310 --> 00:38:56,950 Then I think I multiplied -- 605 00:38:56,950 --> 00:38:57,620 Do you remember? 606 00:38:57,620 --> 00:39:01,890 I multiplied row two by minus one. 607 00:39:01,890 --> 00:39:05,270 Let me just do that. 608 00:39:05,270 --> 00:39:06,610 Then what did I do? 609 00:39:06,610 --> 00:39:14,880 I subtracted two of row two away from row one. 610 00:39:14,880 --> 00:39:15,770 I better do that. 611 00:39:15,770 --> 00:39:17,720 Subtract two of this away from this. 612 00:39:17,720 --> 00:39:24,180 That's minus one, two of these away leaves a plus 2 and 0. 613 00:39:24,180 --> 00:39:28,440 I believe that's E. 614 00:39:28,440 --> 00:39:35,870 The way to check is to see, multiply that E by this A, 615 00:39:35,870 --> 00:39:37,420 just to see did I do it right. 616 00:39:40,570 --> 00:39:49,285 So I believe E was -1 2 0, 1 -1 0, and -1 0 1. 617 00:39:53,020 --> 00:39:53,520 OK. 618 00:39:53,520 --> 00:39:58,030 That's my E, that's my A, and that's R. 619 00:39:58,030 --> 00:40:00,110 All right. 620 00:40:00,110 --> 00:40:02,950 All I'm struggling to do is right, 621 00:40:02,950 --> 00:40:09,660 the reason I wanted this blasted E was so that I could figure 622 00:40:09,660 --> 00:40:14,570 out the left null space, not only its dimension, 623 00:40:14,570 --> 00:40:17,020 which I know -- 624 00:40:17,020 --> 00:40:19,427 actually, what is the dimension of the left null space? 625 00:40:19,427 --> 00:40:20,260 So here's my matrix. 626 00:40:23,180 --> 00:40:24,465 What's the rank of the matrix? 627 00:40:27,560 --> 00:40:30,640 And the dimension of the null -- of the left null space is 628 00:40:30,640 --> 00:40:33,470 supposed to be m-r. 629 00:40:33,470 --> 00:40:34,930 It's 3 -2, 1. 630 00:40:34,930 --> 00:40:39,090 I believe that the left null space is one dimensional. 631 00:40:39,090 --> 00:40:42,990 There is one combination of those three rows 632 00:40:42,990 --> 00:40:46,840 that produces the zero row. 633 00:40:46,840 --> 00:40:52,270 There's a basis -- a basis for the left null space has only 634 00:40:52,270 --> 00:40:54,200 got one vector in it. 635 00:40:54,200 --> 00:40:55,790 And what is that vector? 636 00:40:55,790 --> 00:40:58,710 It's here in the last row of E. 637 00:40:58,710 --> 00:41:01,400 But we could have seen it earlier. 638 00:41:01,400 --> 00:41:05,220 What combination of those rows gives the zero row? 639 00:41:05,220 --> 00:41:09,110 -1 of that plus one of that. 640 00:41:09,110 --> 00:41:14,460 So a basis for the left null space of this matrix -- 641 00:41:14,460 --> 00:41:18,350 I'm looking for combinations of rows that give the zero row 642 00:41:18,350 --> 00:41:22,080 if I'm looking at the left null space. 643 00:41:22,080 --> 00:41:24,980 For the null space, I'm looking at combinations of columns 644 00:41:24,980 --> 00:41:26,780 to get the zero column. 645 00:41:26,780 --> 00:41:29,760 Now I'm looking at combinations of these three rows 646 00:41:29,760 --> 00:41:34,370 to get the zero row, and of course there is my zero row, 647 00:41:34,370 --> 00:41:37,160 and here is my vector that produced it. 648 00:41:37,160 --> 00:41:40,010 -1 of that row and one of that 649 00:41:40,010 --> 00:41:40,510 row. 650 00:41:40,510 --> 00:41:41,650 Obvious. 651 00:41:41,650 --> 00:41:42,210 OK. 652 00:41:42,210 --> 00:41:45,940 So in that example -- and actually in all examples, 653 00:41:45,940 --> 00:41:51,310 we have seen how to produce a basis for the left null space. 654 00:41:51,310 --> 00:41:54,800 I won't ask you that all the time, because -- 655 00:41:54,800 --> 00:41:58,790 it didn't come out immediately from R. 656 00:41:58,790 --> 00:42:03,850 We had to keep track of E for that left null space. 657 00:42:03,850 --> 00:42:07,520 But at least it didn't require us to transpose the matrix 658 00:42:07,520 --> 00:42:10,220 and start all over again. 659 00:42:10,220 --> 00:42:12,390 OK, those are the four subspaces. 660 00:42:12,390 --> 00:42:15,520 Can I review them? 661 00:42:15,520 --> 00:42:18,950 The row space and the null space are in R^n. 662 00:42:18,950 --> 00:42:22,220 Their dimensions add to n. 663 00:42:22,220 --> 00:42:27,470 The column space and the left null space are in R^m, 664 00:42:27,470 --> 00:42:30,900 and their dimensions add to m. 665 00:42:30,900 --> 00:42:33,700 OK. 666 00:42:33,700 --> 00:42:39,080 So let me close these last minutes 667 00:42:39,080 --> 00:42:49,620 by pushing you a little bit more to a new type of vector space. 668 00:42:49,620 --> 00:42:53,230 All our vector spaces, all the ones that we took seriously, 669 00:42:53,230 --> 00:43:01,570 have been subspaces of some real three or n dimensional space. 670 00:43:01,570 --> 00:43:04,880 Now I'm going to write down another vector 671 00:43:04,880 --> 00:43:06,720 space, a new vector space. 672 00:43:14,120 --> 00:43:18,610 Say all three by three matrices. 673 00:43:26,420 --> 00:43:27,990 My matrices are the vectors. 674 00:43:31,830 --> 00:43:33,240 Is that all right? 675 00:43:33,240 --> 00:43:34,130 I'm just naming them. 676 00:43:34,130 --> 00:43:36,570 You can put quotes around vectors. 677 00:43:36,570 --> 00:43:40,050 Every three by three matrix is one of my vectors. 678 00:43:40,050 --> 00:43:43,060 Now how I entitled to call those things vectors? 679 00:43:43,060 --> 00:43:46,380 I mean, they look very much like matrices. 680 00:43:46,380 --> 00:43:49,980 But they are vectors in my vector space because they obey 681 00:43:49,980 --> 00:43:50,640 the rules. 682 00:43:50,640 --> 00:43:55,850 All I'm supposed to be able to do with vectors is add them -- 683 00:43:55,850 --> 00:43:58,190 I can add matrices -- 684 00:43:58,190 --> 00:44:01,580 I'm supposed to be able to multiply them by scalar numbers 685 00:44:01,580 --> 00:44:09,290 like seven -- well, I can multiply a matrix by And that 686 00:44:09,290 --> 00:44:11,960 -- and I can take combinations of matrices, 687 00:44:11,960 --> 00:44:15,060 I can take three of one matrix minus five of another 688 00:44:15,060 --> 00:44:15,990 matrix. 689 00:44:15,990 --> 00:44:21,260 And those combinations, there's a zero matrix, the matrix 690 00:44:21,260 --> 00:44:23,570 that has all zeros in it. 691 00:44:23,570 --> 00:44:26,050 If I add that to another matrix, it doesn't change it. 692 00:44:26,050 --> 00:44:26,960 All the good stuff. 693 00:44:26,960 --> 00:44:30,270 If I multiply a matrix by one it doesn't change it. 694 00:44:30,270 --> 00:44:32,800 All those eight rules for a vector space 695 00:44:32,800 --> 00:44:37,460 that we never wrote down, all easily satisfied. 696 00:44:37,460 --> 00:44:41,150 So now we have a different -- 697 00:44:41,150 --> 00:44:46,450 now of course you can say you can multiply those matrices. 698 00:44:46,450 --> 00:44:47,240 I don't care. 699 00:44:47,240 --> 00:44:50,070 For the moment, I'm only thinking of these matrices 700 00:44:50,070 --> 00:44:57,960 as forming a vector space -- so I only doing A plus B and c 701 00:44:57,960 --> 00:44:59,350 times A. 702 00:44:59,350 --> 00:45:03,050 I'm not interested in A B for now. 703 00:45:06,580 --> 00:45:09,340 The fact that I can multiply is not 704 00:45:09,340 --> 00:45:13,721 relevant to th- to a vector space. 705 00:45:13,721 --> 00:45:14,220 OK. 706 00:45:14,220 --> 00:45:15,636 So I have three by three matrices. 707 00:45:18,180 --> 00:45:21,600 And how about subspaces? 708 00:45:21,600 --> 00:45:26,420 What's -- tell me a subspace of this matrix space. 709 00:45:26,420 --> 00:45:30,270 Let me call this matrix space M. 710 00:45:30,270 --> 00:45:34,270 That's my matrix space, my space of all three by three matrices. 711 00:45:34,270 --> 00:45:37,730 Tell me a subspace of it. 712 00:45:37,730 --> 00:45:40,670 What about the upper triangular matrices? 713 00:45:40,670 --> 00:45:41,260 OK. 714 00:45:41,260 --> 00:45:43,330 So subspaces. 715 00:45:43,330 --> 00:45:50,660 Subspaces of M. 716 00:45:50,660 --> 00:45:53,675 All, all upper triangular matrices. 717 00:46:00,370 --> 00:46:01,750 Another subspace. 718 00:46:01,750 --> 00:46:03,085 All symmetric matrices. 719 00:46:11,610 --> 00:46:13,730 The intersection of two subspaces 720 00:46:13,730 --> 00:46:15,090 is supposed to be a subspace. 721 00:46:15,090 --> 00:46:20,310 We gave a little effort to the proof of that fact. 722 00:46:20,310 --> 00:46:23,150 If I look at the matrices that are in this subspace -- 723 00:46:23,150 --> 00:46:26,420 they're symmetric, and they're also in this subspace, 724 00:46:26,420 --> 00:46:31,280 they're upper triangular, what do they look like? 725 00:46:31,280 --> 00:46:33,550 Well, if they're symmetric but they 726 00:46:33,550 --> 00:46:35,660 have zeros below the diagonal, they better 727 00:46:35,660 --> 00:46:38,650 have zeros above the diagonal, so the intersection 728 00:46:38,650 --> 00:46:40,285 would be diagonal matrices. 729 00:46:44,820 --> 00:46:48,430 That's another subspace, smaller than those. 730 00:46:50,970 --> 00:46:53,740 How can I use the word smaller? 731 00:46:53,740 --> 00:46:56,390 Well, I'm now entitled to use the word smaller. 732 00:46:56,390 --> 00:47:00,410 I mean, well, one way to say is, OK, these 733 00:47:00,410 --> 00:47:02,570 are contained in those. 734 00:47:02,570 --> 00:47:05,210 These are contained in those. 735 00:47:05,210 --> 00:47:09,205 But more precisely, I could give the dimension of these spaces. 736 00:47:11,710 --> 00:47:14,740 So I could -- we can compute -- let's compute it next time, 737 00:47:14,740 --> 00:47:17,870 the dimension of all upper -- of the subspace of upper 738 00:47:17,870 --> 00:47:20,490 triangular three by three matrices. 739 00:47:20,490 --> 00:47:23,790 The dimension of symmetric three by three matrices. 740 00:47:23,790 --> 00:47:27,630 The dimension of diagonal three by three matrices. 741 00:47:27,630 --> 00:47:29,690 Well, to produce dimension, that means 742 00:47:29,690 --> 00:47:33,370 I'm supposed to produce a basis, and then 743 00:47:33,370 --> 00:47:37,230 I just count how many vecto- how many I needed in the basis. 744 00:47:37,230 --> 00:47:39,880 Let me give you the answer for this one. 745 00:47:39,880 --> 00:47:41,430 What's the dimension? 746 00:47:41,430 --> 00:47:44,610 The dimension of this -- say, this subspace, 747 00:47:44,610 --> 00:47:47,520 let me call it D, all diagonal matrices. 748 00:47:47,520 --> 00:47:54,610 The dimension of this subspace is -- 749 00:47:54,610 --> 00:47:57,610 as I write you're working it out -- 750 00:47:57,610 --> 00:47:58,970 three. 751 00:47:58,970 --> 00:48:09,150 Because here's a matrix in this -- it's a diagonal matrix. 752 00:48:09,150 --> 00:48:10,040 Here's another one. 753 00:48:15,560 --> 00:48:16,610 Here's another one. 754 00:48:20,350 --> 00:48:22,755 Better make it diagonal, let me put a seven there. 755 00:48:25,970 --> 00:48:28,130 That was not a very great choice, 756 00:48:28,130 --> 00:48:31,020 but it's three diagonal matrices, 757 00:48:31,020 --> 00:48:33,950 and I believe that they're a basis. 758 00:48:33,950 --> 00:48:37,190 I believe that those three matrices are independent 759 00:48:37,190 --> 00:48:40,680 and I believe that any diagonal matrix is 760 00:48:40,680 --> 00:48:42,220 a combination of those three. 761 00:48:42,220 --> 00:48:47,310 So they span the subspace of diagonal matrices. 762 00:48:47,310 --> 00:48:49,040 Do you see that idea? 763 00:48:49,040 --> 00:48:55,170 It's like stretching the idea from R^n to R^(n by n), 764 00:48:55,170 --> 00:48:57,360 three by three. 765 00:48:57,360 --> 00:49:02,310 But the -- we can still add, we can still multiply by numbers, 766 00:49:02,310 --> 00:49:06,370 and we just ignore the fact that we can multiply two matrices 767 00:49:06,370 --> 00:49:07,550 together. 768 00:49:07,550 --> 00:49:09,380 OK, thank you. 769 00:49:09,380 --> 00:49:11,930 That's lecture ten.