1 00:00:12 --> 00:00:13 OK. 2 00:00:13 --> 00:00:22 Uh this is the review lecture for the first part of the 3 00:00:22 --> 00:00:27 course, the Ax=b part of the course. 4 00:00:27 --> 00:00:34 And the exam will emphasize chapter three. 5 00:00:34 --> 00:00:42 Because those are the --0 chapter three was about the 6 00:00:42 --> 00:00:48 rectangular matrices where we had null spaces and null spaces 7 00:00:48 --> 00:00:52 of A transpose, and ranks, and all the things 8 00:00:52 --> 00:00:58.16 that are so clear when the matrix is square and invertible, 9 00:00:58.16 --> 00:01:04 they became things to think about for rectangular matrices. 10 00:01:04 --> 00:01:10 So, and vector spaces and subspaces and above all those 11 00:01:10 --> 00:01:11.7 four subspaces. 12 00:01:11.7 --> 00:01:17 OK, I'm thinking to start at least I'll just look at old 13 00:01:17 --> 00:01:23 exams, read out questions, write on the board what I need 14 00:01:23 --> 00:01:27 to and we can see what the answers are. 15 00:01:27 --> 00:01:31 The first one I see is one I can just read out. 16 00:01:31 --> 00:01:35 Well, I'll write a little. 17 00:01:35 --> 00:01:40 Suppose u, v and w are nonzero vectors in R^7. 18 00:01:40 --> 00:01:45 What are the possible -- they span a -- a vector space. 19 00:01:45 --> 00:01:50 They span a subspace of R^7, and what are the possible 20 00:01:50 --> 00:01:51 dimensions? 21 00:01:51 --> 00:01:57 So that's a straightforward question, what are the possible 22 00:01:57 --> 00:02:01 dimensions of the subspace spanned by u, 23 00:02:01 --> 00:02:03 v and w? 24 00:02:03 --> 00:02:05 OK, one, two, or three, right. 25 00:02:05 --> 00:02:07 One, two or three. 26 00:02:07 --> 00:02:12 Couldn't be more because we've only got three vectors, 27 00:02:12 --> 00:02:18 and couldn't be zero because -- because I told you the vectors 28 00:02:18 --> 00:02:19 were nonzero. 29 00:02:19 --> 00:02:25.46 Otherwise if I allowed the possibility that those were all 30 00:02:25.46 --> 00:02:31 the zero vector -- then the zero-dimensional 31 00:02:31 --> 00:02:35 subspace would have been in there. 32 00:02:35 --> 00:02:35 OK. 33 00:02:35 --> 00:02:40 Now can I jump to a more serious question? 34 00:02:40 --> 00:02:41 OK. 35 00:02:41 --> 00:02:44 We have a five by three matrix. 36 00:02:44 --> 00:02:47 And I'm calling it U. 37 00:02:47 --> 00:02:51 I'm saying it's in echelon form. 38 00:02:51 --> 00:02:54 And it has three pivots, r=3. 39 00:02:54 --> 00:02:57 Three pivots. 40 00:02:57 --> 00:02:58 OK. 41 00:02:58 --> 00:03:01 First question, what's the null space? 42 00:03:01 --> 00:03:07 What's the null space of this matrix U, so this matrix is five 43 00:03:07 --> 00:03:13.52 by three, and I find it helpful to just see visually what five 44 00:03:13.52 --> 00:03:16 by three means, what that shape is. 45 00:03:16 --> 00:03:18 Three columns. 46 00:03:18 --> 00:03:22 Three columns in U then, five rows, three pivots, 47 00:03:22 --> 00:03:26 and what's the null space? 48 00:03:26 --> 00:03:33 The null space of U is -- and it asks for a spec-of course I'm 49 00:03:33 --> 00:03:39 looking for an answer that isn't just the definition of the null 50 00:03:39 --> 00:03:45.47 space, but is the null space of this matrix, with this 51 00:03:45.47 --> 00:03:47.5 information. 52 00:03:47.5 --> 00:03:48 And what is it? 53 00:03:48 --> 00:03:50 It's only the zero vector. 54 00:03:50 --> 00:03:53 Because we're told that the rank is three, 55 00:03:53 --> 00:03:58 so those three columns must be independent, no combination -- 56 00:03:58 --> 00:04:02 of those columns is the zero vector except -- so the only 57 00:04:02 --> 00:04:07 thing in this null space is the zero vector, and I -- 58 00:04:07 --> 00:04:11 I could even say what that vector is, zero, 59 00:04:11 --> 00:04:12 zero, zero. 60 00:04:12 --> 00:04:13 That's OK. 61 00:04:13 --> 00:04:16 So that's what's in the null space. 62 00:04:16 --> 00:04:19 All right? let me go on with -- this 63 00:04:19 --> 00:04:22 question has multiple parts. 64 00:04:22 --> 00:04:27 What's the -- oh now it asks you about a ten by three matrix, 65 00:04:27 --> 00:04:32.3 B, which is the matrix U and two U. 66 00:04:32.3 --> 00:04:40 It actually -- I would probably be writing R -- and maybe I 67 00:04:40 --> 00:04:44 should be writing R here now. 68 00:04:44 --> 00:04:52 This exam goes back a few years when I emphasized U more than R. 69 00:04:52 --> 00:04:59.67 Now, what's the echelon form for that matrix? 70 00:04:59.67 --> 00:05:03 So the echelon form, what's the rank, 71 00:05:03 --> 00:05:06 and what's the echelon form? 72 00:05:06 --> 00:05:10.92 Let's suppose this is in reduced echelon form, 73 00:05:10.92 --> 00:05:14 so that I could be using the letter R. 74 00:05:14 --> 00:05:20 So I'll ask for the reduced row echelon form so imagine that 75 00:05:20 --> 00:05:25 these are -- U is in reduced row echelon 76 00:05:25 --> 00:05:30 form but now I've doubled the height of the matrix, 77 00:05:30 --> 00:05:35 what will happen when we do row reduction? 78 00:05:35 --> 00:05:40 What row reduction will take us to what matrix here? 79 00:05:40 --> 00:05:43 So you start doing elimination. 80 00:05:43 --> 00:05:48 You're doing elimination on single rows. 81 00:05:48 --> 00:05:53 But of course we're allowed to think of blocks. 82 00:05:53 --> 00:05:57 So what, well, what's the answer look like? 83 00:05:57 --> 00:06:04 U and z- or R -- let's -- I'll stay with this letter U but I'm 84 00:06:04 --> 00:06:09 really thinking it's in reduced form, and zero. 85 00:06:09 --> 00:06:09.76 OK. 86 00:06:09.76 --> 00:06:10 Fine. 87 00:06:10 --> 00:06:14.91 Then it asks oh, further, it asks about this 88 00:06:14.91 --> 00:06:17 matrix. 89 00:06:17 --> 00:06:18 U, U, U, and zero. 90 00:06:18 --> 00:06:21 OK, what's the echelon form of this? 91 00:06:21 --> 00:06:25 So it's just like practice in thinking through what would row 92 00:06:25 --> 00:06:28 elimination, what would row reduction do. 93 00:06:28 --> 00:06:33 Have I thought this through, so what -- what are we -- if we 94 00:06:33 --> 00:06:37 start doing elimination, basically we're going to 95 00:06:37 --> 00:06:40 subtract these rows from these -- 96 00:06:40 --> 00:06:46 so it's going to take us to U, U, zero, and minus U, 97 00:06:46 --> 00:06:48 I guess, right? 98 00:06:48 --> 00:06:55 Take the thing all the way to R -- let's suppose U is really R. 99 00:06:55 --> 00:07:02 Suppose that we're really going for the reduced row echelon 100 00:07:02 --> 00:07:03.06 form. 101 00:07:03.06 --> 00:07:05 Then would we stop there? 102 00:07:05 --> 00:07:07.5 No. 103 00:07:07.5 --> 00:07:11 We would clean out, we would -- we could use this 104 00:07:11 --> 00:07:15.35 to -- is that right, can I so I took this row -- 105 00:07:15.35 --> 00:07:18 these rows away from these to get there. 106 00:07:18 --> 00:07:23 Now I take these rows away from these, so that gives me zero. 107 00:07:23 --> 00:07:24 There. 108 00:07:24 --> 00:07:28 And now what more would I do if I'm really shooting for R, 109 00:07:28 --> 00:07:32 the reduced row echelon form? 110 00:07:32 --> 00:07:36 I would -- then I want plus ones in the pivot, 111 00:07:36 --> 00:07:42 so I would multiply through by minus one to get plus there. 112 00:07:42 --> 00:07:48 So essentially I'm seeing reduced row echelon form there 113 00:07:48 --> 00:07:54.75 and there, and there's just one little twist still to go. 114 00:07:54.75 --> 00:07:58 Do you see what that final twist might be? 115 00:07:58 --> 00:08:02.88 To have if -- if U is in reduced row echelon form and now 116 00:08:02.88 --> 00:08:06 I'm looking at U, U, there's one little step to 117 00:08:06 --> 00:08:11 take, this isn't like a big deal at all, but -- but if I really 118 00:08:11 --> 00:08:17 want this to be in reduced form, what would I still -- might I 119 00:08:17 --> 00:08:19 still have to do? 120 00:08:19 --> 00:08:24 I might have some zero rows here, I might have some zero 121 00:08:24 --> 00:08:28 rows here that strictly should move to the bottom. 122 00:08:28 --> 00:08:32 Well, I'm not going to make a project out of that. 123 00:08:32 --> 00:08:33 OK. 124 00:08:33 --> 00:08:35 What's the rank of that matrix? 125 00:08:35 --> 00:08:38 What's the rank of this matrix C? 126 00:08:38 --> 00:08:43 Given that I know that the original U has rank three, 127 00:08:43 --> 00:08:46.84 what's the rank of this guy? 128 00:08:46.84 --> 00:08:47 Six, right. 129 00:08:47 --> 00:08:50 That has rank six, I can tell. 130 00:08:50 --> 00:08:55 What was -- what's the rank of this B, while -- while we're at 131 00:08:55 --> 00:08:55 it? 132 00:08:55 --> 00:08:58 The rank of B, is that six or three? 133 00:08:58 --> 00:09:00 Three is right. 134 00:09:00 --> 00:09:01 Three is right. 135 00:09:01 --> 00:09:06 Because we actually got it to where we could just see three 136 00:09:06 --> 00:09:08 pivots. 137 00:09:08 --> 00:09:14 OK, and oh, now finally this easy one, what's the dimension 138 00:09:14 --> 00:09:20 of the null space -- of the null space of C transpose? 139 00:09:20 --> 00:09:21 Oh, boy. 140 00:09:21 --> 00:09:27 OK, so what do I -- if I want the dimension of a null space, 141 00:09:27 --> 00:09:33 I want to know the size of the matrix -- 142 00:09:33 --> 00:09:37 so what's the size of the matrix C? 143 00:09:37 --> 00:09:41 It looks like it's ten by six, is it? 144 00:09:41 --> 00:09:46 Ten by six, so C is ten by six, so m is ten, 145 00:09:46 --> 00:09:51 so C has ten rows, C transpose has ten columns, 146 00:09:51 --> 00:09:54 so there are ten columns there. 147 00:09:54 --> 00:10:01.2 So how many free variables have I got, once I -- 148 00:10:01.2 --> 00:10:06 if I start with the ten columns in C transpose, 149 00:10:06 --> 00:10:09 that's the m for the original C. 150 00:10:09 --> 00:10:12 And what do I subtract off? 151 00:10:12 --> 00:10:12 Six. 152 00:10:12 --> 00:10:15 Because we said that was the rank. 153 00:10:15 --> 00:10:18 So I'm left with four. 154 00:10:18 --> 00:10:18 Thanks. 155 00:10:18 --> 00:10:19 OK. 156 00:10:19 --> 00:10:24 So I think that's the right answer -- 157 00:10:24 --> 00:10:31.58 the dimension of the null space of C transpose would be four. 158 00:10:31.58 --> 00:10:32 Right. 159 00:10:32 --> 00:10:32 OK. 160 00:10:32 --> 00:10:33 Yeah. 161 00:10:33 --> 00:10:33 OK. 162 00:10:33 --> 00:10:39 So that's one question, at least it brought in some -- 163 00:10:39 --> 00:10:43 some of the dimension counts. 164 00:10:43 --> 00:10:43 OK. 165 00:10:43 --> 00:10:49 Here's another type of question. 166 00:10:49 --> 00:10:57 I give you an equation, Ax equals two four two. 167 00:10:57 --> 00:11:03 And I give you the complete solution. 168 00:11:03 --> 00:11:09 But I don't give you the matrix. 169 00:11:09 --> 00:11:16 And another -- there's another vector, zero, 170 00:11:16 --> 00:11:18 zero, one. 171 00:11:18 --> 00:11:19 OK. 172 00:11:19 --> 00:11:22 All right. 173 00:11:22 --> 00:11:27 My first question is what's the dimension of the row space? 174 00:11:27 --> 00:11:29 Of the matrix A? 175 00:11:29 --> 00:11:34 So the main thing that you want to get from this question is 176 00:11:34 --> 00:11:37 that a question could start this way. 177 00:11:37 --> 00:11:39 Sort of backward way. 178 00:11:39 --> 00:11:44 By giving you the answer and not telling you what the problem 179 00:11:44 --> 00:11:45 is. 180 00:11:45 --> 00:11:50 But we can get a lot of information, and sometimes we 181 00:11:50 --> 00:11:54 can get complete information about that matrix A. 182 00:11:54 --> 00:11:54 OK. 183 00:11:54 --> 00:11:58.26 So what's the dimension of the row space of A? 184 00:11:58.26 --> 00:11:59 What's the rank? 185 00:11:59 --> 00:12:03 Tell me about what's the size of the matrix, 186 00:12:03 --> 00:10:48 yeah, just -- These are the things we want to 187 00:10:48 --> 00:08:28 think about, what's the -- what's the shape of the matrix, 188 00:08:28 --> 00:07:56 first of all? 189 00:07:56 --> 00:06:47.34 It certainly has three rows. 190 00:06:47.34 --> 00:05:25.98 But is it a- is it three by three? 191 00:05:25.98 --> 00:03:15 So the (x)-s that it multiplies have three components, 192 00:03:15 --> 00:04:08 so that --1 does the matrix have three 193 00:04:08 --> 00:05:00 columns also? 194 00:05:00 --> 00:05:15 Yes. 195 00:05:15 --> 00:08:56 So I'm seeing the same length in b, three, and also in x. 196 00:08:56 --> 00:10:59 So A is a three by three matrix. 197 00:10:59 --> 00:12:50 And what's its rank? 198 00:12:50 --> 00:12:55.09 Its rank -- tell me something about its null space, 199 00:12:55.09 --> 00:13:00 I heard the right answer for the rank, the rank is one in 200 00:13:00 --> 00:13:01 this case. 201 00:13:01 --> 00:13:02 Why? 202 00:13:02 --> 00:13:08 Because the dimension of the null space, so the dimension of 203 00:13:08 --> 00:13:14 the null space of A is from knowing that that's the complete 204 00:13:14 --> 00:13:17 solution, it's two. 205 00:13:17 --> 00:13:22 I'm seeing two vectors here, and they're independent in the 206 00:13:22 --> 00:13:27 null space of A, because they have to be in the 207 00:13:27 --> 00:13:33 null space of A if I'm allowed to throw into the solution any 208 00:13:33 --> 00:13:38 amount of those vectors, that tells me that's the null 209 00:13:38 --> 00:13:41 space part then. 210 00:13:41 --> 00:13:45 So the dimension of the null space is two, 211 00:13:45 --> 00:13:51 and then I -- of course I know the dimensions of all the -- 212 00:13:51 --> 00:13:52 four subspaces. 213 00:13:52 --> 00:13:56.44 Now actually it asks what's the matrix? 214 00:13:56.44 --> 00:14:01 Well, what's the matrix in this case? 215 00:14:01 --> 00:14:05 Do we want to -- shall I try to figure that out? 216 00:14:05 --> 00:14:05 Sure. 217 00:14:05 --> 00:14:09 Let's -- you'd like me to do it, OK. 218 00:14:09 --> 00:14:14 So what about the matrix, or let me at least start it, 219 00:14:14 --> 00:14:14 OK. 220 00:14:14 --> 00:14:19 If A times this x gives two, four, two, what does that tell 221 00:14:19 --> 00:14:23 me about the matrix A? 222 00:14:23 --> 00:14:28 If A times that x solves that equation then it tells me that 223 00:14:28 --> 00:14:33 the first column is -- the first column of A is -- one, 224 00:14:33 --> 00:14:34 two, one, right. 225 00:14:34 --> 00:14:38 The first column of A has to be one, two, one, 226 00:14:38 --> 00:14:43 because if I multiply by x, that's going to multiply just 227 00:14:43 --> 00:14:46 the first column, and give me two, 228 00:14:46 --> 00:14:48 four, two. 229 00:14:48 --> 00:14:52 And then I've got two more columns to find, 230 00:14:52 --> 00:14:56.63 and what information have I got to find them with? 231 00:14:56.63 --> 00:14:58 I've got the null space. 232 00:14:58 --> 00:15:03 So the fact that this is in the null space, what does that tell 233 00:15:03 --> 00:15:06 me about the matrix? 234 00:15:06 --> 00:15:11 A matrix that has zero, zero, one in its null space? 235 00:15:11 --> 00:15:16 That tells me that the last column of the matrix is zeroes. 236 00:15:16 --> 00:15:22 Because this is in the null space, the last column has to be 237 00:15:22 --> 00:15:23 zeroes. 238 00:15:23 --> 00:15:30 And because this is in the null space, what's the second column? 239 00:15:30 --> 00:15:34 Well, this in the null space means that if I multiply A by 240 00:15:34 --> 00:15:38 that vector I must be getting zeroes, so I think that better 241 00:15:38 --> 00:15:41 be minus one, minus two, and minus one. 242 00:15:41 --> 00:15:41 OK. 243 00:15:41 --> 00:15:46.04 That's a type of question that just brings out the information 244 00:15:46.04 --> 00:15:49.2 that's in that complete solution. 245 00:15:49.2 --> 00:15:49 OK. 246 00:15:49 --> 00:15:56 And then actually I go on to ask what vectors -- for what 247 00:15:56 --> 00:16:00 vectors B can Ax=b be solved? 248 00:16:00 --> 00:16:08 Ax=b can be solved if what -- so I'm looking for a condition 249 00:16:08 --> 00:16:10 on b, if any. 250 00:16:10 --> 00:16:17 Can it be solved for every right-hand side b? 251 00:16:17 --> 00:16:07 No. 252 00:16:07 --> 00:15:21 Definitely not. 253 00:15:21 --> 00:14:08 When could it be solved? 254 00:14:08 --> 00:11:35 Well, what's the ---- I actually say in this in the 255 00:11:35 --> 00:10:18 exam, don't just tell me. 256 00:10:18 --> 00:07:23 If b is in the -- and what would -- what does the exam say 257 00:07:23 --> 00:06:38 there?1 258 00:06:38 --> 00:09:10 In the column space, because I do know that it can 259 00:09:10 --> 00:11:37 be solved exactly when B is in the column space, 260 00:11:37 --> 00:14:38 so I guess I'm asking you what is the column space for this 261 00:14:38 --> 00:15:00 matrix? 262 00:15:00 --> 00:17:04 So what is if b has the form -- so I guess I'm asking what's 263 00:17:04 --> 00:17:07 the column space of this matrix, and what is it? 264 00:17:07 --> 00:17:12 It's so the column space of that matrix is all multiples b 265 00:17:12 --> 00:17:15 -- b is a multiple of one, two, one. 266 00:17:15 --> 00:17:16 Right? 267 00:17:16 --> 00:17:19 I can solve the thing if it's a multiple of one, 268 00:17:19 --> 00:17:24 two, one, and of course sure enough -- 269 00:17:24 --> 00:17:30 yeah, that was a multiple of one, two, one, 270 00:17:30 --> 00:17:34 and so I had a solution. 271 00:17:34 --> 00:17:43 So this is a case where we've got lots of null space. 272 00:17:43 --> 00:17:50 Let me just recall rank is big, don't forget those cases, 273 00:17:50 --> 00:17:58 don't forget the other cases when r is as big as it can be, 274 00:17:58 --> 00:18:01 r equal m or r equal n. 275 00:18:01 --> 00:18:06 Those are -- we had a full lecture on that, 276 00:18:06 --> 00:18:10 the full rank, full lecture, 277 00:18:10 --> 00:18:15 and important -- important case. 278 00:18:15 --> 00:18:15 OK. 279 00:18:15 --> 00:18:18 I'll just move on. 280 00:18:18 --> 00:18:22 I think this is the best type of review. 281 00:18:22 --> 00:18:26 It's just brings these ideas out. 282 00:18:26 --> 00:18:33 Apologies to the camera while I recover glasses and exam. 283 00:18:33 --> 00:18:33 OK. 284 00:18:33 --> 00:18:38.5 How about a few true-false ones? 285 00:18:38.5 --> 00:18:42 Actually there won't be a true-false on the quiz. 286 00:18:42 --> 00:18:46 But it gives us a moment of quick review. 287 00:18:46 --> 00:18:47 Here's one. 288 00:18:47 --> 00:18:50 If the null space -- I have a square matrix. 289 00:18:50 --> 00:18:55 If its null space is just the zero vector, what about the null 290 00:18:55 --> 00:18:58 space of A transpose? 291 00:18:58 --> 00:19:03 If the null space of A is just the zero vector, 292 00:19:03 --> 00:19:08 and the matrix is square, what do I know about the null 293 00:19:08 --> 00:19:10 space of A transpose? 294 00:19:10 --> 00:19:12 Also the zero vector. 295 00:19:12 --> 00:19:13 Good. 296 00:19:13 --> 00:19:16 And that's a very very important fact. 297 00:19:16 --> 00:19:17 OK. 298 00:19:17 --> 00:19:19 How about this? 299 00:19:19 --> 00:19:24 These -- look at the space of five by five matrices as a 300 00:19:24 --> 00:19:25 vector space. 301 00:19:25 --> 00:19:30 So it's actually a twenty-five-dimensional vector 302 00:19:30 --> 00:19:31 space. 303 00:19:31 --> 00:19:33 All five by five matrices. 304 00:19:33 --> 00:19:36 Look at the invertible matrices. 305 00:19:36 --> 00:19:38 Do they form a subspace? 306 00:19:38 --> 00:19:44 So I have this five by -- a space of all five by five 307 00:19:44 --> 00:19:45 matrices. 308 00:19:45 --> 00:19:49.64 I can add them, I can multiply by numbers. 309 00:19:49.64 --> 00:19:53.81 But now I narrow down to the invertible ones. 310 00:19:53.81 --> 00:19:56 And I ask are they a subspace? 311 00:19:56 --> 00:20:01 And you -- your answer is -- quiet, but nevertheless 312 00:20:01 --> 00:20:03 definite, no. 313 00:20:03 --> 00:20:04.11 Right? 314 00:20:04.11 --> 00:20:04 No. 315 00:20:04 --> 00:20:08 Because if I add two invertible matrices I have no idea if the 316 00:20:08 --> 00:20:10 answer is invertible. 317 00:20:10 --> 00:20:14 If I multiply that invertible -- well, it doesn't even have 318 00:20:14 --> 00:20:18 the zero matrix in it, it couldn't be a subspace. 319 00:20:18 --> 00:20:21 I have to be able to multiply by zero -- 320 00:20:21 --> 00:20:24 and stay in my subspace, and the invertible ones 321 00:20:24 --> 00:20:25 wouldn't work. 322 00:20:25 --> 00:20:28 Well, the singular ones wouldn't work either. 323 00:20:28 --> 00:20:31 They have zero -- the zero matrix is in the singular 324 00:20:31 --> 00:20:35 matrices, but if I add two singular matrices I don't know 325 00:20:35 --> 00:20:38 if the answer is singular or not. 326 00:20:38 --> 00:20:38 OK. 327 00:20:38 --> 00:20:41 So another true-false. 328 00:20:41 --> 00:20:46 If b squared equals zero then b equals zero. 329 00:20:46 --> 00:20:48 True or false? 330 00:20:48 --> 00:20:52 If b squared equals zero, true, false? 331 00:20:52 --> 00:20:59 b squared equals zero, b has to be a square -- 332 00:20:59 --> 00:21:04 square matrix, so that I can multiply it by 333 00:21:04 --> 00:21:08 itself, does that imply that B is zero? 334 00:21:08 --> 00:21:15 Are there matrices whose square could be the zero matrix? 335 00:21:15 --> 00:21:16 Yes or no? 336 00:21:16 --> 00:21:18 Yes there are. 337 00:21:18 --> 00:21:25 There are matrices whose square is the zero matrix. 338 00:21:25 --> 00:21:27 So this statement is false. 339 00:21:27 --> 00:21:32 If b squared is zero, we don't know that b is zero. 340 00:21:32 --> 00:21:36 For example -- the best example is that matrix. 341 00:21:36 --> 00:21:39 That matrix is a dangerous matrix. 342 00:21:39 --> 00:21:44 It will come up in later parts of this course as an example of 343 00:21:44 --> 00:21:47 what can go wrong. 344 00:21:47 --> 00:22:00 And here is a real simple -- so this -- so if I square that 345 00:22:00 --> 00:22:12 matrix, I do get the zero matrix,and it shows -- 346 00:22:12 --> 00:22:13 OK. 347 00:22:13 --> 00:22:20.86 A system of m equations in m unknowns is solvable for every 348 00:22:20.86 --> 00:22:27.04 right-hand side if the columns are independent. 349 00:22:27.04 --> 00:22:27 OK. 350 00:22:27 --> 00:22:34 So can I say that again, I'm -- I'll write it down then 351 00:22:34 --> 00:20:45 for short. m by m matrix independent 352 00:20:45 --> 00:16:58 columns then the question is does Ax=b, is it always 353 00:16:58 --> 00:16:18 solvable? 354 00:16:18 --> 00:13:25 And the answer is -- yes or no -- right. 355 00:13:25 --> 00:08:53 OK, which -- did you watch that quiz, there was a quiz program 356 00:08:53 --> 00:05:59 on TV for a few weeks, did you see that, 357 00:05:59 --> 00:03:09.8 what was the name of that --2 winning a million dollars or 358 00:03:09.8 --> 00:03:11 something? 359 00:03:11 --> 00:03:14 How to be a millionaire? 360 00:03:14 --> 00:03:19 It was some crazy guy, what was his name? 361 00:03:19 --> 00:03:22 It's -- Regis, right. 362 00:03:22 --> 00:03:23 Regis. 363 00:03:23 --> 00:03:25 OK.2 364 00:03:25 --> 00:07:00 If you saw this, like when you should have been 365 00:07:00 --> 00:11:42.18 doing linear algebra of course -- but I didn't have to do the 366 00:11:42.18 --> 00:16:09 homework, so I was watching it, so there were three -- the 367 00:16:09 --> 00:20:50 interesting -- the novel point was there were three ways that 368 00:20:50 --> 00:23:44.5 you could get help, right -- 369 00:23:44.5 --> 00:23:47 but you could only use each way once, so you couldn't like use 370 00:23:47 --> 00:23:48 them all the time. 371 00:23:48 --> 00:23:49 So remember that? 372 00:23:49 --> 00:23:52 You could -- so you could poll the audience, 373 00:23:52 --> 00:23:55.52 and that was a very -- that was a hundred percent successful 374 00:23:55.52 --> 00:23:57 way, so I'll poll the audience on this. 375 00:23:57 --> 00:24:01 If the other possibility -- another possibility you could 376 00:24:01 --> 00:24:04 call your friend, right, or he's your friend 377 00:24:04 --> 00:24:08 until he gives you the wrong answer, which -- that turned out 378 00:24:08 --> 00:24:11 to be very unreliable, you know, you'd call up your 379 00:24:11 --> 00:24:15.71 brother or something and ask him for the capital of whatever, 380 00:24:15.71 --> 00:24:16 Bosnia. 381 00:24:16 --> 00:24:24 What does he know, he makes some guess, 382 00:24:24 --> 00:24:34 matrix, independent columns, is Ax=b always solvable? 383 00:24:34 --> 00:24:39 Maybe just hands up for that? 384 00:24:39 --> 00:24:40 A few. 385 00:24:40 --> 00:24:42 And who says no? 386 00:24:42 --> 00:24:48 Oh, gosh, this audience is not reliable. 387 00:24:48 --> 00:24:50 Fifty fifty. 388 00:24:50 --> 00:24:56 I guess I'd say, I'd vote yes. 389 00:24:56 --> 00:25:01 Because independent columns, that means that the rank is the 390 00:25:01 --> 00:25:05 full size m, I have a matrix of rank m. 391 00:25:05 --> 00:25:10 That means it's -- I mean it's square, so it's an invertible 392 00:25:10 --> 00:25:13 matrix, and nothing could go wrong. 393 00:25:13 --> 00:25:14 Yes. 394 00:25:14 --> 00:25:22 So that's the good case and we always expect it in chapter two, 395 00:25:22 --> 00:25:29 but of course chapter three is -- only one of the 396 00:25:29 --> 00:25:31 possibilities. 397 00:25:31 --> 00:25:31 OK. 398 00:25:31 --> 00:25:38 Let me move on to another question from an old quiz. 399 00:25:38 --> 00:25:40 OK. 400 00:25:40 --> 00:25:40 OK. 401 00:25:40 --> 00:25:42.42 Let's see. 402 00:25:42.42 --> 00:25:42 OK. 403 00:25:42 --> 00:25:52.28 I'm going to give you a matrix, but I'm going to give it to you 404 00:25:52.28 --> 00:25:58 as a product of a couple of matrices, one, 405 00:25:58 --> 00:26:03 one, zero, zero, one, zero, one, 406 00:26:03 --> 00:26:09 zero, one, times another matrix, one, zero, 407 00:26:09 --> 00:26:14 minus one, two, zero, one, one, 408 00:26:14 --> 00:26:20 minus one, and all zeroes. 409 00:26:20 --> 00:26:05 OK. 410 00:26:05 --> 00:21:55 I would like to ask you questions about that matrix 411 00:21:55 --> 00:17:10 without doing the multiplication and finding the matrix b. 412 00:17:10 --> 00:12:10 Can you tell me something -- so I'll ask questions about this 413 00:12:10 --> 00:06:45 matrix b and I'll answer them without multiplying it out.2 414 00:06:45 --> 00:11:38 For example, I'm going to ask you for a 415 00:11:38 --> 00:14:52 basis for the null space. 416 00:14:52 --> 00:18:21.17 A basis for the null space. 417 00:18:21.17 --> 00:22:59 So I'm going to solve Bx equals zero. 418 00:22:59 --> 00:27:10 So give me a basis for -- for the null space of B. 419 00:27:10 --> 00:27:17 Let's see, what dimension I in -- the null space of B is a 420 00:27:17 --> 00:27:19 subspace of R. 421 00:27:19 --> 00:27:24 What size vectors I looking for here? 422 00:27:24 --> 00:27:32 Because if we don't know the size, we aren't going to find 423 00:27:32 --> 00:27:37.32 it, right? the null -- this matrix is 424 00:27:37.32 --> 00:27:39 three by four obviously. 425 00:27:39 --> 00:27:45 So if we're looking for the null space we're looking for 426 00:27:45 --> 00:27:47 those vectors x in R^4. 427 00:27:47 --> 00:27:48 OK. 428 00:27:48 --> 00:27:53 So the null space of B is certainly a subspace of R^4. 429 00:27:53 --> 00:27:57.94 What do you think its dimension is? 430 00:27:57.94 --> 00:28:02 Of course once we find the basis we would know the 431 00:28:02 --> 00:28:05 dimension immediately, but let's stop first, 432 00:28:05 --> 00:28:08 what's the rank of this matrix B? 433 00:28:08 --> 00:28:12 Let's see, what -- is that matrix invertible, 434 00:28:12 --> 00:28:13 that square one there? 435 00:28:13 --> 00:28:16 Let's say sure, I think it is, 436 00:28:16 --> 00:28:20 yes, that matrix B looks invertible. 437 00:28:20 --> 00:28:22 Is that pretty clear? 438 00:28:22 --> 00:28:23 Yeah. 439 00:28:23 --> 00:28:23 Yeah. 440 00:28:23 --> 00:28:28 So I've gone wrong in this course already, 441 00:28:28 --> 00:28:33 but I'll still hope that that matrix is invertible. 442 00:28:33 --> 00:28:40 Yeah, yeah, because if I look for a combination of those three 443 00:28:40 --> 00:28:44 columns -- well, I couldn't use this 444 00:28:44 --> 00:28:49 middle column because it would have a one and in a position 445 00:28:49 --> 00:28:53 that I -- column is otherwise all zero, so a combination that 446 00:28:53 --> 00:28:58 gives zero can't give us that problem, and then the other two 447 00:28:58 --> 00:29:03 are clearly independent sets -- so that matrix is invertible. 448 00:29:03 --> 00:29:08 Later we could take a determinant or other things. 449 00:29:08 --> 00:29:08.5 OK. 450 00:29:08.5 --> 00:29:09.99 What's the setup? 451 00:29:09.99 --> 00:29:14 If I have an invertible matrix, a nice invertible square 452 00:29:14 --> 00:29:19 matrix, times this guy, times this second factor, 453 00:29:19 --> 00:29:23 and I'm looking for the null space, does this have any 454 00:29:23 --> 00:29:25 effect? 455 00:29:25 --> 00:29:32 Is the null -- so what I'm asking is is the null space of B 456 00:29:32 --> 00:29:37 the same as the null space of just this part? 457 00:29:37 --> 00:29:38 I think so. 458 00:29:38 --> 00:29:39 I think so. 459 00:29:39 --> 00:29:44 If Bx is zero, then multiplying by that guy 460 00:29:44 --> 00:29:48.5 I'll still have zero. 461 00:29:48.5 --> 00:29:53 But also if this times some x give zero, I could always 462 00:29:53 --> 00:29:57 multiply on the left by the inverse of that, 463 00:29:57 --> 00:30:03 because it is invertible, and I would discover that this 464 00:30:03 --> 00:30:05 kind of Bx is zero. 465 00:30:05 --> 00:30:09 You want me to write some of that down -- 466 00:30:09 --> 00:30:15 if I have a product here, C times -- times D, 467 00:30:15 --> 00:30:21 say, and if C is invertible, the null space of CD, 468 00:30:21 --> 00:30:26.6 well, it will the same as the null space of D. 469 00:30:26.6 --> 00:30:28 If C is invertible. 470 00:30:28 --> 00:30:36.04 Multiplying by an invertible matrix on the left can't change 471 00:30:36.04 --> 00:30:37 the null space. 472 00:30:37 --> 00:30:39 OK. 473 00:30:39 --> 00:30:44 So basically I'm asking you for the null space of this. 474 00:30:44 --> 00:30:48 I don't have to do the multiplication because I have C 475 00:30:48 --> 00:30:49 is invertible. 476 00:30:49 --> 00:30:52.52 That first factor C is invertible. 477 00:30:52.52 --> 00:30:55 It's not going to change the null space. 478 00:30:55 --> 00:30:56.9 OK. 479 00:30:56.9 --> 00:31:02 So can we just write down a basis now for the null space? 480 00:31:02 --> 00:31:07 So what's the basis for the null space of -- of that? 481 00:31:07 --> 00:31:13 So basis for the null space I'm looking for the two -- there are 482 00:31:13 --> 00:31:16 two pivot columns obviously. 483 00:31:16 --> 00:31:19.5 It clearly has rank two. 484 00:31:19.5 --> 00:31:23 I'm looking for the two special solutions. 485 00:31:23 --> 00:31:27 They'll come from the third and the fourth. 486 00:31:27 --> 00:31:29 The free variables. 487 00:31:29 --> 00:31:32 OK. so if the third free variable 488 00:31:32 --> 00:31:38 is a one, then I think probably I need a minus one there and a 489 00:31:38 --> 00:31:41 one there, it looks like. 490 00:31:41 --> 00:31:45 Do you agree that if I then do that multiplication I'll get 491 00:31:45 --> 00:31:45 zero? 492 00:31:45 --> 00:31:49.2 And if I have one in the fourth variable, then maybe I need a 493 00:31:49.2 --> 00:31:52 one in the second variable and maybe a minus two in the third. 494 00:31:52 --> 00:31:56 So I just reasoned that through and then if I look back I see 495 00:31:56 --> 00:31:59 sure enough that the free variable part that I sometimes 496 00:31:59 --> 00:32:04.09 call F, that up -- that two by two corner, 497 00:32:04.09 --> 00:32:08 is sitting here with all its signs reversed. 498 00:32:08 --> 00:32:15 So that's -- here I'm seeing minus F, and here I'm seeing the 499 00:32:15 --> 00:32:19.41 identity in the null space matrix. 500 00:32:19.41 --> 00:32:23 OK, so that's the null space. 501 00:32:23 --> 00:32:30 Another question is solve Bx equal one, zero, 502 00:32:30 --> 00:32:31 one. 503 00:32:31 --> 00:32:31 OK. 504 00:32:31 --> 00:32:39 So that's one question, now solve complete solutions. 505 00:32:39 --> 00:32:44 To Bx equal one, zero, one. 506 00:32:44 --> 00:32:44 OK. 507 00:32:44 --> 00:32:52 Yeah, so I guess I'm seeing if I wanted to get one, 508 00:32:52 --> 00:32:58 zero, one - What's our particular solution? 509 00:32:58 --> 00:33:03 So I'm looking for a particular solution and then the null space 510 00:33:03 --> 00:33:03 part. 511 00:33:03 --> 00:33:03.77 OK. 512 00:33:03.77 --> 00:33:08 I-- actually the first column of B, so what's the first column 513 00:33:08 --> 00:33:09 of our matrix B? 514 00:33:09 --> 00:33:11.88 It's the vector one, zero, one. 515 00:33:11.88 --> 00:33:17 The first column of our matrix agrees with the right-hand side. 516 00:33:17 --> 00:33:22 So I guess I'm thinking x particular plus x null space 517 00:33:22 --> 00:33:28 will be the particular solution, since the first column of B is 518 00:33:28 --> 00:33:30.58 exactly right, that's great. 519 00:33:30.58 --> 00:33:35 And then I have C times that first null space vector and D 520 00:33:35 --> 00:33:39 times the other null space vector. 521 00:33:39 --> 00:33:40 Right? 522 00:33:40 --> 00:33:44 The two -- the null space part of the solution, 523 00:33:44 --> 00:33:49 as always has the arbitrary constants, the particular 524 00:33:49 --> 00:33:53 solution doesn't have any arbitrary constants, 525 00:33:53 --> 00:33:59 it's one particular solution, and in this case it'll -- 526 00:33:59 --> 00:34:02 that one would do. 527 00:34:02 --> 00:34:02 OK. 528 00:34:02 --> 00:34:07.03 Fine. so those are questions taken 529 00:34:07.03 --> 00:34:13 from old quizzes, any questions coming to mind? 530 00:34:13 --> 00:34:14 Yeah. 531 00:34:14 --> 00:34:16 Q: value. 532 00:34:16 --> 00:34:16 OK. 533 00:34:16 --> 00:34:24 Well, so that particular x particular, it says that let's 534 00:34:24 --> 00:34:33 see, when I multiply by this guy, I'm going to get the first 535 00:34:33 --> 00:34:34 column of B. 536 00:34:34 --> 00:34:38 That -- if that's a solution, 537 00:34:38 --> 00:34:41 I multiply B, B times this x will be the 538 00:34:41 --> 00:34:45 first column of B, and so I'm saying that the 539 00:34:45 --> 00:34:49 first column of this B agrees with the right-hand side. 540 00:34:49 --> 00:34:55 So I'm saying that look at the first column of that matrix B. 541 00:34:55 --> 00:34:58 If you do the multiplication, it's -- so what's the first 542 00:34:58 --> 00:35:00 column of that matrix? 543 00:35:00 --> 00:35:02 Is that how you do that multiplication? 544 00:35:02 --> 00:35:05 I multiply that matrix by that first column. 545 00:35:05 --> 00:35:07 And it picks out one, zero, one. 546 00:35:07 --> 00:35:11 So the first column of B is exactly that. 547 00:35:11 --> 00:35:22 And therefore a particular solution will be this guy. 548 00:35:22 --> 00:35:23 Yeah. 549 00:35:23 --> 00:35:23.95 OK. 550 00:35:23.95 --> 00:35:24 Yes. 551 00:35:24 --> 00:35:30 Q: particular solution. 552 00:35:30 --> 00:35:40 Any of the solutions can be the particular one that we pick out. 553 00:35:40 --> 00:35:48 So like this plus -- plus this would be another particular 554 00:35:48 --> 00:35:49 solution. 555 00:35:49 --> 00:35:54 It would be another solution. 556 00:35:54 --> 00:35:58.93 The particular is just telling us only take one. 557 00:35:58.93 --> 00:36:03 But it's not telling us which one we have to take. 558 00:36:03 --> 00:36:06 We take the most convenient one. 559 00:36:06 --> 00:36:10 I guess in this -- in this problem that was that one. 560 00:36:10 --> 00:36:11.36 Good. 561 00:36:11.36 --> 00:36:12 Other questions? 562 00:36:12 --> 00:36:14 Yes. 563 00:36:14 --> 00:36:18 And this pattern of particular plus null space, 564 00:36:18 --> 00:36:22 of course, that's going throughout mathematics of linear 565 00:36:22 --> 00:36:23 systems. 566 00:36:23 --> 00:36:28 We're really doing mathematics of linear systems here. 567 00:36:28 --> 00:36:33 Our systems are discrete and they're finite-dimensional -- 568 00:36:33 --> 00:36:38 and so it's linear algebra, but this particular plus null 569 00:36:38 --> 00:36:43 space goes -- that doesn't depend on having finite matrices 570 00:36:43 --> 00:36:47 -- that spreads much -- that spreads everywhere. 571 00:36:47 --> 00:36:51.92 OK, I'm going to just like to encourage you to take problems 572 00:36:51.92 --> 00:36:56 out of the book, let me do the same myself. 573 00:36:56 --> 00:37:00 OK well here's some easy true or falses. 574 00:37:00 --> 00:37:05 I don't know why the author put these in here. 575 00:37:05 --> 00:37:05 OK. 576 00:37:05 --> 00:37:11 If m=n, then the row space equals the column space. 577 00:37:11 --> 00:37:15 So these are true or falses. 578 00:37:15 --> 00:37:18 If m equals n, so that means the matrix is 579 00:37:18 --> 00:37:23 square, then the row space equals the column space? 580 00:37:23 --> 00:37:24 False, good. 581 00:37:24 --> 00:37:27.49 Good, what is equal there? 582 00:37:27.49 --> 00:37:31 What can I say is equal, if M -- well, 583 00:37:31 --> 00:37:31 yeah. 584 00:37:31 --> 00:37:36 Yeah it -- so that's definitely false -- 585 00:37:36 --> 00:37:41 the row space and the column space, and this matrix is like 586 00:37:41 --> 00:37:43 always a good example to consider. 587 00:37:43 --> 00:37:48 So there's a square matrix but it's row space is the multiples 588 00:37:48 --> 00:37:51 of zero, one, and its column space is the 589 00:37:51 --> 00:37:53 multiples of one, zero. 590 00:37:53 --> 00:37:55 Very different. 591 00:37:55 --> 00:37:59 The row space and the column space are totally different for 592 00:37:59 --> 00:38:00 that matrix. 593 00:38:00 --> 00:38:04 Now of course if the matrix was symmetric, well, 594 00:38:04 --> 00:38:08 then clearly the row space equals the column space. 595 00:38:08 --> 00:38:08 OK. 596 00:38:08 --> 00:38:10 How about this question? 597 00:38:10 --> 00:38:15 The matrices A and minus A share the same four subspaces? 598 00:38:15 --> 00:38:20 Do the matrices A and minus A have the same column space, 599 00:38:20 --> 00:38:25 do they have the same null space, do they have the same row 600 00:38:25 --> 00:38:25 space? 601 00:38:25 --> 00:38:28 What's the answer on that? 602 00:38:28 --> 00:38:29 Yes or no. 603 00:38:29 --> 00:38:29 Yes. 604 00:38:29 --> 00:38:30 Good. 605 00:38:30 --> 00:38:30 OK. 606 00:38:30 --> 00:38:31 How about this? 607 00:38:31 --> 00:38:37.2 If A and B have the same four subspaces, then A is a multiple 608 00:38:37.2 --> 00:38:38 of B. 609 00:38:38 --> 00:38:44 If -- suppose those subspaces are the same. 610 00:38:44 --> 00:38:47 Then is A a multiple of B? 611 00:38:47 --> 00:38:48.13 OK. 612 00:38:48.13 --> 00:38:53 How how do you answer a question like that? 613 00:38:53 --> 00:39:01 Of course if you want to answer it yes, then I would -- 614 00:39:01 --> 00:39:04 then they'd have to think of a reason why. 615 00:39:04 --> 00:39:09 If you want to answer no way, then you would -- and I would 616 00:39:09 --> 00:39:13 sort of like first I would try to think no, I would say can I 617 00:39:13 --> 00:39:17 come up with an example where it isn't true? 618 00:39:17 --> 00:39:20 Let me repeat the question. 619 00:39:20 --> 00:35:44 And then write the answer. 620 00:35:44 --> 00:35:19 OK. 621 00:35:19 --> 00:31:19 So I'll repeat that question. 622 00:31:19 --> 00:24:07 If -- so true or false, if A and B have the same four 623 00:24:07 --> 00:18:44 subspaces, then A is some multiple of B. 624 00:18:44 --> 00:13:04 True or false, how do you feel about it at 625 00:13:04 --> 00:09:54 this instant?3 626 00:09:54 --> 00:15:12.39 There's quite a few trues, shall I take a poll, 627 00:15:12.39 --> 00:17:51 so how many think true? 628 00:17:51 --> 00:18:12 OK. 629 00:18:12 --> 00:23:09 I gave you every chance to think about that. 630 00:23:09 --> 00:24:19.19 Let's see. 631 00:24:19.19 --> 00:30:39 So what -- I would just take extreme cases if it was me, 632 00:30:39 --> 00:37:07 so when do I know -- well, I would say suppose the matrix 633 00:37:07 --> 00:40:23 is invertible -- suppose A is an invertible 634 00:40:23 --> 00:40:28 matrix, then what -- suppose it's six by six invertible 635 00:40:28 --> 00:40:32 matrix, then what's its row space, and its column space is 636 00:40:32 --> 00:40:37 all of R^6, and the null space, and the null space of A 637 00:40:37 --> 00:40:40 transpose would be the zero vector. 638 00:40:40 --> 00:40:46 So every invertible matrix is going to give that answer. 639 00:40:46 --> 00:40:49 If I have a six by six invertible matrix, 640 00:40:49 --> 00:40:52.8 I know what those subspaces are. 641 00:40:52.8 --> 00:40:58 Heck, that was back in chapter two, when I didn't even know 642 00:40:58 --> 00:41:00 what subspaces were. 643 00:41:00 --> 00:41:05 The row space and column space are both all six-dimensional 644 00:41:05 --> 00:41:08 space -- the whole space, 645 00:41:08 --> 00:41:11 and the rank is six, in other words, 646 00:41:11 --> 00:41:14 and the null spaces have zero dimension. 647 00:41:14 --> 00:41:16 So do you see now the answer? 648 00:41:16 --> 00:41:18 So A and B could be. 649 00:41:18 --> 00:41:23 So A and B could be for example any -- so I'm going to say 650 00:41:23 --> 00:41:23 false. 651 00:41:23 --> 00:41:28 Because A and B for example -- So an example: 652 00:41:28 --> 00:41:34 A and B any invertible six by six, six by six. 653 00:41:34 --> 00:41:41 So those would have the same four subspaces but they wouldn't 654 00:41:41 --> 00:41:42 be the same. 655 00:41:42 --> 00:41:50 Of course th- there should be something about those matrices 656 00:41:50 --> 00:41:54 that would be the same. 657 00:41:54 --> 00:41:58 It's sort of a natural problem, so now actually we're getting 658 00:41:58 --> 00:41:59 to a math question. 659 00:41:59 --> 00:42:02 The answer is this is not true. 660 00:42:02 --> 00:42:06 One matrix doesn't have to be a multiple of the other. 661 00:42:06 --> 00:42:09 But there must be something that's true. 662 00:42:09 --> 00:42:14 And that would be sort of like a natural question to ask. 663 00:42:14 --> 00:42:27 If they have the same subspaces, same four subspaces, 664 00:42:27 --> 00:42:38 then what -- what could you -- instinct wasn't necessarily 665 00:42:38 --> 00:42:39 right. 666 00:42:39 --> 00:42:43 But I hope you now see that the correct answer is false. 667 00:42:43 --> 00:42:47 And then you might think OK, well, they certainly do have 668 00:42:47 --> 00:42:48 the same rank. 669 00:42:48 --> 00:42:52.62 But do -- obviously if they have the same four subspaces, 670 00:42:52.62 --> 00:42:55 they have the same rank. 671 00:42:55 --> 00:43:00 I might say if they have the well, I could extend that 672 00:43:00 --> 00:43:06 question and think about other possibilities and finally come 673 00:43:06 --> 00:43:10 up with something that was true. 674 00:43:10 --> 00:43:12 But I won't press that one. 675 00:43:12 --> 00:43:17 Let me keep going with practice questions. 676 00:43:17 --> 00:43:23 And these practice questions are quite appropriate I think 677 00:43:23 --> 00:43:24 for the exam. 678 00:43:24 --> 00:43:26.21 OK. let's see. 679 00:43:26.21 --> 00:43:31.96 If I exchange two rows of A which subspaces stay the same? 680 00:43:31.96 --> 00:43:37 So I'm trying to take out questions that we can answer 681 00:43:37 --> 00:43:42.25 without you know we can answer quickly. 682 00:43:42.25 --> 00:43:47 If I have a matrix A, and I exchange two of its rows, 683 00:43:47 --> 00:43:50 which subspaces stay the same? 684 00:43:50 --> 00:43:53 The row space does stay the same. 685 00:43:53 --> 00:43:57 And the null space stays the same. 686 00:43:57 --> 00:43:57 Good. 687 00:43:57 --> 00:43:58 Good. 688 00:43:58 --> 00:43:59 Correct. 689 00:43:59 --> 00:44:03 Column space would be a wrong answer. 690 00:44:03 --> 00:44:07.85 OK. all right, here's a question. 691 00:44:07.85 --> 00:44:12 Oh, this leads into the next chapter. 692 00:44:12 --> 00:44:18 Why can the vector one, two, three not be a row and 693 00:44:18 --> 00:44:21 also in the null space? 694 00:44:21 --> 00:44:25 Fitting we close with this question. 695 00:44:25 --> 00:44:30 Close is -- so V equal this one, 696 00:44:30 --> 00:44:38 two, three can't be in the null space of a matrix and the row 697 00:44:38 --> 00:44:39 space. 698 00:44:39 --> 00:44:42 And my question is why not? 699 00:44:42 --> 00:44:43 Why not? 700 00:44:43 --> 00:44:51.71 So this is a question that we can because it's sort of asked 701 00:44:51.71 --> 00:45:00 in a straightforward way, we can figure out an answer. 702 00:45:00 --> 00:45:05 Well, actually yeah -- I'll even pin it down, 703 00:45:05 --> 00:45:10 it can't be in the null space -- and be a row. 704 00:45:10 --> 00:45:14 I'll even pin it down further. 705 00:45:14 --> 00:45:17 Ask it to be a row of A. 706 00:45:17 --> 00:45:18 Why not? 707 00:45:18 --> 00:45:25 So I'm -- we know the dimensions of these spaces. 708 00:45:25 --> 00:45:30 But now I'm asking you sort of like the overlap between -- so 709 00:45:30 --> 00:45:36 the null space and the row space, those are in the same 710 00:45:36 --> 00:45:38 n-dimensional space. 711 00:45:38 --> 00:45:42 Those are -- well, those are both subspaces of 712 00:45:42 --> 00:45:47 n-dimensional space, and I'm basically saying they 713 00:45:47 --> 00:45:50 can't overlap. 714 00:45:50 --> 00:45:54 I can't have a vector like this, a typical vector, 715 00:45:54 --> 00:45:59 that's in the null space and it's also a row of the matrix. 716 00:45:59 --> 00:46:00 Why is that? 717 00:46:00 --> 00:46:03 So that's a new sort of idea. 718 00:46:03 --> 00:46:07 Let's just see what it would mean. 719 00:46:07 --> 00:46:11 I mean that A times this V, why can this A times this V it 720 00:46:11 --> 00:46:13.04 can't be zero. 721 00:46:13.04 --> 00:46:17 Oh well, if it's zero, so this is -- I'm getting it 722 00:46:17 --> 00:46:19 into the null space here. 723 00:46:19 --> 00:46:24 So this is -- now let's put that vector's in the null space, 724 00:46:24 --> 00:46:27 why can't the first row of a matrix be one, 725 00:46:27 --> 00:46:29 two, three? 726 00:46:29 --> 00:46:34 I can fill out the matrix as I like. 727 00:46:34 --> 00:46:38 Why is that impossible? 728 00:46:38 --> 00:46:45.25 Well, you're seeing it's impossible, right? 729 00:46:45.25 --> 00:46:49 That if that was a row of the matrix and in the null space, 730 00:46:49 --> 00:46:53 that number would not be zero, it would be fourteen. 731 00:46:53 --> 00:46:53 Right. 732 00:46:53 --> 00:46:58 So now we actually are beginning to get a more complete 733 00:46:58 --> 00:47:00 picture of these four subspaces. 734 00:47:00 --> 00:47:03 The two that are over in n-dimensional space, 735 00:47:03 --> 00:47:07 they actually only share the zero vector. 736 00:47:07 --> 00:47:12 The intersection of the null space and the row space is only 737 00:47:12 --> 00:47:14 the zero vector. 738 00:47:14 --> 00:47:19 And in fact the null space is perpendicular to the row space. 739 00:47:19 --> 00:47:24 That'll be the first topic let's see, we have a holiday 740 00:47:24 --> 00:47:28 Monday -- and I'll see you Wednesday with 741 00:47:28 --> 00:47:29.66 perpendiculars. 742 00:47:29.66 --> 00:47:31 And I'll see you Friday. 743 00:47:31 --> 00:47:34 So good luck on the quiz.