1 00:00:06 --> 00:00:07 Okay. 2 00:00:07 --> 00:00:13.85 This is lecture six in linear algebra, and we're at the start 3 00:00:13.85 --> 00:00:18 of this new chapter, chapter three in the text, 4 00:00:18 --> 00:00:24 which is really getting to the center of linear algebra. 5 00:00:24 --> 00:00:30 And I had time to make a first start on it at the end of 6 00:00:30 --> 00:00:32 lecture five. 7 00:00:32 --> 00:00:37 But now is lecture six is officially the lecture on vector 8 00:00:37 --> 00:00:39.62 spaces and subspaces. 9 00:00:39.62 --> 00:00:44 And then especially -- there are two subspaces that we're 10 00:00:44 --> 00:00:46.54 specially interested in. 11 00:00:46.54 --> 00:00:51.47 One is the column space of a matrix, the other is the null 12 00:00:51.47 --> 00:00:53.2 space of the matrix. 13 00:00:53.2 --> 00:00:57 So, I got to tell you what those are. 14 00:00:57 --> 00:00:57 Okay. 15 00:00:57 --> 00:01:01 So, first to remember from lecture five, 16 00:01:01 --> 00:01:03.77 what is a vector space? 17 00:01:03.77 --> 00:01:09 It's a bunch of vectors that -- where I'm allowed -- where I can 18 00:01:09 --> 00:01:15 add -- I can add any two vectors in the space and the answer 19 00:01:15 --> 00:01:17.46 stays in the space. 20 00:01:17.46 --> 00:01:23 Or I can multiply any vector in the space by any constant and 21 00:01:23 --> 00:01:27 the result stays in the space. 22 00:01:27 --> 00:01:31 So that's -- in fact if I combine those two into one, 23 00:01:31 --> 00:01:35 you can see that -- if I can add and I can multiply by 24 00:01:35 --> 00:01:40 numbers, that really means that I can take linear combinations. 25 00:01:40 --> 00:01:45 So the quick way to say it is that all linear combinations, 26 00:01:45 --> 00:01:49 C -- any multiple of V plus any 27 00:01:49 --> 00:01:52 multiple of W stay in the space. 28 00:01:52 --> 00:01:58 So, can I give you examples that are vector spaces and also 29 00:01:58 --> 00:02:03 some examples that are not, to make that point clear? 30 00:02:03 --> 00:02:06 So, suppose I'm in three dimensions. 31 00:02:06 --> 00:02:11 Then one way to get us one space is the whole three 32 00:02:11 --> 00:02:14 dimensional space. 33 00:02:14 --> 00:02:18 So the whole space R^3, three dimensional space, 34 00:02:18 --> 00:02:23 would be a vector space, because if I have a couple of 35 00:02:23 --> 00:02:29 vectors I can add them and I'm certainly okay and they follow 36 00:02:29 --> 00:02:30 all the rules. 37 00:02:30 --> 00:02:31 So R^3 is easy. 38 00:02:31 --> 00:02:36 Now I'm interested also in subspaces. 39 00:02:36 --> 00:02:39 So there's this key word, subspaces. 40 00:02:39 --> 00:02:45 That's a space -- that's some vectors inside the given space, 41 00:02:45 --> 00:02:51.66 inside R three that still make up a vector space of their own. 42 00:02:51.66 --> 00:02:56 It's a vector space inside a vector space. 43 00:02:56 --> 00:02:59 And the simplest example was a plane. 44 00:02:59 --> 00:03:04 So, like, can I just sketch it -- there is a plane. 45 00:03:04 --> 00:03:08 It's got to go through the origin, and of course it goes 46 00:03:08 --> 00:03:10 infinitely far. 47 00:03:10 --> 00:03:12 That's of that's a subspace now. 48 00:03:12 --> 00:03:18.12 Do you see that if I have two vectors on the plane and I add 49 00:03:18.12 --> 00:03:22 them, the result stays in the plane. 50 00:03:22 --> 00:03:28 If I take a vector in the plane and I multiply by minus two, 51 00:03:28 --> 00:03:31 I'm still in the plane. 52 00:03:31 --> 00:03:34 So that plane is a subspace. 53 00:03:34 --> 00:03:37 So let me just make that point. 54 00:03:37 --> 00:03:43 Plane through zero, through that zero zero zero is 55 00:03:43 --> 00:03:44 a subspace. 56 00:03:44 --> 00:03:44 Okay. 57 00:03:44 --> 00:03:50 And also, another subspace would be a line. 58 00:03:50 --> 00:03:55 A line through zero zero zero -- yeah, the line has to go 59 00:03:55 --> 00:03:57 through the origin. 60 00:03:57 --> 00:04:01 All subspaces have got to contain the origin, 61 00:04:01 --> 00:04:04 contain zero -- the zero vector. 62 00:04:04 --> 00:04:07 So this line is a subspace. 63 00:04:07 --> 00:04:11 Really, if I want to say it really correctly, 64 00:04:11 --> 00:04:15 I should say a subspace of R^3. 65 00:04:15 --> 00:04:21 That of R^3 was, like, understood there. 66 00:04:21 --> 00:04:26 Now -- so let me call this plane P. 67 00:04:26 --> 00:04:30 And let me call this line L. 68 00:04:30 --> 00:04:37 And let me ask you about other sets of vectors. 69 00:04:37 --> 00:04:44 Suppose I took -- yeah -- so here's a first 70 00:04:44 --> 00:04:45 question. 71 00:04:45 --> 00:04:49 Suppose I take two subspaces, like P and L. 72 00:04:49 --> 00:04:53 And I just put them together, take their union, 73 00:04:53 --> 00:04:59 take all the vectors -- so now you've got P and L in mind, 74 00:04:59 --> 00:05:00 here. 75 00:05:00 --> 00:05:03 So I have two subspaces. 76 00:05:03 --> 00:05:09 I have two subspaces and, for example, 77 00:05:09 --> 00:05:13 P -- a plane and L a line. 78 00:05:13 --> 00:05:14 Okay. 79 00:05:14 --> 00:05:21 Now I want to ask you about the union of those. 80 00:05:21 --> 00:05:23 So P union L. 81 00:05:23 --> 00:05:29 This is all vectors in P or L or both. 82 00:05:29 --> 00:05:32 Is that a subspace? 83 00:05:32 --> 00:05:37.81 Is this a subspace? 84 00:05:37.81 --> 00:05:41 This is or is not a subspace? 85 00:05:41 --> 00:05:47 Because we're -- I just want to be sure that I've got the 86 00:05:47 --> 00:05:49 central idea. 87 00:05:49 --> 00:05:56 Suppose I take the vectors in the plane and also the vectors 88 00:05:56 --> 00:06:00 on that line, put them together, 89 00:06:00 --> 00:06:06 so I've got a bunch of vectors, is it a subspace? 90 00:06:06 --> 00:06:10 Can you give me, like, so the camera can hear it 91 00:06:10 --> 00:06:12 or maybe the tape. 92 00:06:12 --> 00:06:14 Can you say yes or no? 93 00:06:14 --> 00:06:19 Do I have a subspace if I put -- if I take all the vectors on 94 00:06:19 --> 00:06:24 the plane plus all -- and all the ones on the line and just 95 00:06:24 --> 00:06:29.05 join them together -- but I'm not taking this guy 96 00:06:29.05 --> 00:06:32 that's -- actually, I'm not taking most of them, 97 00:06:32 --> 00:06:36 because most vectors are not on the line or the plane, 98 00:06:36 --> 00:06:38 they're off somewhere else. 99 00:06:38 --> 00:06:39.65 Do I have a subspace? 100 00:06:39.65 --> 00:06:40 STUDENTS: No. 101 00:06:40 --> 00:06:41 STRANG: Right. 102 00:06:41 --> 00:06:42 Thank you. 103 00:06:42 --> 00:06:42 No. 104 00:06:42 --> 00:06:44 Because -- why not? 105 00:06:44 --> 00:06:46 Because I can't add. 106 00:06:46 --> 00:06:51 Because if I that requirement isn't satisfied. 107 00:06:51 --> 00:06:57 If I take one vector like this guy and another vector that 108 00:06:57 --> 00:07:03 happens to come from L and add, I'm off somewhere else. 109 00:07:03 --> 00:07:07 You see that I've gone outside the union if I just add 110 00:07:07 --> 00:07:11 something from P and something from L, then normally what'll 111 00:07:11 --> 00:07:16 happen is I'm outside the union -- and I don't have a subspace. 112 00:07:16 --> 00:07:18 So the correct answer is -- is not. 113 00:07:18 --> 00:07:19 Okay. 114 00:07:19 --> 00:07:23 Now let me ask you about -- the other thing we do is take the 115 00:07:23 --> 00:07:25.24 intersection. 116 00:07:25.24 --> 00:07:28 So what does intersection mean? 117 00:07:28 --> 00:07:34 Intersection means all vectors that are in both P and L. 118 00:07:34 --> 00:07:36 Is this a subspace. 119 00:07:36 --> 00:07:42 Yeah, so I guess I want to go back up to the same question. 120 00:07:42 --> 00:07:45 This is or is not a subspace? 121 00:07:45 --> 00:07:51 And you can answer me -- answer the question first for this 122 00:07:51 --> 00:07:56 particular example, this picture I drew. 123 00:07:56 --> 00:07:59 What is P intersect L for this case? 124 00:07:59 --> 00:08:01 STUDENT: It's only zero. 125 00:08:01 --> 00:08:04 STRANG: It's only zero. 126 00:08:04 --> 00:08:09 At least, sort of this was the artist's idea as he drew it 127 00:08:09 --> 00:08:13 that, that that line L was not in the plane and, 128 00:08:13 --> 00:08:18 went off somewhere else -- and then the only point that 129 00:08:18 --> 00:08:21 was in common was the zero vector. 130 00:08:21 --> 00:08:23 Is the zero vector by itself a subspace? 131 00:08:23 --> 00:08:24 STUDENT: Yes. 132 00:08:24 --> 00:08:26.43 STRANG: Yes, absolutely. 133 00:08:26.43 --> 00:08:29.54 And what about, if I don't have this plane and 134 00:08:29.54 --> 00:08:33 this line but any subspace and any other subspace? 135 00:08:33 --> 00:08:40 So now -- can I ask that question for any two subspaces? 136 00:08:40 --> 00:08:44.54 So maybe I'll write it up here. 137 00:08:44.54 --> 00:08:47 If I'm strong enough. 138 00:08:47 --> 00:08:47.8 Okay. 139 00:08:47.8 --> 00:08:52 So this is the general question. 140 00:08:52 --> 00:08:56.1 I have subspaces, say S and T. 141 00:08:56.1 --> 00:09:03 And I want to ask you about their intersection S intersect T 142 00:09:03 --> 00:09:07.07 and I want -- it is a subspace. 143 00:09:07.07 --> 00:09:08 Do you see why? 144 00:09:08 --> 00:09:15 Do you see why if I take the vectors that are in both one- 145 00:09:15 --> 00:09:22 th- that are in both of the subspaces -- so that's like a 146 00:09:22 --> 00:09:29 smaller set of vectors, probably, because it's -- 147 00:09:29 --> 00:09:32 we've added requirements. 148 00:09:32 --> 00:09:34.46 It has to be in S and in T. 149 00:09:34.46 --> 00:09:37 How do I know that's a subspace? 150 00:09:37 --> 00:09:42 Can we just think through that abstract stuff and then I get to 151 00:09:42 --> 00:09:43 the examples. 152 00:09:43 --> 00:09:44 Okay. 153 00:09:44 --> 00:09:44 So why? 154 00:09:44 --> 00:09:48 Suppose I take a couple of vectors that are in the 155 00:09:48 --> 00:09:50 intersection. 156 00:09:50 --> 00:09:55 Why is the sum also in the intersection? 157 00:09:55 --> 00:09:59 Okay, so let me give a name to these vectors, 158 00:09:59 --> 00:10:01 say V and W. 159 00:10:01 --> 00:10:04 They're in the intersection. 160 00:10:04 --> 00:10:07 So that means they're both in S. 161 00:10:07 --> 00:10:10 Also means they're both in T. 162 00:10:10 --> 00:10:14 So what can I say about V plus W? 163 00:10:14 --> 00:10:15 Is it in S? 164 00:10:15 --> 00:10:15 Yes. 165 00:10:15 --> 00:10:18 Right? 166 00:10:18 --> 00:10:22 If I take two vectors, V and W that are both in S, 167 00:10:22 --> 00:10:25 then the sum is in S, because S was a subspace. 168 00:10:25 --> 00:10:30 And if they're both in T and I add them, then the result is 169 00:10:30 --> 00:10:33 also in T, because T was a subspace. 170 00:10:33 --> 00:10:37 So the result V plus W is in the intersection. 171 00:10:37 --> 00:10:42 It's in both and requirement one is satisfied. 172 00:10:42 --> 00:10:43 Requirement two's the same. 173 00:10:43 --> 00:10:47 If I take a vector that's in both, I multiply by seven. 174 00:10:47 --> 00:10:52 Seven times that vector is in S, because the vector was in S. 175 00:10:52 --> 00:10:56 Seven times that vector's in T because the original one was in 176 00:10:56 --> 00:10:56 T. 177 00:10:56 --> 00:11:00 So seven times that vector is in the intersection. 178 00:11:00 --> 00:11:05 In other words, when you take the intersection 179 00:11:05 --> 00:11:09 of two subspaces, you get probably a smaller 180 00:11:09 --> 00:11:12 subspace, but it is a subspace. 181 00:11:12 --> 00:11:13 Okay. 182 00:11:13 --> 00:11:18 So that's like sort of just emphasizing what these two 183 00:11:18 --> 00:11:20 requirements mean. 184 00:11:20 --> 00:11:23 Again -- Let me circle those, 185 00:11:23 --> 00:11:26 because those are so important. 186 00:11:26 --> 00:11:30.5 The sum and the scale of multiplication which combines 187 00:11:30.5 --> 00:11:32 into linear combinations. 188 00:11:32 --> 00:11:36 That's what you have to do inside the subspace. 189 00:11:36 --> 00:11:36 Okay. 190 00:11:36 --> 00:11:38 On to the column space. 191 00:11:38 --> 00:11:38 Okay. 192 00:11:38 --> 00:11:44 So my lecture last time started that and I want to continue it. 193 00:11:44 --> 00:11:45 Okay. 194 00:11:45 --> 00:11:50 Column space of a matrix. 195 00:11:50 --> 00:11:51 Of A. 196 00:11:51 --> 00:11:52 Okay. 197 00:11:52 --> 00:11:56 Can I take an example? 198 00:11:56 --> 00:12:01 Say one two three four. 199 00:12:01 --> 00:12:04 One one one one. 200 00:12:04 --> 00:12:09 Two three four five. 201 00:12:09 --> 00:12:10 Okay. 202 00:12:10 --> 00:12:13 That's my matrix A. 203 00:12:13 --> 00:12:23 So, it's got columns, three columns. 204 00:12:23 --> 00:12:28.35 Those columns are vectors, so the column space of this A, 205 00:12:28.35 --> 00:12:32 of this A -- let's stay with this example for a while. 206 00:12:32 --> 00:12:38 The column space of this matrix is a subspace of R -- R what? 207 00:12:38 --> 00:12:43 So what space are we in if I'm looking at the columns of this 208 00:12:43 --> 00:12:44 matrix? 209 00:12:44 --> 00:12:45.82 R^4 , right? 210 00:12:45.82 --> 00:12:50.54 These are vectors in R^4, they're four dimensional 211 00:12:50.54 --> 00:12:51 vectors. 212 00:12:51 --> 00:12:56 So it's this column space of A is a subspace of R^4 here, 213 00:12:56 --> 00:13:01 because A was four by -- A is a four by three matrix. 214 00:13:01 --> 00:13:07 This tells me how many rows there are, how many components 215 00:13:07 --> 00:13:11 in a column, and so we're in R^4. 216 00:13:11 --> 00:13:15 Okay, now what's in that subspace? 217 00:13:15 --> 00:13:21 So the column space of A, it's a subspace of R^4. 218 00:13:21 --> 00:13:26 I call it the column space of A, like that. 219 00:13:26 --> 00:13:34.21 So that's my little symbol for some subspace of R^4. 220 00:13:34.21 --> 00:13:36 What's in that subspace? 221 00:13:36 --> 00:13:38 Well, that column certainly is. 222 00:13:38 --> 00:13:40.18 One two three four. 223 00:13:40.18 --> 00:13:41 This column is in. 224 00:13:41 --> 00:13:44 This column is in, and what else? 225 00:13:44 --> 00:13:48 So it's got the columns of A in it, but that's not enough, 226 00:13:48 --> 00:13:49.62 certainly. 227 00:13:49.62 --> 00:13:50 Right? 228 00:13:50 --> 00:13:55 I don't have a subspace if I just put in three vectors. 229 00:13:55 --> 00:13:59 So how do I fill that out to be a subspace? 230 00:13:59 --> 00:14:02 I take their linear combinations. 231 00:14:02 --> 00:14:06 So the column space of A is all linear combinations -- 232 00:14:06 --> 00:14:09 combinations of the columns. 233 00:14:09 --> 00:14:12 And that does give me a subspace. 234 00:14:12 --> 00:14:17 It does give me a vector space, because if I have one linear 235 00:14:17 --> 00:14:23 combination and I multiply by eleven, I've got another linear 236 00:14:23 --> 00:14:25 combination. 237 00:14:25 --> 00:14:29 If I have a linear combination, I add to another linear 238 00:14:29 --> 00:14:32.66 combination I get a third combination. 239 00:14:32.66 --> 00:14:36 So that -- this is like the smallest space -- like, 240 00:14:36 --> 00:14:39 it's got to have those three columns in it, 241 00:14:39 --> 00:14:44 and it has to have their combinations and that's where we 242 00:14:44 --> 00:14:45 stop. 243 00:14:45 --> 00:14:45 Okay. 244 00:14:45 --> 00:14:49 Now I'm going to be interested in that space. 245 00:14:49 --> 00:14:54 So I, like -- get some idea of what's in that space. 246 00:14:54 --> 00:14:56 How big is that space? 247 00:14:56 --> 00:15:02 Is that space the whole four dimensional space? 248 00:15:02 --> 00:15:04 Or is it a subspace inside? 249 00:15:04 --> 00:15:09 Can you -- let me just see if we can get a yes or no answer 250 00:15:09 --> 00:15:13 sometimes without being ready for the complete proof. 251 00:15:13 --> 00:15:15 What do you think? 252 00:15:15 --> 00:15:18 Is the subspace that I'm talking about here, 253 00:15:18 --> 00:15:24 the combinations of those three guys, does that fill the full 254 00:15:24 --> 00:15:26 four dimensional space? 255 00:15:26 --> 00:15:29 Maybe yes or no on that one. 256 00:15:29 --> 00:15:29 No. 257 00:15:29 --> 00:15:29 No. 258 00:15:29 --> 00:15:34 Somehow our feeling is, and it happens to be right, 259 00:15:34 --> 00:15:38 that if we start with three vectors and take their 260 00:15:38 --> 00:15:41 combinations, we can't get the whole four 261 00:15:41 --> 00:15:44 dimensional space. 262 00:15:44 --> 00:15:48 Now -- so somehow we get a smaller space. 263 00:15:48 --> 00:15:51 But how much smaller? 264 00:15:51 --> 00:15:54.62 That's going to come up here. 265 00:15:54.62 --> 00:15:57 That's not so immediate. 266 00:15:57 --> 00:16:03 Let me first make this critical connection with -- 267 00:16:03 --> 00:16:10.86 with, linear equations, because behind our abstract 268 00:16:10.86 --> 00:16:15 definition, we have a purpose. 269 00:16:15 --> 00:16:19 And that is to understand Ax=b. 270 00:16:19 --> 00:16:26 So suppose I make the connection -- w- w- does A x=b 271 00:16:26 --> 00:16:32.74 always have a solution for every b? 272 00:16:32.74 --> 00:16:37 Have a solution for every right-hand side? 273 00:16:37 --> 00:16:43 I guess that's going to be a yes or no question. 274 00:16:43 --> 00:16:50 And then I'm going to ask which right-hand sides are okay? 275 00:16:50 --> 00:16:58 That's really the question I'm after, is which right-hand sides 276 00:16:58 --> 00:17:05 (b) do make up -- you can see from the way I'm speaking what 277 00:17:05 --> 00:17:10 the question -- As it is. 278 00:17:10 --> 00:17:13 The answer is no. 279 00:17:13 --> 00:17:19 A x=b does not have a solution for every b. 280 00:17:19 --> 00:17:22 Why do I say no? 281 00:17:22 --> 00:17:29 Because A x=b is -- like, this is four equations, 282 00:17:29 --> 00:17:34 and only three unknowns. 283 00:17:34 --> 00:17:35 Right? 284 00:17:35 --> 00:17:44 X is -- let me right out that whole -- what the whole thing 285 00:17:44 --> 00:17:46 looks like. 286 00:17:46 --> 00:17:46 Yeah. 287 00:17:46 --> 00:17:50 Let me write out A x=b. 288 00:17:50 --> 00:17:56 A x is -- these columns are one two three four. 289 00:17:56 --> 00:18:04 One one one one and two three four five. 290 00:18:04 --> 00:18:08 Then x, of course, has three components, 291 00:18:08 --> 00:18:10 x1, x2, x3. 292 00:18:10 --> 00:18:16 And I'm trying to get the -- hit the right-hand side, 293 00:18:16 --> 00:18:18 b1,b2,b3 and b4. 294 00:18:18 --> 00:18:23 So my first point is, I can't always do it. 295 00:18:23 --> 00:18:29 In a way, that just says again what you told me five minutes 296 00:18:29 --> 00:18:34 ago -- that the combinations of these 297 00:18:34 --> 00:18:37 columns don't fill the whole four dimensional space. 298 00:18:37 --> 00:18:41 There's going to be some vectors b, a lot of vectors b, 299 00:18:41 --> 00:18:45 that are not combinations of these three columns, 300 00:18:45 --> 00:18:48 because the combinations of those columns are, 301 00:18:48 --> 00:18:53 like, going to be just a little plane or something inside -- 302 00:18:53 --> 00:18:54 inside R^4. 303 00:18:54 --> 00:19:00 Now, so and you see that I do have four equations and only 304 00:19:00 --> 00:19:01 three unknowns. 305 00:19:01 --> 00:19:06 So, like anybody is going to say, no you dope, 306 00:19:06 --> 00:19:11 you can't usually solve four equations with only three 307 00:19:11 --> 00:19:13 unknowns. 308 00:19:13 --> 00:19:16 But now I want to say sometimes you can. 309 00:19:16 --> 00:19:20 For some right-hand sides, I can solve this. 310 00:19:20 --> 00:19:25 So that's the bunch of right-hand sides that I'm 311 00:19:25 --> 00:19:27 interested in right now. 312 00:19:27 --> 00:19:31.45 Which right-hand sides allow me to solve this? 313 00:19:31.45 --> 00:19:35 This is the question for today. 314 00:19:35 --> 00:19:41 It's going to have, like, a nice clear answer. 315 00:19:41 --> 00:19:49 So my question is -- is which bs, which vectors b, 316 00:19:49 --> 00:19:53.7 allow this system to be solved? 317 00:19:53.7 --> 00:19:59 And I want to ask you -- so that's, like, 318 00:19:59 --> 00:20:07 gets two question marks to indicate that's -- 319 00:20:07 --> 00:20:10.23 this is the important question. 320 00:20:10.23 --> 00:20:14 Okay, first, before we give a total answer, 321 00:20:14 --> 00:20:16 give me just a partial answer. 322 00:20:16 --> 00:20:21 Tell me one right-hand side that I know I can solve this 323 00:20:21 --> 00:20:22 thing for. 324 00:20:22 --> 00:20:24 So -- all zeroes. 325 00:20:24 --> 00:20:24 Okay. 326 00:20:24 --> 00:20:28 That's the, like, guaranteed. 327 00:20:28 --> 00:20:32 If these were all zero, then I know I can solve it, 328 00:20:32 --> 00:20:36.43 let the x-s all be zero, no problem. 329 00:20:36.43 --> 00:20:36 Okay. 330 00:20:36 --> 00:20:39 So that's always a -- okay. 331 00:20:39 --> 00:20:42 A x=0 I can always solve. 332 00:20:42 --> 00:20:47 Now tell me another right-hand side, just a specific set of 333 00:20:47 --> 00:20:53 numbers for which I can solve these three -- 334 00:20:53 --> 00:20:57 these four equations with only three unknowns, 335 00:20:57 --> 00:21:00 but if you give me a good right-hand side, 336 00:21:00 --> 00:21:01 I can do it. 337 00:21:01 --> 00:21:03 So tell me one? 338 00:21:03 --> 00:21:04 STUDENT:1 2 3 4. 339 00:21:04 --> 00:21:06 STRANG: 1 2 3 4? 340 00:21:06 --> 00:21:10 If I -- can I solve -- is that a good right-hand side? 341 00:21:10 --> 00:21:15 Can you solve -- can you find a solution that -- 342 00:21:15 --> 00:21:18 X one plus X two plus two X three is one, 343 00:21:18 --> 00:21:22.9 two X one plus X two plus three X three is two and two more 344 00:21:22.9 --> 00:21:27.22 equations -- so I'm asking you to solve in your head in -- 345 00:21:27.22 --> 00:21:30 within five seconds, four equations and three 346 00:21:30 --> 00:21:35 unknowns, but you can do it, because the right-hand side is, 347 00:21:35 --> 00:21:40 like, showing up here is -- it's one of the columns. 348 00:21:40 --> 00:21:45 So tell me what's the X that does solve it? 349 00:21:45 --> 00:21:47 One zero zero. 350 00:21:47 --> 00:21:51.89 One zero zero solves it, because -- well, 351 00:21:51.89 --> 00:21:57 so you can multiply this out by rows, but oh God, 352 00:21:57 --> 00:22:04 it's much nicer to say -- okay, this is one of this 353 00:22:04 --> 00:22:07 column, zero of this, zero of this, 354 00:22:07 --> 00:22:12 so it's one of that column, which is exactly what we 355 00:22:12 --> 00:22:13 wanted. 356 00:22:13 --> 00:22:14 Okay. 357 00:22:14 --> 00:22:17 So there is a b that's okay. 358 00:22:17 --> 00:22:23 Now tell me another B that's okay, another right-hand side 359 00:22:23 --> 00:22:25 that would be all right? 360 00:22:25 --> 00:22:28 Well -- all ones? 361 00:22:28 --> 00:22:33 Actually -- and then what's the solution in that case? 362 00:22:33 --> 00:22:34 0 1 0, thanks. 363 00:22:34 --> 00:22:37 And, in fact, it's much e- like, 364 00:22:37 --> 00:22:42 one way to do it is think of a solution first, 365 00:22:42 --> 00:22:47 right, and then just see what b turns out to be. 366 00:22:47 --> 00:22:51.34 What b turns out to be, right. 367 00:22:51.34 --> 00:22:52 Okay. 368 00:22:52 --> 00:22:58 So I think of a solution -- so I think of an x, 369 00:22:58 --> 00:23:03 I think of any -- x1, x2, x3, I do this 370 00:23:03 --> 00:23:09 multiplication and what have I got? 371 00:23:09 --> 00:23:13 Now I'm ready to answer the big question. 372 00:23:13 --> 00:23:19 I can solve A x=b exactly when the right-hand side B is a 373 00:23:19 --> 00:23:22 vector in the column space. 374 00:23:22 --> 00:23:22 Good. 375 00:23:22 --> 00:23:28 I can solve A x=b when b is a combination of the columns, 376 00:23:28 --> 00:23:34 when it's in the column space -- so let me write that answer 377 00:23:34 --> 00:23:36 down. 378 00:23:36 --> 00:23:43 I can solve Ax=b exactly when B is in the column space. 379 00:23:43 --> 00:23:48.67 Let me just say again why that is. 380 00:23:48.67 --> 00:23:57.33 Because it -- the column space by its definition contains all 381 00:23:57.33 --> 00:23:59 the combinations. 382 00:23:59 --> 00:24:03 It contains all the Ax-s. 383 00:24:03 --> 00:24:11 The column space really consists of all vectors A times 384 00:24:11 --> 00:24:13.64 any X. 385 00:24:13.64 --> 00:24:17 So those are the bs that I can deal with. 386 00:24:17 --> 00:24:23 If b is a combination of the columns, then that combination 387 00:24:23 --> 00:24:26 tells me what X should be. 388 00:24:26 --> 00:24:32 If b is not a combination of the columns, then there is no x. 389 00:24:32 --> 00:24:36 There's no way to solve A x equal b. 390 00:24:36 --> 00:24:37 Okay. 391 00:24:37 --> 00:24:44 So the column space -- that's really why we're interested in 392 00:24:44 --> 00:24:50 this column space, because it's the central guy. 393 00:24:50 --> 00:24:56 It says when we can solve, and that -- 394 00:24:56 --> 00:24:59 we got to understand this column space better. 395 00:24:59 --> 00:25:00 Let's see. 396 00:25:00 --> 00:25:03 Do I want to think -- yeah, somehow -- oh, 397 00:25:03 --> 00:25:06 well, let's just -- as long as we've got it here, 398 00:25:06 --> 00:25:10 what do I get for this particular example? 399 00:25:10 --> 00:25:13.59 If I take combinations of this and this and this, 400 00:25:13.59 --> 00:25:17 I'll tell you the question that's in my mind. 401 00:25:17 --> 00:25:23 It's not even proper to use this word yet, 402 00:25:23 --> 00:25:27 but you'll know what it means. 403 00:25:27 --> 00:25:32.41 Are those three columns independent? 404 00:25:32.41 --> 00:25:40 If I take the combinations of the three columns -- does each 405 00:25:40 --> 00:25:46 column contribute something new or now? 406 00:25:46 --> 00:25:51 So that if I take the combinations of those three 407 00:25:51 --> 00:25:53 columns, do I, like, get some three 408 00:25:53 --> 00:25:58 dimensional subspace -- do I have three vectors that are, 409 00:25:58 --> 00:26:00 like, you know, independent, 410 00:26:00 --> 00:26:02.68 whatever that means? 411 00:26:02.68 --> 00:26:06 Or do I -- is one of those columns, like, 412 00:26:06 --> 00:26:11 contributing nothing new -- So that actually only two of 413 00:26:11 --> 00:26:15 the columns would have given the same column space? 414 00:26:15 --> 00:26:18.55 Yeah -- that's a good way to ask the question. 415 00:26:18.55 --> 00:26:20 Finally I think of it. 416 00:26:20 --> 00:26:25 Can I throw away any columns -- and have the same column space? 417 00:26:25 --> 00:26:26 STUDENT: Yes. 418 00:26:26 --> 00:26:27 STRANG: Yes. 419 00:26:27 --> 00:26:31 And which one do you suggest I throw away? 420 00:26:31 --> 00:26:33 STUDENT: Column three -- three. 421 00:26:33 --> 00:26:36.71 STRANG: Well, three is the natural, 422 00:26:36.71 --> 00:26:38 like, guy to target. 423 00:26:38 --> 00:26:40 So if I -- and why? 424 00:26:40 --> 00:26:44 Because -- what's so bad about three here? 425 00:26:44 --> 00:26:45 Column three? 426 00:26:45 --> 00:26:47 It's the sum of these, right? 427 00:26:47 --> 00:26:51 So it's not -- if I'm taking -- if I have combinations of these 428 00:26:51 --> 00:26:55 two and I put in this one, actually, I don't get anything 429 00:26:55 --> 00:26:55.57 more. 430 00:26:55.57 --> 00:26:59 So later on I will call these pivot columns. 431 00:26:59 --> 00:27:05 And the third guy will not be a pivot column in this -- with 432 00:27:05 --> 00:27:06 those numbers. 433 00:27:06 --> 00:27:12 Now actually -- honesty makes me ask you this question. 434 00:27:12 --> 00:27:16 Could I have thrown away column one? 435 00:27:16 --> 00:27:17 Yes, I could. 436 00:27:17 --> 00:27:19 I could. 437 00:27:19 --> 00:27:22 So when I say pivot columns, my convention is, 438 00:27:22 --> 00:27:26 okay, I'll keep the first ones as long as they're not 439 00:27:26 --> 00:27:26 dependent. 440 00:27:26 --> 00:27:29 So I keep this guy, he's fine, he's a line. 441 00:27:29 --> 00:27:30 I keep the second guy. 442 00:27:30 --> 00:27:32 It's in a second direction. 443 00:27:32 --> 00:27:35 But the third one, which is in the same plane as 444 00:27:35 --> 00:27:38.51 the first two gives me nothing new. 445 00:27:38.51 --> 00:27:44 It's dependent in the language that we will use and I don't 446 00:27:44 --> 00:27:45 need it. 447 00:27:45 --> 00:27:45 Okay. 448 00:27:45 --> 00:27:51 So I would describe the column space of this matrix as a two 449 00:27:51 --> 00:27:54 dimensional subspace of R^4. 450 00:27:54 --> 00:27:58 A two dimensional subspace of R^4. 451 00:27:58 --> 00:27:58 Okay. 452 00:27:58 --> 00:28:05 So you're seeing how these vector spaces work and you -- 453 00:28:05 --> 00:28:09 you're seeing that we -- some idea of dependence or 454 00:28:09 --> 00:28:12 independence is in our future. 455 00:28:12 --> 00:28:13 Okay. 456 00:28:13 --> 00:28:17 Now I want to speak about another vector space, 457 00:28:17 --> 00:28:18 the null space. 458 00:28:18 --> 00:28:24 So again I'm getting a little ahead because it's in section 459 00:28:24 --> 00:28:26 three point two, but that's okay. 460 00:28:26 --> 00:28:28 All right. 461 00:28:28 --> 00:28:32 Now I'm ready for the null space. 462 00:28:32 --> 00:28:35 Let me keep the same matrix. 463 00:28:35 --> 00:28:41 And this is going to be a different -- totally different 464 00:28:41 --> 00:28:42 subspace. 465 00:28:42 --> 00:28:44 Totally different. 466 00:28:44 --> 00:28:45 Okay. 467 00:28:45 --> 00:28:49 Now -- so let me make space for it. 468 00:28:49 --> 00:28:54 Now -- here comes a completely 469 00:28:54 --> 00:28:59 different subspace, the null space of A. 470 00:28:59 --> 00:29:00 What's in it? 471 00:29:00 --> 00:29:05 It contains not right-hand sides b. 472 00:29:05 --> 00:29:07 It contains x-s. 473 00:29:07 --> 00:29:15 It contains all x-s that solve -- this word null is going to -- 474 00:29:15 --> 00:29:21 I mean, that's the key word here, meaning zero. 475 00:29:21 --> 00:29:28 So this contains -- this is all solutions x, 476 00:29:28 --> 00:29:33 and of course x is our vectors, x1, x2 and x3, 477 00:29:33 --> 00:29:36 to the equation A x=0. 478 00:29:36 --> 00:29:41 Well, four equations, because we've got -- so, 479 00:29:41 --> 00:29:44 do you see what I'm doing? 480 00:29:44 --> 00:29:49 I'm now saying, okay, columns were great, 481 00:29:49 --> 00:29:53 the column space we understood. 482 00:29:53 --> 00:29:56 Now I'm interested in x-s. 483 00:29:56 --> 00:30:02 I'm not -- the only b I'm interested in now is the b of 484 00:30:02 --> 00:30:03 all zeroes. 485 00:30:03 --> 00:30:06 The right-hand side is now zeroes. 486 00:30:06 --> 00:30:09 And I'm interested in solutions. 487 00:30:09 --> 00:30:10 x-s. 488 00:30:10 --> 00:30:15 So t- where is this null space for this example? 489 00:30:15 --> 00:30:20 These x-s are -- have three components. 490 00:30:20 --> 00:30:29.58 So the null space is a subspace -- we still have to show it is a 491 00:30:29.58 --> 00:30:32 subspace -- of R^3. 492 00:30:32 --> 00:30:39 So this is -- and we will show -- these vectors x, 493 00:30:39 --> 00:30:45 this is in R^3, where the column space was in 494 00:30:45 --> 00:30:49 R^4 in our example. 495 00:30:49 --> 00:30:54 For an m by n matrix, this is m and this is n, 496 00:30:54 --> 00:31:00 because the number of columns, n, tells me how many unknowns, 497 00:31:00 --> 00:31:06 how many x-s multiply those columns, so it tells me the big 498 00:31:06 --> 00:31:10 space, in this case R three that I'm in. 499 00:31:10 --> 00:31:15 Now tell me -- why don't we figure out what 500 00:31:15 --> 00:31:20 the null space is for this example, just by looking at it. 501 00:31:20 --> 00:31:23 I mean, that's the beauty of small examples, 502 00:31:23 --> 00:31:28.6 that my official way to find null spaces and column spaces 503 00:31:28.6 --> 00:31:32 and get all the facts straight would be elimination, 504 00:31:32 --> 00:31:34 and we'll do that. 505 00:31:34 --> 00:31:39 But with a small example, we can see that -- 506 00:31:39 --> 00:31:44 see what's going on without going through the mechanics of 507 00:31:44 --> 00:31:46 elimination. 508 00:31:46 --> 00:31:51 So this null space -- so I'm talking about -- again, 509 00:31:51 --> 00:31:55 the null space, and let me copy again the 510 00:31:55 --> 00:31:56 matrix. 511 00:31:56 --> 00:32:01 One two three four, one one one one and two three 512 00:32:01 --> 00:32:03 four five. 513 00:32:03 --> 00:32:05 What's in the null space? 514 00:32:05 --> 00:32:09 So I'm taking A times x, so let me right it again, 515 00:32:09 --> 00:32:13 and I want you to solve those four equations. 516 00:32:13 --> 00:32:18 In fact, I want you to find all solutions to those four 517 00:32:18 --> 00:32:19 equations. 518 00:32:19 --> 00:32:23 Well, actually, just first of all find one. 519 00:32:23 --> 00:32:27 Why should I ask you for all of them? 520 00:32:27 --> 00:32:31 Tell me one -- well, tell me one solution that y- 521 00:32:31 --> 00:32:35 you don't even have to look at the matrix to know one solution 522 00:32:35 --> 00:32:37 to this set of equations. 523 00:32:37 --> 00:32:38 It is zero vector. 524 00:32:38 --> 00:32:43 Whatever that matrix is, its null space contains zero -- 525 00:32:43 --> 00:32:49 because A times the zero vector sure gives the zero right-hand 526 00:32:49 --> 00:32:49 side. 527 00:32:49 --> 00:32:53 So the null space certainly contains zero. 528 00:32:53 --> 00:32:57.78 A- so it's got a chance to be a vector space now, 529 00:32:57.78 --> 00:33:00 and it will turn out it is. 530 00:33:00 --> 00:33:00 Okay. 531 00:33:00 --> 00:33:03 Tell me another solution. 532 00:33:03 --> 00:33:08 So this particular null space -- and of course I'm going to 533 00:33:08 --> 00:33:13.63 call it N(A) for null space -- this contains-- well we've 534 00:33:13.63 --> 00:33:19 already located the zero vector, and now you're going to tell me 535 00:33:19 --> 00:33:23 another vector that's in the null space, another solution, 536 00:33:23 --> 00:33:28 another x, another -- you see what I'm asking you for 537 00:33:28 --> 00:33:31 is a combination of those columns. 538 00:33:31 --> 00:33:35 That's what I'm always looking at combinations of columns, 539 00:33:35 --> 00:33:40 but now I'm looking at the weights, the coefficients in the 540 00:33:40 --> 00:33:41 combination. 541 00:33:41 --> 00:33:45 So tell me a good set of numbers to put in there. 542 00:33:45 --> 00:33:48 One one -- STUDENTS: Minus one. 543 00:33:48 --> 00:33:50 STRANG: One one minus one. 544 00:33:50 --> 00:33:51.22 Thanks. 545 00:33:51.22 --> 00:33:52 One one minus one. 546 00:33:52 --> 00:33:55 So there's a vector that's in it. 547 00:33:55 --> 00:33:55.99 Okay. 548 00:33:55.99 --> 00:33:59 But have I got a subspace at this point? 549 00:33:59 --> 00:34:01 Certainly not, right? 550 00:34:01 --> 00:34:05 I've got just a couple of vectors. 551 00:34:05 --> 00:34:07 No way they make a subspace. 552 00:34:07 --> 00:34:11 Tell me -- actually, why don't I jump the whole way 553 00:34:11 --> 00:34:12 now? 554 00:34:12 --> 00:34:15 Tell me -- well, tell me one more solution, 555 00:34:15 --> 00:34:18 one more X that would work. 556 00:34:18 --> 00:34:19 Student: 2 2 -2. 557 00:34:19 --> 00:34:20 STRANG: 2 2 -2? 558 00:34:20 --> 00:34:26 Oh, well, tell me all of them, that would have been easier. 559 00:34:26 --> 00:34:29 Tell me the whole lot, now. 560 00:34:29 --> 00:34:32 What is the null space for this matrix? 561 00:34:32 --> 00:34:37 It's all vectors of the form -- what could this be? 562 00:34:37 --> 00:34:42 It could be one one minus one, it could be it could be any 563 00:34:42 --> 00:34:48 number C, any number -- the same number again and -- 564 00:34:48 --> 00:34:50 STUDENTS: Minus. 565 00:34:50 --> 00:34:51 STRANG: Minus C. 566 00:34:51 --> 00:34:57 In other words -- actually, any multiple of this guy. 567 00:34:57 --> 00:35:02 Oh, that's the perfect description, because now the 568 00:35:02 --> 00:35:08 zero vector's automatically included because C could be 569 00:35:08 --> 00:35:09 zero. 570 00:35:09 --> 00:35:14 The vector I had is included, because C could be one. 571 00:35:14 --> 00:35:16 But now any vector. 572 00:35:16 --> 00:35:18.69 And that's actually it. 573 00:35:18.69 --> 00:35:21.01 And do I have a subspace? 574 00:35:21.01 --> 00:35:23 And what does it look like? 575 00:35:23 --> 00:35:27 It's in -- how would you describe this, 576 00:35:27 --> 00:35:30 the null space, this -- 577 00:35:30 --> 00:35:34 all these vectors of this form C C minus C, like, 578 00:35:34 --> 00:35:36 seven seven minus seven. 579 00:35:36 --> 00:35:39 Minus eleven minus eleven plus eleven. 580 00:35:39 --> 00:35:41 What have I got here? 581 00:35:41 --> 00:35:46 If -- describe that whole null space of -- what -- if I drew 582 00:35:46 --> 00:35:47 it, what do I draw? 583 00:35:47 --> 00:35:48.95 A line, right? 584 00:35:48.95 --> 00:35:52 The null space is a line. 585 00:35:52 --> 00:35:57 It's the line through -- in R^3 and the vector one one negative 586 00:35:57 --> 00:36:02 one maybe goes down here, I don't know where it goes, 587 00:36:02 --> 00:36:04 say, down here. 588 00:36:04 --> 00:36:09 There's the vector one one negative one that you gave me. 589 00:36:09 --> 00:36:14 And where is the vector C C negative C? 590 00:36:14 --> 00:36:16 It's on this line. 591 00:36:16 --> 00:36:20 Of course, there's zero zero zero that we had. 592 00:36:20 --> 00:36:25.05 And what we've got is that whole -- oops -- that whole 593 00:36:25.05 --> 00:36:28 line, going both ways, through the origin. 594 00:36:28 --> 00:36:31 The null space is a line in R^3. 595 00:36:31 --> 00:36:32 Okay. 596 00:36:32 --> 00:36:35 For that example, we could find all the 597 00:36:35 --> 00:36:41 combinations of the columns that gave zero at sight. 598 00:36:41 --> 00:36:48 Now, can I just take one more time, to go back to the 599 00:36:48 --> 00:36:53 definition of subspace, vector space, 600 00:36:53 --> 00:37:01 and ask you -- how do I know that the null space is a vector 601 00:37:01 --> 00:37:02 space? 602 00:37:02 --> 00:37:08 How I entitled to use this word space? 603 00:37:08 --> 00:37:15 I'll never use that word space without meaning that the two 604 00:37:15 --> 00:37:19 requirements are satisfied. 605 00:37:19 --> 00:37:23 Can we just check that they are? 606 00:37:23 --> 00:37:29 So I'm going to check that -- can I just continue here? 607 00:37:29 --> 00:37:36 Check that -- that the solutions to A x=0 always give a 608 00:37:36 --> 00:37:37 subspace. 609 00:37:37 --> 00:37:45 And, of course, the key word is that= "Space." 610 00:37:45 --> 00:37:48 So what do I have to check? 611 00:37:48 --> 00:37:53 I have to show that if I have one solution, 612 00:37:53 --> 00:37:58 call it x, and another solution, call it x*, 613 00:37:58 --> 00:38:02 that their sum is also a solution, right? 614 00:38:02 --> 00:38:05.44 That's a requirement. 615 00:38:05.44 --> 00:38:11 To use that word space, I have to say -- 616 00:38:11 --> 00:38:18 I have to convince myself that if A x is zero and also -- and A 617 00:38:18 --> 00:38:26 x* is zero, or maybe I should have said if A v is zero and A w 618 00:38:26 --> 00:38:30 is zero, then what about v plus w? 619 00:38:30 --> 00:38:34 Shall I -- let me use those letters. 620 00:38:34 --> 00:38:41 If A v is zero and A w is zero, then what -- 621 00:38:41 --> 00:38:45 if that and that, then what's my point here? 622 00:38:45 --> 00:38:48 That A times (v+w) must be zero. 623 00:38:48 --> 00:38:53 That says that if v is in the null space and w's in the null 624 00:38:53 --> 00:38:57.91 space, then their sum v+w is in the null space. 625 00:38:57.91 --> 00:39:02 And of course, now that I've written it down, 626 00:39:02 --> 00:39:07 it's totally absurd, ridiculously simple -- 627 00:39:07 --> 00:39:12 because matrix multiplication allows me to separate that out 628 00:39:12 --> 00:39:14 into A v plus A w. 629 00:39:14 --> 00:39:17 I shouldn't say absurdly simple. 630 00:39:17 --> 00:39:19 That was a dumb thing to say. 631 00:39:19 --> 00:39:24 Could -- we've used, here, a basic law of matrix 632 00:39:24 --> 00:39:25 multiplication. 633 00:39:25 --> 00:39:32 Actually, we've used it without proving it, but that's okay. 634 00:39:32 --> 00:39:36 We only live so long, we just skip that proof. 635 00:39:36 --> 00:39:40.88 I think it's called the distributive law that I can 636 00:39:40.88 --> 00:39:44 split these -- split this into two pieces. 637 00:39:44 --> 00:39:49 But now you see the point, that A v is zero and A w is 638 00:39:49 --> 00:39:54 zero so I have zero plus zero and I do get zero. 639 00:39:54 --> 00:39:55 It checks. 640 00:39:55 --> 00:39:59 And, similarly, I have to show that if A v is 641 00:39:59 --> 00:40:04.54 zero, then A times any multiple, say 12v is also zero. 642 00:40:04.54 --> 00:40:06.67 And how do I know that? 643 00:40:06.67 --> 00:40:11 Because I'm allowed to s- bring that twelve outside. 644 00:40:11 --> 00:40:16 A number, a scaler can move outside, so I have twelve A vs, 645 00:40:16 --> 00:40:20 twelve zeroes -- I have zero. 646 00:40:20 --> 00:40:21 Okay. 647 00:40:21 --> 00:40:27 Just to -- it's really critical to understand the -- oh yeah. 648 00:40:27 --> 00:40:34 Here -- I was going to say, understand what's the point of 649 00:40:34 --> 00:40:35 a vector space? 650 00:40:35 --> 00:40:42 Let me make that point by changing the right-hand side. 651 00:40:42 --> 00:40:43 Oops. 652 00:40:43 --> 00:40:43 Okay. 653 00:40:43 --> 00:40:48 Let me change the right-hand side to one two three four. 654 00:40:48 --> 00:40:49 Oh, okay. 655 00:40:49 --> 00:40:54 Why don't we do all of linear algebra in one lecture, 656 00:40:54 --> 00:40:56 then we -- okay. 657 00:40:56 --> 00:41:00 I would like to know the solutions to this equation. 658 00:41:00 --> 00:41:03 For those four equations. 659 00:41:03 --> 00:41:06 So I have four equations. 660 00:41:06 --> 00:41:10 I have only three unknowns, so if I don't have a pretty 661 00:41:10 --> 00:41:14 special right-hand side there won't be any solution at all. 662 00:41:14 --> 00:41:17 But that is a very special right-hand side. 663 00:41:17 --> 00:41:20 And we know that there is a solution, one zero zero. 664 00:41:20 --> 00:41:22 Were there any more solutions? 665 00:41:22 --> 00:41:25 And did they form a vector space? 666 00:41:25 --> 00:41:26 Okay. 667 00:41:26 --> 00:41:28 So I'm asking two questions there. 668 00:41:28 --> 00:41:32 One is, do -- so my right-hand side now is not zero anymore. 669 00:41:32 --> 00:41:36 I'm not looking at the null space because I changed from 670 00:41:36 --> 00:41:37 zeroes. 671 00:41:37 --> 00:41:40 So my first question is, do the solutions, 672 00:41:40 --> 00:41:44 if there are any and there are, do they form a subspace? 673 00:41:44 --> 00:41:48 Let's answer that question first. 674 00:41:48 --> 00:41:49 Yes or no. 675 00:41:49 --> 00:41:56.84 Do I get a subspace if I look at the solutions to -- let me go 676 00:41:56.84 --> 00:41:58 back to x1 x2 x3. 677 00:41:58 --> 00:42:05 I'm looking at all the x-s, at all those vectors in R^3 678 00:42:05 --> 00:42:08.11 that solve A x -b. 679 00:42:08.11 --> 00:42:18 The only thing I've changed is b isn't zero anymore. 680 00:42:18 --> 00:42:28 Do the x-s, the solutions, form a vector space? 681 00:42:28 --> 00:42:32 The solutions to this do not form a subspace. 682 00:42:32 --> 00:42:35 The solutions don't, because -- how shall I see 683 00:42:35 --> 00:42:36 that? 684 00:42:36 --> 00:42:40 The zero vector is not a solution, so I never even got 685 00:42:40 --> 00:42:40 started. 686 00:42:40 --> 00:42:44 The zero vector doesn't solve this system. 687 00:42:44 --> 00:42:48 I can't -- solutions can't be a vector space. 688 00:42:48 --> 00:42:50 Now what are they like? 689 00:42:50 --> 00:42:54.82 Well, we'll see this, but let's do it for this 690 00:42:54.82 --> 00:42:55 example. 691 00:42:55 --> 00:42:58 So one zero zero was a solution. 692 00:42:58 --> 00:43:00 You saw that right away. 693 00:43:00 --> 00:43:03 Are there any other solutions? 694 00:43:03 --> 00:43:07 Can you tell me a second solution to this system of 695 00:43:07 --> 00:43:09 equations? 696 00:43:09 --> 00:43:13 STUDENTS: 0 -1 1 STRANG: 0 -1 1. 697 00:43:13 --> 00:43:16 Boy, that's -- 0 -1 1. 698 00:43:16 --> 00:43:17 Yes. 699 00:43:17 --> 00:43:24 Because that says I take minus this column plus this one and 700 00:43:24 --> 00:43:26 sure enough. 701 00:43:26 --> 00:43:28 That's right. 702 00:43:28 --> 00:43:32 So there are -- there's a bunch of solutions 703 00:43:32 --> 00:43:33 here. 704 00:43:33 --> 00:43:34 But they're not a subspace. 705 00:43:34 --> 00:43:36.35 I'll tell you what it's like. 706 00:43:36.35 --> 00:43:39 It's like a plane that doesn't go through the origin, 707 00:43:39 --> 00:43:42 or a line that doesn't go through the origin. 708 00:43:42 --> 00:43:45 Maybe in this case it's a line that doesn't go through the 709 00:43:45 --> 00:43:48 origin, if I graft the solutions to A x equal B. 710 00:43:48 --> 00:43:52.3 So you -- I think you've got the idea. 711 00:43:52.3 --> 00:43:55 Subspaces have to go through the origin. 712 00:43:55 --> 00:44:00 If I'm looking at x-s, then they'd better solve Ax=0. 713 00:44:00 --> 00:44:05 In a way I've got -- my two subspaces that I -- 714 00:44:05 --> 00:44:10 talking about today are kind of the two ways I can tell you what 715 00:44:10 --> 00:44:12.46 a -- about subspace. 716 00:44:12.46 --> 00:44:15 If I want to tell you about the column space, 717 00:44:15 --> 00:44:20 I tell you a few columns and I say take their combinations. 718 00:44:20 --> 00:44:22.91 Like I build up this subspace. 719 00:44:22.91 --> 00:44:26.71 I put in a few vectors, their combinations make a 720 00:44:26.71 --> 00:44:28 subspace. 721 00:44:28 --> 00:44:33 Now, when I went to -- let me come back to the one that is a 722 00:44:33 --> 00:44:34 subspace here. 723 00:44:34 --> 00:44:39 Here, when I talked about the null space, I didn't tell you 724 00:44:39 --> 00:44:40 what's in it. 725 00:44:40 --> 00:44:43 We had to figure out what was in it. 726 00:44:43 --> 00:44:48 What I told you was the equations that I'm -- that has 727 00:44:48 --> 00:44:50 to be satisfied. 728 00:44:50 --> 00:44:56 You see those -- like, those are the two natural ways 729 00:44:56 --> 00:44:59.99 to tell you what's in a subspace. 730 00:44:59.99 --> 00:45:05.88 I can either give you a few vectors and say fill it out, 731 00:45:05.88 --> 00:45:11 take combinations -- or I can give you a system of 732 00:45:11 --> 00:45:15 equations, the requirements that the x-s have to satisfy. 733 00:45:15 --> 00:45:20 And both of those ways produce subspaces and they're the 734 00:45:20 --> 00:45:23.54 important ways to construct subspaces. 735 00:45:23.54 --> 00:45:26 Okay, so today's lecture actually got, 736 00:45:26 --> 00:45:32 the essentials of three point two, the idea of the null space. 737 00:45:32 --> 00:45:37 Now we have to tackle, Wednesday, the job of how do we 738 00:45:37 --> 00:45:41 actually get hold of that subspace in an example that's 739 00:45:41 --> 00:45:44 bigger and we can't see it just by eye. 740 00:45:44 --> 00:45:45 Okay. 741 00:45:45 --> 00:45:46 See you Wednesday. 742 00:45:46 --> 00:45:49 Thanks.