1 00:00:00 --> 00:00:05 OK, this is linear algebra lecture nine. 2 00:00:05 --> 00:00:11 And this is a key lecture, this is where we get these 3 00:00:11 --> 00:00:18 ideas of linear independence, when a bunch of vectors are 4 00:00:18 --> 00:00:24 independent -- or dependent, that's the opposite. 5 00:00:24 --> 00:00:27 The space they span. 6 00:00:27 --> 00:00:33.65 A basis for a subspace or a basis for a vector space, 7 00:00:33.65 --> 00:00:37.85 that's a central idea. 8 00:00:37.85 --> 00:00:42 And then the dimension of that subspace. 9 00:00:42 --> 00:00:47 So this is the day that those words get assigned clear 10 00:00:47 --> 00:00:48 meanings. 11 00:00:48 --> 00:00:54.91 And emphasize that we talk about a bunch of vectors being 12 00:00:54.91 --> 00:00:56 independent. 13 00:00:56 --> 00:01:02 Wouldn't talk about a matrix being independent. 14 00:01:02 --> 00:01:05 A bunch of vectors being independent. 15 00:01:05 --> 00:01:08 A bunch of vectors spanning a space. 16 00:01:08 --> 00:01:11 A bunch of vectors being a basis. 17 00:01:11 --> 00:01:14 And the dimension is some number. 18 00:01:14 --> 00:01:17 OK, so what are the definitions? 19 00:01:17 --> 00:01:21 Can I begin with a fact, a highly important fact, 20 00:01:21 --> 00:01:27 that, I didn't call directly attention to earlier. 21 00:01:27 --> 00:01:32 Suppose I have a matrix and I look at Ax equals zero. 22 00:01:32 --> 00:01:38 Suppose the matrix has a lot of columns, so that n is bigger 23 00:01:38 --> 00:01:39 than m. 24 00:01:39 --> 00:01:43 So I'm looking at n equations -- I mean, sorry, 25 00:01:43 --> 00:01:47 m equations, a small number of equations m, 26 00:01:47 --> 00:01:49 and more unknowns. 27 00:01:49 --> 00:01:53 I have more unknowns than equations. 28 00:01:53 --> 00:01:56 Let me write that down. 29 00:01:56 --> 00:02:00 More unknowns than equations. 30 00:02:00 --> 00:02:03 More unknown x-s than equations. 31 00:02:03 --> 00:02:10 Then the conclusion is that there's something in the null 32 00:02:10 --> 00:02:16 space of A, other than just the zero vector. 33 00:02:16 --> 00:02:23 The conclusion is there are some non-zero x-s such that Ax 34 00:02:23 --> 00:02:24 is zero. 35 00:02:24 --> 00:02:28.26 There are some special solutions. 36 00:02:28.26 --> 00:02:29.25 And why? 37 00:02:29.25 --> 00:02:32.47 We know why. 38 00:02:32.47 --> 00:02:36 I mean, it sort of like seems like a reasonable thing, 39 00:02:36 --> 00:02:41 more unknowns than equations, then it seems reasonable that 40 00:02:41 --> 00:02:43 we can solve them. 41 00:02:43 --> 00:02:46 But we have a, a clear algorithm which starts 42 00:02:46 --> 00:02:51 with a system and does elimination, gets the thing into 43 00:02:51 --> 00:02:55.62 an echelon form with some pivots and pivot columns, 44 00:02:55.62 --> 00:03:01 and possibly some free columns that don't have pivots. 45 00:03:01 --> 00:03:09 And the point is here there will be some free columns. 46 00:03:09 --> 00:03:17 The reason, so the reason is there must -- there will be free 47 00:03:17 --> 00:03:21 variables, at least one. 48 00:03:21 --> 00:03:24 That's the reason. 49 00:03:24 --> 00:03:30 That we now have this -- a complete, algorithm, 50 00:03:30 --> 00:03:36 a complete systematic way to say, OK, we take the system Ax 51 00:03:36 --> 00:03:41 equals zero, we row reduce, we identify the free variables, 52 00:03:41 --> 00:03:46 and, since there are n variables and at most m pivots, 53 00:03:46 --> 00:03:50 there will be some free variables, at least one, 54 00:03:50 --> 00:03:54 at least n-m in fact, left over. 55 00:03:54 --> 00:03:57 And those variables I can assign non-zero values to. 56 00:03:57 --> 00:04:00 I don't have to set those to zero. 57 00:04:00 --> 00:04:03 I can take them to be one or whatever I like, 58 00:04:03 --> 00:04:06 and then I can solve for the pivot variables. 59 00:04:06 --> 00:04:09 So then it gives me a solution to Ax equals zero. 60 00:04:09 --> 00:04:12 And it's a solution that isn't all zeros. 61 00:04:12 --> 00:04:19 So, that's an important point that we'll use now in this 62 00:04:19 --> 00:04:20 lecture. 63 00:04:20 --> 00:04:27 So now I want to say what does it mean for a bunch of vectors 64 00:04:27 --> 00:04:29 to be independent. 65 00:04:29 --> 00:04:29 OK. 66 00:04:29 --> 00:04:35 So this is like the background that we know. 67 00:04:35 --> 00:04:42 Now I want to speak about independence. 68 00:04:42 --> 00:04:42 OK. 69 00:04:42 --> 00:04:44 Let's see. 70 00:04:44 --> 00:04:53 I can give you the abstract definition, and I will, 71 00:04:53 --> 00:05:02 but I would also like to give you the direct meaning. 72 00:05:02 --> 00:05:08.86 So the question is, when vectors x1, 73 00:05:08.86 --> 00:05:15 x2 up to -- Suppose I have n vectors are 74 00:05:15 --> 00:05:16 independent if. 75 00:05:16 --> 00:05:22.5 Now I have to give you -- or linearly independent -- I'll 76 00:05:22.5 --> 00:05:27 often just say and write independent for short. 77 00:05:27 --> 00:05:27 OK. 78 00:05:27 --> 00:05:30 I'll give you the full definition. 79 00:05:30 --> 00:05:35 These are just vectors in some vector space. 80 00:05:35 --> 00:05:39 I can take combinations of them. 81 00:05:39 --> 00:05:45 The question is, do any combinations give zero? 82 00:05:45 --> 00:05:52 If some combination of those vectors gives the zero vector, 83 00:05:52 --> 00:05:58 other than the combination of all zeros, then they're 84 00:05:58 --> 00:05:59 dependent. 85 00:05:59 --> 00:06:06 They're independent if no combination gives the zero 86 00:06:06 --> 00:06:14 vector -- and then I have, I'll have to put in an except 87 00:06:14 --> 00:06:17 the zero combination. 88 00:06:17 --> 00:06:20 So what do I mean by that? 89 00:06:20 --> 00:06:25 No combination gives the zero vector. 90 00:06:25 --> 00:06:33 Any combination c1 x1+c2 x2 plus, plus cn xn is not zero 91 00:06:33 --> 00:06:37 except for the zero combination. 92 00:06:37 --> 00:06:44 This is when all the c-s, all the c-s are zero. 93 00:06:44 --> 00:06:46 Then of course. 94 00:06:46 --> 00:06:49 That combination -- I know I'll get zero. 95 00:06:49 --> 00:06:53 But the question is, does any other combination give 96 00:06:53 --> 00:06:54 zero? 97 00:06:54 --> 00:06:57.57 If not, then the vectors are independent. 98 00:06:57.57 --> 00:07:02 If some other combination does give zero, the vectors are 99 00:07:02 --> 00:07:04 dependent. 100 00:07:04 --> 00:07:04.33 OK. 101 00:07:04.33 --> 00:07:06 Let's just take examples. 102 00:07:06 --> 00:07:11 Suppose I'm in, say, in two dimensional space. 103 00:07:11 --> 00:07:12 OK. 104 00:07:12 --> 00:07:18 I give you -- I'd like to first take an example -- let me take 105 00:07:18 --> 00:07:25 an example where I have a vector and twice that vector. 106 00:07:25 --> 00:07:27 So that's two vectors, V and 2V. 107 00:07:27 --> 00:07:30 Are those dependent or independent? 108 00:07:30 --> 00:07:34 Those are dependent for sure, right, because there's one 109 00:07:34 --> 00:07:36 vector is twice the other. 110 00:07:36 --> 00:07:41.15 One vector is twice as long as the other, so if the word 111 00:07:41.15 --> 00:07:45 dependent means anything, these should be dependent. 112 00:07:45 --> 00:07:47 And they are. 113 00:07:47 --> 00:07:51 And in fact, I would take two of the first 114 00:07:51 --> 00:07:54 -- so here's, here is a vector V and the 115 00:07:54 --> 00:07:59 other guy is a vector 2V, that's my -- so there's a 116 00:07:59 --> 00:08:03 vector V1 and my next vector V2 is 2V1. 117 00:08:03 --> 00:08:08.42 Of course those are dependent, because two of these first 118 00:08:08.42 --> 00:08:13 vectors minus the second vector is zero. 119 00:08:13 --> 00:08:18 That's a combination of these two vectors that gives the zero 120 00:08:18 --> 00:08:18 vector. 121 00:08:18 --> 00:08:20.12 OK, that was clear. 122 00:08:20.12 --> 00:08:24 Suppose, suppose I have a vector -- here's another 123 00:08:24 --> 00:08:24 example. 124 00:08:24 --> 00:08:26 It's easy example. 125 00:08:26 --> 00:08:30 Suppose I have a vector and the other guy is the zero vector. 126 00:08:30 --> 00:08:36 Suppose I have a vector V1 and V2 is the zero vector. 127 00:08:36 --> 00:08:40 Then are those vectors dependent or independent? 128 00:08:40 --> 00:08:42 They're dependent again. 129 00:08:42 --> 00:08:46 You could say, well, this guy is zero times 130 00:08:46 --> 00:08:47 that one. 131 00:08:47 --> 00:08:51 This one is some combination of those. 132 00:08:51 --> 00:08:54 But let me write it the other way. 133 00:08:54 --> 00:08:58 Let me say -- what combination, 134 00:08:58 --> 00:09:03 how many V1s and how many V2s shall I take to get the zero 135 00:09:03 --> 00:09:03 vector? 136 00:09:03 --> 00:09:08 If, if V1 is like the vector two one and V2 is the zero 137 00:09:08 --> 00:09:12 vector, zero zero, then I would like to show that 138 00:09:12 --> 00:09:16 some combination of those gives the zero vector. 139 00:09:16 --> 00:09:18 What shall I take? 140 00:09:18 --> 00:09:21 How many V1s shall I take? 141 00:09:21 --> 00:09:23 Zero of them. 142 00:09:23 --> 00:09:26 Yeah, no, take no V1s. 143 00:09:26 --> 00:09:28 But how many V2s? 144 00:09:28 --> 00:09:29 Six. 145 00:09:29 --> 00:09:29 OK. 146 00:09:29 --> 00:09:30 Or five. 147 00:09:30 --> 00:09:36 Then -- in other words, the point is if the zero 148 00:09:36 --> 00:09:42 vector's in there, if the zero -- 149 00:09:42 --> 00:09:46 if one of these vectors is the zero vector, independence is 150 00:09:46 --> 00:09:47 dead, right? 151 00:09:47 --> 00:09:51 If one of those vectors is the zero vector then I could always 152 00:09:51 --> 00:09:54 take -- include that one and none of the others, 153 00:09:54 --> 00:09:57 and I would get the zero answer, and I would show 154 00:09:57 --> 00:09:58 dependence. 155 00:09:58 --> 00:09:58.32 OK. 156 00:09:58.32 --> 00:10:01 Now, let me, let me finally draw an example 157 00:10:01 --> 00:10:04 where they will be independent. 158 00:10:04 --> 00:10:07 Suppose that's V1 and that's V2. 159 00:10:07 --> 00:10:10 Those are surely independent, right? 160 00:10:10 --> 00:10:15 Any combination of V1 and V2, will not be zero except, 161 00:10:15 --> 00:10:17.85 the zero combination. 162 00:10:17.85 --> 00:10:20 So those would be independent. 163 00:10:20 --> 00:10:25 But now let me, let me stick in a third vector, 164 00:10:25 --> 00:10:25 V3. 165 00:10:25 --> 00:10:31 Independent or dependent now, those three vectors? 166 00:10:31 --> 00:10:33 So now n is three here. 167 00:10:33 --> 00:10:39 I'm in two dimensional space, whatever, I'm in the plane. 168 00:10:39 --> 00:10:43 I have three vectors that I didn't draw so carefully. 169 00:10:43 --> 00:10:47 I didn't even tell you what exactly they were. 170 00:10:47 --> 00:10:53 But what's this answer on dependent or independent? 171 00:10:53 --> 00:10:54 Dependent. 172 00:10:54 --> 00:10:57 How do I know those are dependent? 173 00:10:57 --> 00:11:02 How do I know that some combination of V1, 174 00:11:02 --> 00:11:05 V2, and V3 gives me the zero vector? 175 00:11:05 --> 00:11:08 I know because of that. 176 00:11:08 --> 00:11:14 That's the key fact that tells me that three vectors in the 177 00:11:14 --> 00:11:18 plane have to be dependent. 178 00:11:18 --> 00:11:19 Why's that? 179 00:11:19 --> 00:11:24 What's the connection between the dependence of these three 180 00:11:24 --> 00:11:26 vectors and that fact? 181 00:11:26 --> 00:11:26 OK. 182 00:11:26 --> 00:11:29 So here's the connection. 183 00:11:29 --> 00:11:33 I take the matrix A that has V1 in its first column, 184 00:11:33 --> 00:11:39 V2 in its second column, V3 in its third column. 185 00:11:39 --> 00:11:42 So it's got three columns. 186 00:11:42 --> 00:11:48 And V1 -- I don't know, that looks like about two one 187 00:11:48 --> 00:11:48 to me. 188 00:11:48 --> 00:11:52 V2 looks like it might be one two. 189 00:11:52 --> 00:11:59 V3 looks like it might be maybe two, maybe two and a half, 190 00:11:59 --> 00:12:00 minus one. 191 00:12:00 --> 00:12:00 OK. 192 00:12:00 --> 00:12:07 Those are my three vectors, and I put them in the columns 193 00:12:07 --> 00:12:09 of A. 194 00:12:09 --> 00:12:13 Now that matrix A is two by three. 195 00:12:13 --> 00:12:20.26 It fits this pattern, that where we know we've got 196 00:12:20.26 --> 00:12:25 extra variables, we know we have some free 197 00:12:25 --> 00:12:33 variables, we know that there's some combination -- 198 00:12:33 --> 00:12:36 and let me instead of x-s, let me call them c1, 199 00:12:36 --> 00:12:39 c2, and c3 -- that gives the zero vector. 200 00:12:39 --> 00:12:42 Sorry that my little bit of art got in the way. 201 00:12:42 --> 00:12:43 Do you see the point? 202 00:12:43 --> 00:12:47 When I have a matrix, I'm interested in whether its 203 00:12:47 --> 00:12:49 columns are dependent or independent. 204 00:12:49 --> 00:12:53 The columns are dependent if there is something in the null 205 00:12:53 --> 00:12:55.23 space. 206 00:12:55.23 --> 00:12:59.31 The columns are dependent because this, 207 00:12:59.31 --> 00:13:05 this thing in the null space says that c1 of that plus c2 of 208 00:13:05 --> 00:13:08.76 that plus c3 of this is zero. 209 00:13:08.76 --> 00:13:12 So in other words, I can go out some V1, 210 00:13:12 --> 00:13:17.78 out some more V2, back on V3, and end up zero. 211 00:13:17.78 --> 00:13:19 OK. 212 00:13:19 --> 00:13:27 So let -- here I've give the general, abstract definition, 213 00:13:27 --> 00:13:35 but let me repeat that definition -- this is like 214 00:13:35 --> 00:13:40 repeat -- let me call them Vs now. 215 00:13:40 --> 00:13:47 V1 up to Vn are the columns of a matrix A. 216 00:13:47 --> 00:13:53 In other words, this is telling me that if I'm 217 00:13:53 --> 00:14:00 in m dimensional space, like two dimensional space in 218 00:14:00 --> 00:14:07 the example, I can answer the dependence-independence question 219 00:14:07 --> 00:14:14.7 directly by putting those vectors in the columns of a 220 00:14:14.7 --> 00:14:16 matrix. 221 00:14:16 --> 00:14:22 They are independent if the null space of A, 222 00:14:22 --> 00:14:24 of A, is what? 223 00:14:24 --> 00:14:32 If I have a bunch of columns in a matrix, I'm looking at their 224 00:14:32 --> 00:14:38 combinations, but that's just A times the 225 00:14:38 --> 00:14:39 vector of c-s. 226 00:14:39 --> 00:14:47 And these columns will be independent if the null space of 227 00:14:47 --> 00:14:51 A is the zero vector. 228 00:14:51 --> 00:15:02 They are dependent if there's something else in there. 229 00:15:02 --> 00:15:11 If there's something else in the null space, 230 00:15:11 --> 00:15:23 if A times c gives the zero vector for some non-zero vector 231 00:15:23 --> 00:15:29 c in the null space. 232 00:15:29 --> 00:15:32 Then they're dependent, because that's telling me a 233 00:15:32 --> 00:15:36 combination of the columns gives the zero column. 234 00:15:36 --> 00:15:38.96 I think you're with be, because we've seen, 235 00:15:38.96 --> 00:15:42 like, lecture after lecture, we're looking at the 236 00:15:42 --> 00:15:46 combinations of the columns and asking, do we get zero or don't 237 00:15:46 --> 00:15:47.62 we? 238 00:15:47.62 --> 00:15:50 And now we're giving the official name, 239 00:15:50 --> 00:15:54 dependent if we do, independent if we don't. 240 00:15:54 --> 00:15:57 So I could express this in other words now. 241 00:15:57 --> 00:16:02 I could say the rank -- what's the rank in this independent 242 00:16:02 --> 00:16:03 case? 243 00:16:03 --> 00:16:05 The rank r of the, of the matrix, 244 00:16:05 --> 00:16:10 in the case of independent columns, is? 245 00:16:10 --> 00:16:12 So the columns are independent. 246 00:16:12 --> 00:16:16 So how many pivot columns have I got. 247 00:16:16 --> 00:16:16 All n. 248 00:16:16 --> 00:16:21 All the columns would be pivot columns, because free columns 249 00:16:21 --> 00:16:27 are telling me that they're a combination of earlier columns. 250 00:16:27 --> 00:16:32.24 So this would be the case where the rank is n. 251 00:16:32.24 --> 00:16:38 This would be the case where the rank is smaller than n. 252 00:16:38 --> 00:16:45 So in this case the rank is n and the null space of A is only 253 00:16:45 --> 00:16:47 the zero vector. 254 00:16:47 --> 00:16:50 And no free variables. 255 00:16:50 --> 00:16:52 No free variables. 256 00:16:52 --> 00:16:57 And this is the case yes free variables. 257 00:16:57 --> 00:17:05 If you'll allow me to stretch the English language that far. 258 00:17:05 --> 00:17:11 That's the case where we have, a combination that gives the 259 00:17:11 --> 00:17:12 zero column. 260 00:17:12 --> 00:17:19 I'm often interested in the case when my vectors are popped 261 00:17:19 --> 00:17:20 into a matrix. 262 00:17:20 --> 00:17:26 So the, the definition over there of independence didn't 263 00:17:26 --> 00:17:30 talk about any matrix. 264 00:17:30 --> 00:17:35 The vectors didn't have to be vectors in N dimensional space. 265 00:17:35 --> 00:17:40 And I want to give you some examples of vectors that aren't 266 00:17:40 --> 00:17:44 what you think of immediately as vectors. 267 00:17:44 --> 00:17:48 But most of the time, this is -- the vectors we think 268 00:17:48 --> 00:17:50 of are columns. 269 00:17:50 --> 00:17:53 And we can put them in a matrix. 270 00:17:53 --> 00:17:59 And then independence or dependence comes back to the 271 00:17:59 --> 00:18:00 null space. 272 00:18:00 --> 00:18:00 OK. 273 00:18:00 --> 00:18:04 So that's the idea of independence. 274 00:18:04 --> 00:18:09.15 Can I just, yeah, let me go on to spanning a 275 00:18:09.15 --> 00:18:09 space. 276 00:18:09 --> 00:18:15 What does it mean for a bunch of vectors to span a space? 277 00:18:15 --> 00:18:21 Well, actually, we've seen it already. 278 00:18:21 --> 00:18:25 You remember, if we had a columns in a 279 00:18:25 --> 00:18:32 matrix, we took all their combinations and that gave us 280 00:18:32 --> 00:18:34 the column space. 281 00:18:34 --> 00:18:41 Those vectors that we started with span that column space. 282 00:18:41 --> 00:18:49 So spanning a space means -- so let me move that important stuff 283 00:18:49 --> 00:18:51 right up. 284 00:18:51 --> 00:18:52 OK. 285 00:18:52 --> 00:19:02 So vectors -- let me call them, say, V1 up to -- call you some 286 00:19:02 --> 00:19:10 different letter, say Vl -- span a space, 287 00:19:10 --> 00:19:18 a subspace, or just a vector space I could say, 288 00:19:18 --> 00:19:26 span a space means, means the space consists of all 289 00:19:26 --> 00:19:34 combinations of those vectors. 290 00:19:34 --> 00:19:39 That's exactly what we did with the column space. 291 00:19:39 --> 00:19:46 So now I could say in shorthand the columns of a matrix span the 292 00:19:46 --> 00:19:48 column space. 293 00:19:48 --> 00:19:55 So you remember it's a bunch of vectors that have this property 294 00:19:55 --> 00:20:01 that they span a space, and actually if I give you a 295 00:20:01 --> 00:20:07.72 bunch of vectors and say -- OK, let S be the space that 296 00:20:07.72 --> 00:20:13 they span, in other words let S contain all their combinations, 297 00:20:13 --> 00:20:17 that space S will be the smallest space with those 298 00:20:17 --> 00:20:18 vectors in it, right? 299 00:20:18 --> 00:20:24 Because any space with those vectors in it must have all the 300 00:20:24 --> 00:20:27 combinations of those vectors in it. 301 00:20:27 --> 00:20:32 And if I stop there, then I've got the smallest 302 00:20:32 --> 00:20:37 space, and that's the space that they span. 303 00:20:37 --> 00:20:37 OK. 304 00:20:37 --> 00:20:41 So I'm just -- rather than, needing to say, 305 00:20:41 --> 00:20:47.25 take all linear combinations and put them in a space, 306 00:20:47.25 --> 00:20:51 I'm compressing that into the word span. 307 00:20:51 --> 00:20:54.24 Straightforward. 308 00:20:54.24 --> 00:20:54 OK. 309 00:20:54 --> 00:20:58.78 So if I think of a, of the column space of a 310 00:20:58.78 --> 00:20:59 matrix. 311 00:20:59 --> 00:21:03 I've got their -- so I start with the columns. 312 00:21:03 --> 00:21:06 I take all their combinations. 313 00:21:06 --> 00:21:09 That gives me the columns space. 314 00:21:09 --> 00:21:12 They span the column space. 315 00:21:12 --> 00:21:15 Now are they independent? 316 00:21:15 --> 00:21:18 Maybe yes, maybe no. 317 00:21:18 --> 00:21:23 It depends on the particular columns that went into that 318 00:21:23 --> 00:21:23 matrix. 319 00:21:23 --> 00:21:28 But obviously I'm highly interested in a set of vectors 320 00:21:28 --> 00:21:31 that spans a space and is independent. 321 00:21:31 --> 00:21:37.29 That's, that means like I've got the right number of vectors. 322 00:21:37.29 --> 00:21:42 If I didn't have all of them, I wouldn't have my whole space. 323 00:21:42 --> 00:21:48 If I had more than that, they probably wouldn't -- 324 00:21:48 --> 00:21:52 they wouldn't be independent. 325 00:21:52 --> 00:22:00 So, like, basis -- and that's the word that's coming -- is 326 00:22:00 --> 00:22:02 just right. 327 00:22:02 --> 00:22:08 So here let me put what that word means. 328 00:22:08 --> 00:22:16 A basis for a vector space is, is a, is a sequence of vectors 329 00:22:16 --> 00:22:20 -- shall I call them V1, 330 00:22:20 --> 00:22:26 V2, up to let me say Vd now, I'll stop with that letters -- 331 00:22:26 --> 00:22:28 that has two properties. 332 00:22:28 --> 00:22:32 I've got enough vectors and not too many. 333 00:22:32 --> 00:22:36 It's a natural idea of a basis. 334 00:22:36 --> 00:22:41 So a basis is a bunch of vectors in the space and it's a 335 00:22:41 --> 00:22:46 so it's a sequence of vectors with two properties, 336 00:22:46 --> 00:22:49 with two properties. 337 00:22:49 --> 00:22:55 One, they are independent. 338 00:22:55 --> 00:23:02 And two -- you know what's coming? 339 00:23:02 --> 00:23:07 -- they span the space. 340 00:23:07 --> 00:23:07 OK. 341 00:23:07 --> 00:23:17 Let me take -- so time for examples, of course. 342 00:23:17 --> 00:23:26 So I'm asking you now to put definition one, 343 00:23:26 --> 00:23:37 the definition of independence, together with definition two, 344 00:23:37 --> 00:23:41 and let's look at examples, because this is -- this 345 00:23:41 --> 00:23:45 combination means the set I've -- of vectors I have is just 346 00:23:45 --> 00:23:50 right, and the -- so that this idea of a basis will be central. 347 00:23:50 --> 00:23:54 I'll always be asking you now for a basis. 348 00:23:54 --> 00:23:58 Whenever I look at a subspace, if I ask you for -- if you give 349 00:23:58 --> 00:24:02 me a basis for that subspace, you've told me what it is. 350 00:24:02 --> 00:24:05 You've told me everything I need to know about that 351 00:24:05 --> 00:24:06 subspace. 352 00:24:06 --> 00:24:09 Those -- I take their combinations and I know that I 353 00:24:09 --> 00:24:11 need all the combinations. 354 00:24:11 --> 00:24:12 OK. 355 00:24:12 --> 00:24:13 Examples. 356 00:24:13 --> 00:24:17 OK, so examples of a basis. 357 00:24:17 --> 00:24:23.02 Let me start with two dimensional space. 358 00:24:23.02 --> 00:24:27 Suppose the space -- say example. 359 00:24:27 --> 00:24:32.44 The space is, oh, let's make it R^3. 360 00:24:32.44 --> 00:24:36 Real three dimensional space. 361 00:24:36 --> 00:24:39 Give me one basis. 362 00:24:39 --> 00:24:42 One basis is? 363 00:24:42 --> 00:24:47 So I want some vectors, because if I ask you for a 364 00:24:47 --> 00:24:52 basis, I'm asking you for vectors, a little list of 365 00:24:52 --> 00:24:53 vectors. 366 00:24:53 --> 00:24:56 And it should be just right. 367 00:24:56 --> 00:25:01 So what would be a basis for three dimensional space? 368 00:25:01 --> 00:25:05 Well, the first basis that comes to mind, 369 00:25:05 --> 00:25:10 why don't we write that down. 370 00:25:10 --> 00:25:15 The first basis that comes to mind is this vector, 371 00:25:15 --> 00:25:18 this vector, and this vector. 372 00:25:18 --> 00:25:18 OK. 373 00:25:18 --> 00:25:20 That's one basis. 374 00:25:20 --> 00:25:25 Not the only basis, that's going to be my point. 375 00:25:25 --> 00:25:29 But let's just see -- yes, that's a basis. 376 00:25:29 --> 00:25:34 Are, are those vectors independent? 377 00:25:34 --> 00:25:38 So that's the like the x, y, z axes, so if those are not 378 00:25:38 --> 00:25:40 independent, we're in trouble. 379 00:25:40 --> 00:25:42 Certainly, they are. 380 00:25:42 --> 00:25:46 Take a combination c1 of this vector plus c2 of this vector 381 00:25:46 --> 00:25:50 plus c3 of that vector and try to make it give the zero vector. 382 00:25:50 --> 00:25:52 What are the c-s? 383 00:25:52 --> 00:25:56 If c1 of that plus c2 of that plus c3 of that gives me 0 0 0, 384 00:25:56 --> 00:25:59 then the c-s are all -- 0, right. 385 00:25:59 --> 00:26:03 So that's the test for independence. 386 00:26:03 --> 00:26:09 In the language of matrices, which was under that board, 387 00:26:09 --> 00:26:13 I could make those the columns of a matrix. 388 00:26:13 --> 00:26:17 Well, it would be the identity matrix. 389 00:26:17 --> 00:26:21 Then I would ask, what's the null space of the 390 00:26:21 --> 00:26:24 identity matrix? 391 00:26:24 --> 00:26:27 And you would say it's only the zero vector. 392 00:26:27 --> 00:26:31 And I would say, fine, then the columns are 393 00:26:31 --> 00:26:32.19 independent. 394 00:26:32.19 --> 00:26:36 The only thing -- the identity times a vector giving zero, 395 00:26:36 --> 00:26:40 the only vector that does that is zero. 396 00:26:40 --> 00:26:40 OK. 397 00:26:40 --> 00:26:45 Now that's not the only basis. 398 00:26:45 --> 00:26:47 Far from it. 399 00:26:47 --> 00:26:53 Tell me another basis, a second basis, 400 00:26:53 --> 00:26:55 another basis. 401 00:26:55 --> 00:27:02 So, give me -- well, I'll just start it out. 402 00:27:02 --> 00:27:04 One one two. 403 00:27:04 --> 00:27:06 Two two five. 404 00:27:06 --> 00:27:12.69 Suppose I stopped there. 405 00:27:12.69 --> 00:27:18 Has that little bunch of vectors got the properties that 406 00:27:18 --> 00:27:21 I'm asking for in a basis for R^3? 407 00:27:21 --> 00:27:25 We're looking for a basis for R^3. 408 00:27:25 --> 00:27:29 Are they independent, those two column vectors? 409 00:27:29 --> 00:27:30 Yes. 410 00:27:30 --> 00:27:33 Do they span R^3? 411 00:27:33 --> 00:27:33.35 No. 412 00:27:33.35 --> 00:27:34 Our feeling is no. 413 00:27:34 --> 00:27:36 Our feeling is no. 414 00:27:36 --> 00:27:41 Our feeling is that there're some vectors in R3 that are not 415 00:27:41 --> 00:27:43 combinations of those. 416 00:27:43 --> 00:27:43.29 OK. 417 00:27:43.29 --> 00:27:47 So suppose I add in -- I need another vector then, 418 00:27:47 --> 00:27:51 because these two don't span the space. 419 00:27:51 --> 00:27:51 OK. 420 00:27:51 --> 00:27:56 Now it would be foolish for me to put in three three seven, 421 00:27:56 --> 00:27:58 right, as the third vector. 422 00:27:58 --> 00:28:00 That would be a goof. 423 00:28:00 --> 00:28:04 Because that, if I put in three three seven, 424 00:28:04 --> 00:28:08.65 those vectors would be dependent, right? 425 00:28:08.65 --> 00:28:12 If I put in three three seven, it would be the sum of those 426 00:28:12 --> 00:28:16 two, it would lie in the same plane as those. 427 00:28:16 --> 00:28:18 It wouldn't be independent. 428 00:28:18 --> 00:28:21.31 My attempt to create a basis would be dead. 429 00:28:21.31 --> 00:28:25 But if I take -- so what vector can I take? 430 00:28:25 --> 00:28:29 I can take any vector that's not in that plane. 431 00:28:29 --> 00:28:33 Let me try -- I hope that 3 3 8 would do it. 432 00:28:33 --> 00:28:37 At least it's not the sum of those two vectors. 433 00:28:37 --> 00:28:40 But I believe that's a basis. 434 00:28:40 --> 00:28:45 And what's the test then, for that to be a basis? 435 00:28:45 --> 00:28:51.57 Because I just picked those numbers, and if I had picked, 436 00:28:51.57 --> 00:28:57 5 7 -14 how would we know do we have a basis or don't we? 437 00:28:57 --> 00:29:02 You would put them in the columns of a matrix, 438 00:29:02 --> 00:29:08 and you would do elimination, row reduction -- 439 00:29:08 --> 00:29:15 and you would see do you get any free variables or are all 440 00:29:15 --> 00:29:18 the columns pivot columns. 441 00:29:18 --> 00:29:25 Well now actually we have a square -- the matrix would be 442 00:29:25 --> 00:29:27 three by three. 443 00:29:27 --> 00:29:33 So, what's the test on the matrix then? 444 00:29:33 --> 00:29:42 The matrix -- so in this case, when my space is R^3 and I have 445 00:29:42 --> 00:29:48 three vectors, my matrix is square and what I 446 00:29:48 --> 00:29:57 asking about that matrix in order for those columns to be a 447 00:29:57 --> 00:29:58.34 basis? 448 00:29:58.34 --> 00:30:05 So in this -- for R^n, if I have -- n vectors give a 449 00:30:05 --> 00:30:12 basis if the n by n matrix with those columns, 450 00:30:12 --> 00:30:18 with those columns, is what? 451 00:30:18 --> 00:30:21 What's the requirement on that matrix? 452 00:30:21 --> 00:30:24 Invertible, right, right. 453 00:30:24 --> 00:30:27 The matrix should be invertible. 454 00:30:27 --> 00:30:32 For a square matrix, that's the, that's the perfect 455 00:30:32 --> 00:30:33 answer. 456 00:30:33 --> 00:30:34 Is invertible. 457 00:30:34 --> 00:30:39 So that's when, that's when the space is the 458 00:30:39 --> 00:30:42 whole space R^n. 459 00:30:42 --> 00:30:46 Let me, let me be sure you're with me here. 460 00:30:46 --> 00:30:48 Let me remove that. 461 00:30:48 --> 00:30:54 Are those two vectors a basis for any space at all? 462 00:30:54 --> 00:31:00 Is there a vector space that those really are a basis for, 463 00:31:00 --> 00:31:06 those, that pair of vectors, this guy and this 1, 464 00:31:06 --> 00:31:08 1 1 2 and 2 2 5? 465 00:31:08 --> 00:31:12 Is there a space for which that's a basis? 466 00:31:12 --> 00:31:12 Sure. 467 00:31:12 --> 00:31:16 They're independent, so they satisfy the first 468 00:31:16 --> 00:31:22 requirement, so what space shall I take for them to be a basis 469 00:31:22 --> 00:31:22 of? 470 00:31:22 --> 00:31:26 What spaces will they be a basis for? 471 00:31:26 --> 00:31:28 The one they span. 472 00:31:28 --> 00:31:30 Their combinations. 473 00:31:30 --> 00:31:31 It's a plane, right? 474 00:31:31 --> 00:31:34 It'll be a plane inside R^3. 475 00:31:34 --> 00:31:39 So if I take this vector 1 1 2, say it goes there, 476 00:31:39 --> 00:31:43 and this vector 2 2 5, say it goes there, 477 00:31:43 --> 00:31:49 those are a basis for -- because they span a plane. 478 00:31:49 --> 00:31:52 And they're a basis for the plane, because they're 479 00:31:52 --> 00:31:52 independent. 480 00:31:52 --> 00:31:56 If I stick in some third guy, like 3 3 7, which is in the 481 00:31:56 --> 00:31:58 plane -- suppose I put in, try to put in 3 3 7, 482 00:31:58 --> 00:32:01.45 then the three vectors would still span the plane, 483 00:32:01.45 --> 00:32:04 but they wouldn't be a basis anymore because they're not 484 00:32:04 --> 00:32:06.32 independent anymore. 485 00:32:06.32 --> 00:32:06 OK. 486 00:32:06 --> 00:32:15 So, we're looking at the question of -- again, 487 00:32:15 --> 00:32:24.81 the case with independent columns is the case where the 488 00:32:24.81 --> 00:32:32 column vectors span the column space. 489 00:32:32 --> 00:32:36.93 They're independent, so they're a basis for the 490 00:32:36.93 --> 00:32:38 column space. 491 00:32:38 --> 00:32:38 OK. 492 00:32:38 --> 00:32:41 So now there's one bit of intuition. 493 00:32:41 --> 00:32:44 Let me go back to all of R^n. 494 00:32:44 --> 00:32:46 So I -- where I put 3 3 8. 495 00:32:46 --> 00:32:47.18 OK. 496 00:32:47.18 --> 00:32:51 The first message is that the basis is not unique, 497 00:32:51 --> 00:32:53.48 right. 498 00:32:53.48 --> 00:32:56 There's zillions of bases. 499 00:32:56 --> 00:33:00 I take any invertible three by three matrix, 500 00:33:00 --> 00:33:03 its columns are a basis for R^3. 501 00:33:03 --> 00:33:07 The column space is R^3, and if those, 502 00:33:07 --> 00:33:13 if that matrix is invertible, those columns are independent, 503 00:33:13 --> 00:33:16 I've got a basis for R^3. 504 00:33:16 --> 00:33:19 So there're many, many bases. 505 00:33:19 --> 00:33:24 But there is something in common for all those bases. 506 00:33:24 --> 00:33:31 There's something that this basis shares with that basis and 507 00:33:31 --> 00:33:33.87 every other basis for R^3. 508 00:33:33.87 --> 00:33:35 And what's that? 509 00:33:35 --> 00:33:41 Well, you saw it coming, because when I stopped here and 510 00:33:41 --> 00:33:47 asked if that was a basis for R^3, you said no. 511 00:33:47 --> 00:33:53 And I know that you said no because you knew there weren't 512 00:33:53 --> 00:33:55 enough vectors there. 513 00:33:55 --> 00:33:59 And the great fact is that there're many, 514 00:33:59 --> 00:34:04 many bases, but -- let me put in somebody else, 515 00:34:04 --> 00:34:06 just for variety. 516 00:34:06 --> 00:34:11 There are many, many bases, but they all have 517 00:34:11 --> 00:34:15 the same number of vectors. 518 00:34:15 --> 00:34:20 If we're talking about the space R^3, then that number of 519 00:34:20 --> 00:34:22 vectors is three. 520 00:34:22 --> 00:34:28 If we're talking about the space R^n, then that number of 521 00:34:28 --> 00:34:29 vectors is n. 522 00:34:29 --> 00:34:35 If we're talking about some other space, the column space of 523 00:34:35 --> 00:34:40 some matrix, or the null space of some matrix, 524 00:34:40 --> 00:34:47 or some other space that we haven't even thought of, 525 00:34:47 --> 00:34:52 then that still is true that every basis -- that there're 526 00:34:52 --> 00:34:58.51 lots of bases but every basis has the same number of vectors. 527 00:34:58.51 --> 00:35:01.77 Let me write that great fact down. 528 00:35:01.77 --> 00:35:05 Every basis -- we're given a space. 529 00:35:05 --> 00:35:06 Given a space. 530 00:35:06 --> 00:35:12 R^3 or R^n or some other column space of a matrix or the null 531 00:35:12 --> 00:35:18.07 space of a matrix or some other vector space. 532 00:35:18.07 --> 00:35:26 Then the great fact is that every basis for this, 533 00:35:26 --> 00:35:33 for the space has the same number of vectors. 534 00:35:33 --> 00:35:43 If one basis has six vectors, then every other basis has six 535 00:35:43 --> 00:35:44 vectors. 536 00:35:44 --> 00:35:54 So that number six is telling me like it's telling me how big 537 00:35:54 --> 00:35:58 is the space. 538 00:35:58 --> 00:36:02 It's telling me how many vectors do I have to have to 539 00:36:02 --> 00:36:03 have a basis. 540 00:36:03 --> 00:36:05 And of course we're seeing it this way. 541 00:36:05 --> 00:36:08 That number six, if we had seven vectors, 542 00:36:08 --> 00:36:10 then we've got too many. 543 00:36:10 --> 00:36:13 If we have five vectors we haven't got enough. 544 00:36:13 --> 00:36:18 Sixes are like just right for whatever space that is. 545 00:36:18 --> 00:36:21.47 And what do we call that number? 546 00:36:21.47 --> 00:36:27 That number is -- now I'm ready for the last definition today. 547 00:36:27 --> 00:36:30 It's the dimension of that space. 548 00:36:30 --> 00:36:36 So every basis for a space has the same number of vectors in 549 00:36:36 --> 00:36:36 it. 550 00:36:36 --> 00:36:42 Not the same vectors, all sorts of bases -- 551 00:36:42 --> 00:36:47.68 but the same number of vectors is always the same, 552 00:36:47.68 --> 00:36:51 and that number is the dimension. 553 00:36:51 --> 00:36:53 This is definitional. 554 00:36:53 --> 00:36:58 This number is the dimension of the space. 555 00:36:58 --> 00:36:58 OK. 556 00:36:58 --> 00:36:59 OK. 557 00:36:59 --> 00:37:01 Let's do some examples. 558 00:37:01 --> 00:37:05.71 Because now we've got definitions. 559 00:37:05.71 --> 00:37:12 Let me repeat the four things, the four words that have now 560 00:37:12 --> 00:37:15 got defined. 561 00:37:15 --> 00:37:19 Independence, that looks at combinations not 562 00:37:19 --> 00:37:19 being zero. 563 00:37:19 --> 00:37:23 Spanning, that looks at all the combinations. 564 00:37:23 --> 00:37:27 Basis, that's the one that combines independence and 565 00:37:27 --> 00:37:28 spanning. 566 00:37:28 --> 00:37:33 And now we've got the idea of the dimension of a space. 567 00:37:33 --> 00:37:40 It's the number of vectors in any basis, because all bases 568 00:37:40 --> 00:37:42.79 have the same number. 569 00:37:42.79 --> 00:37:43 OK. 570 00:37:43 --> 00:37:45.47 Let's take examples. 571 00:37:45.47 --> 00:37:50 Suppose I take, my space is -- examples now -- 572 00:37:50 --> 00:37:55 space is the, say, the column space of this 573 00:37:55 --> 00:37:57 matrix. 574 00:37:57 --> 00:38:01 Let me write down a matrix. 1 1 1, 2 1 2, 575 00:38:01 --> 00:38:07 and I'll -- just to make it clear, I'll take the sum there, 576 00:38:07 --> 00:38:12 3 2 3, and let me take the sum of all -- oh, 577 00:38:12 --> 00:38:17 let me put in one -- yeah, I'll put in one one one again. 578 00:38:17 --> 00:38:19 OK. 579 00:38:19 --> 00:38:21 So that's four vectors. 580 00:38:21 --> 00:38:25.74 OK, do they span the column space of that matrix? 581 00:38:25.74 --> 00:38:29 Let me repeat, do they span the column space 582 00:38:29 --> 00:38:31.15 of that matrix? 583 00:38:31.15 --> 00:38:31 Yes. 584 00:38:31 --> 00:38:35.72 By definition, that's what the column space -- 585 00:38:35.72 --> 00:38:38 where it comes from. 586 00:38:38 --> 00:38:41 Are they a basis for the column space? 587 00:38:41 --> 00:38:42 Are they independent? 588 00:38:42 --> 00:38:44 No, they're not independent. 589 00:38:44 --> 00:38:47 There's something in that null space. 590 00:38:47 --> 00:38:51 Maybe we can -- so let's look at the null space of the matrix. 591 00:38:51 --> 00:38:56 Tell me a vector that's in the null space of that matrix. 592 00:38:56 --> 00:39:03 So I'm looking for some vector that combines those columns and 593 00:39:03 --> 00:39:05 produces the zero column. 594 00:39:05 --> 00:39:11 Or in other words, I'm looking for solutions to A 595 00:39:11 --> 00:39:12 X equals zero. 596 00:39:12 --> 00:39:17 So tell me a vector in the null space. 597 00:39:17 --> 00:39:20 Maybe -- well, this was, this column was that 598 00:39:20 --> 00:39:24 one plus that one, so maybe if I have one of those 599 00:39:24 --> 00:39:28 and minus one of those that would be a vector in the null 600 00:39:28 --> 00:39:28 space. 601 00:39:28 --> 00:39:32 So, you've already told me now, are those vectors independent, 602 00:39:32 --> 00:39:37 the answer is -- those column vectors, the answer is -- no. 603 00:39:37 --> 00:39:38 Right? 604 00:39:38 --> 00:39:40 They're not independent. 605 00:39:40 --> 00:39:43.33 Because -- you knew they weren't independent. 606 00:39:43.33 --> 00:39:47 Anyway, minus one of this minus one of this plus one of this 607 00:39:47 --> 00:39:50.03 zero of that is the zero vector. 608 00:39:50.03 --> 00:39:50 OK. 609 00:39:50 --> 00:39:52 OK, so they're not independent. 610 00:39:52 --> 00:39:56 They span, but they're not independent. 611 00:39:56 --> 00:39:59 Tell me a basis for that column space. 612 00:39:59 --> 00:40:02.73 What's a basis for the column space? 613 00:40:02.73 --> 00:40:07 These are all the questions that the homework asks, 614 00:40:07 --> 00:40:10 the quizzes ask, the final exam will ask. 615 00:40:10 --> 00:40:16 Find a basis for the column space of this matrix. 616 00:40:16 --> 00:40:16 OK. 617 00:40:16 --> 00:40:21 Now there's many answers, but give me the most natural 618 00:40:21 --> 00:40:21.67 answer. 619 00:40:21.67 --> 00:40:23 Columns one and two. 620 00:40:23 --> 00:40:25 Columns one and two. 621 00:40:25 --> 00:40:27 That's the natural answer. 622 00:40:27 --> 00:40:31 Those are the pivot columns, because, I mean, 623 00:40:31 --> 00:40:33 we s- we begin systematically. 624 00:40:33 --> 00:40:37 We look at the first column, it's OK. 625 00:40:37 --> 00:40:40 We can put that in the basis. 626 00:40:40 --> 00:40:43 We look at the second column, it's OK. 627 00:40:43 --> 00:40:46 We can put that in the basis. 628 00:40:46 --> 00:40:49 The third column we can't put in the basis. 629 00:40:49 --> 00:40:52 The fourth column we can't, again. 630 00:40:52 --> 00:40:58 So the rank of the matrix is -- what's the rank of our matrix? 631 00:40:58 --> 00:40:59 Two. 632 00:40:59 --> 00:41:00.03 Two. 633 00:41:00.03 --> 00:41:08 And, and now that rank is also -- we also have another word. 634 00:41:08 --> 00:41:12 We, we have a great theorem here. 635 00:41:12 --> 00:41:15 The rank of A, that rank r, 636 00:41:15 --> 00:41:22 is the number of pivot columns and it's also -- well, 637 00:41:22 --> 00:41:27 so now please use my new word. 638 00:41:27 --> 00:41:34 This, it's the number two, of course, two is the rank of 639 00:41:34 --> 00:41:41.13 my matrix, it's the number of pivot 640 00:41:41.13 --> 00:41:46.9 columns, those pivot columns form a basis, 641 00:41:46.9 --> 00:41:50 of course, so what's two? 642 00:41:50 --> 00:41:53 It's the dimension. 643 00:41:53 --> 00:41:59 The rank of A, the number of pivot columns, 644 00:41:59 --> 00:42:04.07 is the dimension of the column space. 645 00:42:04.07 --> 00:42:06 Of course, you say. 646 00:42:06 --> 00:42:08 It had to be. 647 00:42:08 --> 00:42:11 Right. 648 00:42:11 --> 00:42:15 But just watch, look for one moment at the, 649 00:42:15 --> 00:42:19 the language, the way the English words get 650 00:42:19 --> 00:42:20 involved here. 651 00:42:20 --> 00:42:25 I take the rank of a matrix, the rank of a matrix. 652 00:42:25 --> 00:42:31 It's a number of columns and it's the dimension of -- not the 653 00:42:31 --> 00:42:37 dimension of the matrix, that's what I want to say. 654 00:42:37 --> 00:42:41 It's the dimension of a space, a subspace, the column space. 655 00:42:41 --> 00:42:45 Do you see, I don't take the dimension of A. 656 00:42:45 --> 00:42:47 That's not what I want. 657 00:42:47 --> 00:42:51.56 I'm looking for the dimension of the column space of A. 658 00:42:51.56 --> 00:42:56 If you use those words right, it shows you've got the idea 659 00:42:56 --> 00:42:56 right. 660 00:42:56 --> 00:42:58 Similarly here. 661 00:42:58 --> 00:43:01 I don't talk about the rank of a subspace. 662 00:43:01 --> 00:43:03 It's a matrix that has a rank. 663 00:43:03 --> 00:43:05 I talk about the rank of a matrix. 664 00:43:05 --> 00:43:09 And the beauty is that these definitions just merge so that 665 00:43:09 --> 00:43:13.76 the rank of a matrix is the dimension of its column space. 666 00:43:13.76 --> 00:43:15 And in this example it's two. 667 00:43:15 --> 00:43:19 And then the further question is, what's a basis? 668 00:43:19 --> 00:43:22 And the first two columns are a basis. 669 00:43:22 --> 00:43:24.14 Tell me another basis. 670 00:43:24.14 --> 00:43:26 Another basis for the columns space. 671 00:43:26 --> 00:43:29 You see I just keep hammering away. 672 00:43:29 --> 00:43:32 I apologize, but it's, I have to be sure you 673 00:43:32 --> 00:43:33 have the idea of basis. 674 00:43:33 --> 00:43:37 Tell me another basis for the column space. 675 00:43:37 --> 00:43:42 Well, you could take columns one and three. 676 00:43:42 --> 00:43:46 That would be a basis for the column space. 677 00:43:46 --> 00:43:50 Or columns two and three would be a basis. 678 00:43:50 --> 00:43:53 Or columns two and four. 679 00:43:53 --> 00:44:00 Or tell me another basis that's not made out of those columns at 680 00:44:00 --> 00:44:01 all? 681 00:44:01 --> 00:44:05 So -- I guess I'm giving you infinitely many possibilities, 682 00:44:05 --> 00:44:07 so I can't expect a unanimous answer here. 683 00:44:07 --> 00:44:10 I'll tell you -- but let's look at another basis, 684 00:44:10 --> 00:44:11 though. 685 00:44:11 --> 00:44:14 I'll just -- because it's only one out of zillions, 686 00:44:14 --> 00:44:18 I'm going to put it down and I'm going to erase it. 687 00:44:18 --> 00:44:23 Another basis for the column space would be -- let's see. 688 00:44:23 --> 00:44:27 I'll put in some things that are not there. 689 00:44:27 --> 00:44:31 Say, oh well, just to make it -- my life 690 00:44:31 --> 00:44:33 easy, 2 2 2. 691 00:44:33 --> 00:44:35 That's in the column space. 692 00:44:35 --> 00:44:39 And, that was sort of obvious. 693 00:44:39 --> 00:44:43 Let me take the sum of those, say 6 4 6. 694 00:44:43 --> 00:44:47.45 Or the sum of all of the columns, 7 5 7, 695 00:44:47.45 --> 00:44:48 why not. 696 00:44:48 --> 00:44:50 That's in the column space. 697 00:44:50 --> 00:44:55 Those are independent and I've got the number right, 698 00:44:55 --> 00:44:57 I've got two. 699 00:44:57 --> 00:45:00 Actually, this is a key point. 700 00:45:00 --> 00:45:06.04 If you know the dimension of the space you're working with, 701 00:45:06.04 --> 00:45:13 and we know that this column -- we know that the dimension, 702 00:45:13 --> 00:45:18 DIM, the dimension of the column space is two. 703 00:45:18 --> 00:45:25.55 If you know the dimension, then -- and we have a couple of 704 00:45:25.55 --> 00:45:32 vectors that are independent, they'll automatically be a 705 00:45:32 --> 00:45:32 basis. 706 00:45:32 --> 00:45:38 If we've got the number of vectors right, 707 00:45:38 --> 00:45:42 two vectors in this case, then if they're independent, 708 00:45:42 --> 00:45:45 they can't help but span the space. 709 00:45:45 --> 00:45:50.38 Because if they didn't span the space, there'd be a third guy to 710 00:45:50.38 --> 00:45:54 help span the space, but it couldn't be independent. 711 00:45:54 --> 00:45:58.25 So, it just has to be independent if we've got the 712 00:45:58.25 --> 00:46:00 numbers right. 713 00:46:00 --> 00:46:01 And they span. 714 00:46:01 --> 00:46:01 OK. 715 00:46:01 --> 00:46:02 Very good. 716 00:46:02 --> 00:46:05 So you got the dimension of a space. 717 00:46:05 --> 00:46:09 So this was another basis that I just invented. 718 00:46:09 --> 00:46:09 OK. 719 00:46:09 --> 00:46:13 Now, now I get to ask about the null space. 720 00:46:13 --> 00:46:17.82 What's the dimension of the null space? 721 00:46:17.82 --> 00:46:22 So we, we got a great fact there, the dimension of the 722 00:46:22 --> 00:46:25 column space is the rank. 723 00:46:25 --> 00:46:29.34 Now I want to ask you about the null space. 724 00:46:29.34 --> 00:46:35 That's the other part of the lecture, and it'll go on to the 725 00:46:35 --> 00:46:36 next lecture. 726 00:46:36 --> 00:46:36.54 OK. 727 00:46:36.54 --> 00:46:41 So we know the dimension of the column space is two, 728 00:46:41 --> 00:46:43 the rank. 729 00:46:43 --> 00:46:45 What about the null space? 730 00:46:45 --> 00:46:48 This is a vector in the null space. 731 00:46:48 --> 00:46:51 Are there other vectors in the null space? 732 00:46:51 --> 00:46:51 Yes or no? 733 00:46:51 --> 00:46:52 Yes. 734 00:46:52 --> 00:46:55 So this isn't a basis because it's doesn't span, 735 00:46:55 --> 00:46:56 right? 736 00:46:56 --> 00:47:01 There's more in the null space than we've got so far. 737 00:47:01 --> 00:47:03 I need another vector at least. 738 00:47:03 --> 00:47:06 So tell me another vector in the null space. 739 00:47:06 --> 00:47:10 Well, the natural choice, the choice you naturally think 740 00:47:10 --> 00:47:14 of is I'm going on to the fourth column, I'm letting that free 741 00:47:14 --> 00:47:18 variable be a one, and that free variable be a 742 00:47:18 --> 00:47:22 zero, and I'm asking is that fourth column a combination of 743 00:47:22 --> 00:47:24 my pivot columns? 744 00:47:24 --> 00:47:25 Yes, it is. 745 00:47:25 --> 00:47:28 And it's -- that will do. 746 00:47:28 --> 00:47:34 So what I've written there are actually the two special 747 00:47:34 --> 00:47:36 solutions, right? 748 00:47:36 --> 00:47:42 I took the two free variables, free and free. 749 00:47:42 --> 1. I gave them the values 1 0 or 0 750 1. --> 00:47:46 751 00:47:46 --> 00:47:48 I figured out the rest. 752 00:47:48 --> 00:47:52 So do you see, let me just say it in words. 753 00:47:52 --> 00:47:55 This vector, these vectors in the null space 754 00:47:55 --> 00:47:58 are telling me, they're telling me the 755 00:47:58 --> 00:48:01 combinations of the columns that give zero. 756 00:48:01 --> 00:48:07 They're telling me in what way the, the columns are dependent. 757 00:48:07 --> 00:48:10 That's what the null space is doing. 758 00:48:10 --> 00:48:11 Have I got enough now? 759 00:48:11 --> 00:48:13 And what's the null space now? 760 00:48:13 --> 00:48:16.35 We have to think about the null space. 761 00:48:16.35 --> 00:48:19 These are two vectors in the null space. 762 00:48:19 --> 00:48:20.46 They're independent. 763 00:48:20.46 --> 00:48:23 Are they a basis for the null space? 764 00:48:23 --> 00:48:26.35 What's the dimension of the null space? 765 00:48:26.35 --> 00:48:30 You see that those questions just keep coming up all the 766 00:48:30 --> 00:48:30 time. 767 00:48:30 --> 00:48:33 Are they a basis for the null space? 768 00:48:33 --> 00:48:37 You can tell me the answer even though we haven't written out a 769 00:48:37 --> 00:48:38 proof of that. 770 00:48:38 --> 00:48:39 Can you? 771 00:48:39 --> 00:48:40 Yes or no? 772 00:48:40 --> 00:48:45 Do these two special solutions form a basis for the null space? 773 00:48:45 --> 00:48:48 In other words, does the null space consist of 774 00:48:48 --> 00:48:50 all combinations of those two guys? 775 00:48:50 --> 00:48:51 Yes or no? 776 00:48:51 --> 00:48:51.87 Yes. 777 00:48:51.87 --> 00:48:52 Yes. 778 00:48:52 --> 00:48:54.57 The null space is two dimensional. 779 00:48:54.57 --> 00:48:57 The null space, the dimension of the null 780 00:48:57 --> 00:49:01 space, is the number of free variables. 781 00:49:01 --> 00:49:08 So the dimension of the null space is the number of free 782 00:49:08 --> 00:49:09 variables. 783 00:49:09 --> 00:49:15 And at the last second, give me the formula. 784 00:49:15 --> 00:49:20 This is then the key formula that we know. 785 00:49:20 --> 00:49:26 How many free variables are there in terms of R, 786 00:49:26 --> 00:49:31 the rank, m -- the number of rows, 787 00:49:31 --> 00:49:34 n, the number of columns? 788 00:49:34 --> 00:49:36.09 What do we get? 789 00:49:36.09 --> 00:49:40 We have n columns, r of them are pivot columns, 790 00:49:40 --> 00:49:46 so n-r is the number of free columns, free variables. 791 00:49:46 --> 00:49:50 And now it's the dimension of the null space. 792 00:49:50 --> 00:49:52 OK. 793 00:49:52 --> 00:49:53.6 That's great. 794 00:49:53.6 --> 00:49:57 That's the key spaces, their bases, 795 00:49:57 --> 00:49:59 and their dimensions. 796 00:49:59 --> 00:50:02 Thanks.