1 00:00:09,840 --> 00:00:14,610 OK, this is linear algebra lecture nine. 2 00:00:14,610 --> 00:00:21,130 And this is a key lecture, this is where we get these ideas 3 00:00:21,130 --> 00:00:27,480 of linear independence, when a bunch of vectors are 4 00:00:27,480 --> 00:00:29,600 independent -- 5 00:00:29,600 --> 00:00:33,890 or dependent, that's the opposite. 6 00:00:33,890 --> 00:00:36,680 The space they span. 7 00:00:36,680 --> 00:00:40,550 A basis for a subspace or a basis for a vector 8 00:00:40,550 --> 00:00:42,730 space, that's a central idea. 9 00:00:42,730 --> 00:00:47,060 And then the dimension of that subspace. 10 00:00:47,060 --> 00:00:50,240 So this is the day that those words 11 00:00:50,240 --> 00:00:52,960 get assigned clear meanings. 12 00:00:52,960 --> 00:00:58,380 And emphasize that we talk about a bunch of vectors 13 00:00:58,380 --> 00:00:59,550 being independent. 14 00:00:59,550 --> 00:01:02,780 Wouldn't talk about a matrix being independent. 15 00:01:02,780 --> 00:01:05,150 A bunch of vectors being independent. 16 00:01:05,150 --> 00:01:07,840 A bunch of vectors spanning a space. 17 00:01:07,840 --> 00:01:11,480 A bunch of vectors being a basis. 18 00:01:11,480 --> 00:01:15,070 And the dimension is some number. 19 00:01:15,070 --> 00:01:17,480 OK, so what are the definitions? 20 00:01:17,480 --> 00:01:21,840 Can I begin with a fact, a highly important fact, 21 00:01:21,840 --> 00:01:28,240 that, I didn't call directly attention to earlier. 22 00:01:28,240 --> 00:01:35,900 Suppose I have a matrix and I look at Ax equals zero. 23 00:01:35,900 --> 00:01:39,530 Suppose the matrix has a lot of columns, 24 00:01:39,530 --> 00:01:43,340 so that n is bigger than m. 25 00:01:43,340 --> 00:01:46,320 So I'm looking at n equations -- 26 00:01:46,320 --> 00:01:48,920 I mean, sorry, m equations, a small number 27 00:01:48,920 --> 00:01:52,650 of equations m, and more unknowns. 28 00:01:52,650 --> 00:01:55,960 I have more unknowns than equations. 29 00:01:55,960 --> 00:01:57,040 Let me write that down. 30 00:01:57,040 --> 00:01:59,850 More unknowns than equations. 31 00:01:59,850 --> 00:02:03,170 More unknown x-s than equations. 32 00:02:03,170 --> 00:02:15,440 Then the conclusion is that there's 33 00:02:15,440 --> 00:02:19,330 something in the null space of A, other 34 00:02:19,330 --> 00:02:21,350 than just the zero vector. 35 00:02:21,350 --> 00:02:26,320 The conclusion is there are some non-zero x-s 36 00:02:26,320 --> 00:02:28,220 such that Ax is zero. 37 00:02:28,220 --> 00:02:30,030 There are some special solutions. 38 00:02:30,030 --> 00:02:31,060 And why? 39 00:02:31,060 --> 00:02:33,590 We know why. 40 00:02:33,590 --> 00:02:38,120 I mean, it sort of like seems like a reasonable thing, more 41 00:02:38,120 --> 00:02:42,150 unknowns than equations, then it seems reasonable 42 00:02:42,150 --> 00:02:43,540 that we can solve them. 43 00:02:43,540 --> 00:02:49,210 But we have a, a clear algorithm which starts with a system 44 00:02:49,210 --> 00:02:55,440 and does elimination, gets the thing into an echelon 45 00:02:55,440 --> 00:03:01,630 form with some pivots and pivot columns, 46 00:03:01,630 --> 00:03:06,050 and possibly some free columns that don't have pivots. 47 00:03:06,050 --> 00:03:09,290 And the point is here there will be some free columns. 48 00:03:09,290 --> 00:03:14,940 The reason, so the reason is there must -- 49 00:03:14,940 --> 00:03:20,080 there will be free variables, at least one. 50 00:03:25,010 --> 00:03:27,580 That's the reason. 51 00:03:27,580 --> 00:03:30,130 That we now have this -- 52 00:03:30,130 --> 00:03:36,200 a complete, algorithm, a complete systematic way to say, 53 00:03:36,200 --> 00:03:38,860 OK, we take the system Ax equals zero, 54 00:03:38,860 --> 00:03:42,370 we row reduce, we identify the free variables, 55 00:03:42,370 --> 00:03:48,690 and, since there are n variables and at most m pivots, 56 00:03:48,690 --> 00:03:51,840 there will be some free variables, at least one, 57 00:03:51,840 --> 00:03:55,090 at least n-m in fact, left over. 58 00:03:55,090 --> 00:04:00,480 And those variables I can assign non-zero values to. 59 00:04:00,480 --> 00:04:02,110 I don't have to set those to zero. 60 00:04:02,110 --> 00:04:04,580 I can take them to be one or whatever I like, 61 00:04:04,580 --> 00:04:07,970 and then I can solve for the pivot variables. 62 00:04:07,970 --> 00:04:11,870 So then it gives me a solution to Ax equals zero. 63 00:04:11,870 --> 00:04:15,820 And it's a solution that isn't all zeros. 64 00:04:15,820 --> 00:04:21,769 So, that's an important point that we'll 65 00:04:21,769 --> 00:04:26,350 use now in this lecture. 66 00:04:26,350 --> 00:04:30,110 So now I want to say what does it mean for a bunch of vectors 67 00:04:30,110 --> 00:04:31,470 to be independent. 68 00:04:31,470 --> 00:04:32,370 OK. 69 00:04:32,370 --> 00:04:36,276 So this is like the background that we know. 70 00:04:36,276 --> 00:04:37,900 Now I want to speak about independence. 71 00:04:41,500 --> 00:04:42,000 OK. 72 00:04:46,110 --> 00:04:48,740 Let's see. 73 00:04:48,740 --> 00:04:55,430 I can give you the abstract definition, and I will, 74 00:04:55,430 --> 00:05:00,325 but I would also like to give you the direct meaning. 75 00:05:03,700 --> 00:05:12,700 So the question is, when vectors x1, x2 up to -- 76 00:05:12,700 --> 00:05:19,860 Suppose I have n vectors are independent if. 77 00:05:19,860 --> 00:05:22,350 Now I have to give you -- 78 00:05:22,350 --> 00:05:25,560 or linearly independent -- 79 00:05:25,560 --> 00:05:30,470 I'll often just say and write independent for short. 80 00:05:30,470 --> 00:05:31,300 OK. 81 00:05:31,300 --> 00:05:33,580 I'll give you the full definition. 82 00:05:33,580 --> 00:05:36,920 These are just vectors in some vector space. 83 00:05:36,920 --> 00:05:39,360 I can take combinations of them. 84 00:05:39,360 --> 00:05:45,070 The question is, do any combinations give zero? 85 00:05:45,070 --> 00:05:47,210 If some combination of those vectors 86 00:05:47,210 --> 00:05:51,030 gives the zero vector, other than the combination 87 00:05:51,030 --> 00:05:55,520 of all zeros, then they're dependent. 88 00:05:55,520 --> 00:06:06,100 They're independent if no combination gives the zero 89 00:06:06,100 --> 00:06:08,550 vector -- 90 00:06:08,550 --> 00:06:17,070 and then I have, I'll have to put in an except the zero 91 00:06:17,070 --> 00:06:18,840 combination. 92 00:06:18,840 --> 00:06:21,160 So what do I mean by that? 93 00:06:21,160 --> 00:06:23,720 No combination gives the zero vector. 94 00:06:23,720 --> 00:06:29,950 Any combination c1 x1+c2 x2 plus, 95 00:06:29,950 --> 00:06:39,260 plus cn xn is not zero except for the zero combination. 96 00:06:39,260 --> 00:06:45,610 This is when all the c-s, all the c-s are zero. 97 00:06:45,610 --> 00:06:46,315 Then of course. 98 00:06:49,290 --> 00:06:52,100 That combination -- I know I'll get zero. 99 00:06:52,100 --> 00:06:55,950 But the question is, does any other combination give zero? 100 00:06:55,950 --> 00:06:59,970 If not, then the vectors are independent. 101 00:06:59,970 --> 00:07:02,630 If some other combination does give zero, 102 00:07:02,630 --> 00:07:04,880 the vectors are dependent. 103 00:07:04,880 --> 00:07:05,510 OK. 104 00:07:05,510 --> 00:07:08,420 Let's just take examples. 105 00:07:08,420 --> 00:07:14,510 Suppose I'm in, say, in two dimensional space. 106 00:07:14,510 --> 00:07:15,590 OK. 107 00:07:15,590 --> 00:07:16,940 I give you -- 108 00:07:16,940 --> 00:07:19,900 I'd like to first take an example -- 109 00:07:19,900 --> 00:07:24,480 let me take an example where I have a vector and twice that 110 00:07:24,480 --> 00:07:26,290 vector. 111 00:07:26,290 --> 00:07:29,720 So that's two vectors, V and 2V. 112 00:07:29,720 --> 00:07:33,830 Are those dependent or independent? 113 00:07:33,830 --> 00:07:35,820 Those are dependent for sure, right, 114 00:07:35,820 --> 00:07:40,380 because there's one vector is twice the other. 115 00:07:40,380 --> 00:07:42,250 One vector is twice as long as the other, 116 00:07:42,250 --> 00:07:44,780 so if the word dependent means anything, 117 00:07:44,780 --> 00:07:46,670 these should be dependent. 118 00:07:46,670 --> 00:07:47,910 And they are. 119 00:07:47,910 --> 00:07:51,860 And in fact, I would take two of the first -- 120 00:07:51,860 --> 00:07:56,750 so here's, here is a vector V and the other guy is a vector 121 00:07:56,750 --> 00:07:59,350 2V, that's my -- 122 00:07:59,350 --> 00:08:04,730 so there's a vector V1 and my next vector V2 is 2V1. 123 00:08:04,730 --> 00:08:08,630 Of course those are dependent, because two 124 00:08:08,630 --> 00:08:14,120 of these first vectors minus the second vector is zero. 125 00:08:14,120 --> 00:08:17,620 That's a combination of these two vectors that gives the zero 126 00:08:17,620 --> 00:08:18,120 vector. 127 00:08:18,120 --> 00:08:20,060 OK, that was clear. 128 00:08:20,060 --> 00:08:24,430 Suppose, suppose I have a vector -- 129 00:08:24,430 --> 00:08:25,060 here's another 130 00:08:25,060 --> 00:08:25,800 example. 131 00:08:25,800 --> 00:08:26,950 It's easy example. 132 00:08:26,950 --> 00:08:31,010 Suppose I have a vector and the other guy is the zero vector. 133 00:08:31,010 --> 00:08:36,820 Suppose I have a vector V1 and V2 is the zero vector. 134 00:08:36,820 --> 00:08:43,309 Then are those vectors dependent or independent? 135 00:08:43,309 --> 00:08:46,230 They're dependent again. 136 00:08:46,230 --> 00:08:50,480 You could say, well, this guy is zero times that one. 137 00:08:50,480 --> 00:08:52,970 This one is some combination of those. 138 00:08:52,970 --> 00:08:54,900 But let me write it the other way. 139 00:08:54,900 --> 00:09:01,110 Let me say -- what combination, how many V1s and how many V2s 140 00:09:01,110 --> 00:09:02,671 shall I take to get the zero 141 00:09:02,671 --> 00:09:03,170 vector? 142 00:09:05,810 --> 00:09:11,490 If, if V1 is like the vector two one and V2 is the zero vector, 143 00:09:11,490 --> 00:09:15,740 zero zero, then I would like to show 144 00:09:15,740 --> 00:09:18,640 that some combination of those gives the zero vector. 145 00:09:18,640 --> 00:09:19,820 What shall I take? 146 00:09:19,820 --> 00:09:22,660 How many V1s shall I take? 147 00:09:22,660 --> 00:09:23,540 Zero of them. 148 00:09:23,540 --> 00:09:25,310 Yeah, no, take no V1s. 149 00:09:25,310 --> 00:09:28,290 But how many V2s? 150 00:09:28,290 --> 00:09:28,970 Six. 151 00:09:28,970 --> 00:09:31,700 OK. 152 00:09:31,700 --> 00:09:34,790 Or five. 153 00:09:34,790 --> 00:09:38,590 Then -- in other words, the point is if the zero 154 00:09:38,590 --> 00:09:41,340 vector's in there, if the zero -- 155 00:09:41,340 --> 00:09:44,760 if one of these vectors is the zero vector, 156 00:09:44,760 --> 00:09:47,490 independence is dead, right? 157 00:09:47,490 --> 00:09:50,440 If one of those vectors is the zero vector then I could always 158 00:09:50,440 --> 00:09:51,970 take -- 159 00:09:51,970 --> 00:09:54,120 include that one and none of the others, 160 00:09:54,120 --> 00:09:59,370 and I would get the zero answer, and I would show dependence. 161 00:09:59,370 --> 00:10:00,230 OK. 162 00:10:00,230 --> 00:10:04,890 Now, let me, let me finally draw an example 163 00:10:04,890 --> 00:10:06,860 where they will be independent. 164 00:10:06,860 --> 00:10:10,900 Suppose that's V1 and that's V2. 165 00:10:10,900 --> 00:10:14,820 Those are surely independent, right? 166 00:10:14,820 --> 00:10:19,400 Any combination of V1 and V2, will not 167 00:10:19,400 --> 00:10:22,780 be zero except, the zero combination. 168 00:10:22,780 --> 00:10:24,390 So those would be independent. 169 00:10:24,390 --> 00:10:28,040 But now let me, let me stick in a third vector, V3. 170 00:10:31,100 --> 00:10:34,200 Independent or dependent now, those three vectors? 171 00:10:34,200 --> 00:10:37,390 So now n is three here. 172 00:10:37,390 --> 00:10:42,150 I'm in two dimensional space, whatever, I'm in the plane. 173 00:10:42,150 --> 00:10:47,390 I have three vectors that I didn't draw so carefully. 174 00:10:47,390 --> 00:10:50,520 I didn't even tell you what exactly they were. 175 00:10:50,520 --> 00:10:55,820 But what's this answer on dependent or independent? 176 00:10:55,820 --> 00:10:57,200 Dependent. 177 00:10:57,200 --> 00:11:00,570 How do I know those are dependent? 178 00:11:00,570 --> 00:11:05,570 How do I know that some combination of V1, V2, and V3 179 00:11:05,570 --> 00:11:09,010 gives me the zero vector? 180 00:11:09,010 --> 00:11:12,130 I know because of that. 181 00:11:12,130 --> 00:11:20,160 That's the key fact that tells me 182 00:11:20,160 --> 00:11:23,990 that three vectors in the plane have to be dependent. 183 00:11:23,990 --> 00:11:25,120 Why's that? 184 00:11:25,120 --> 00:11:28,650 What's the connection between the dependence of these three 185 00:11:28,650 --> 00:11:30,620 vectors and that fact? 186 00:11:30,620 --> 00:11:31,140 OK. 187 00:11:31,140 --> 00:11:33,350 So here's the connection. 188 00:11:33,350 --> 00:11:42,260 I take the matrix A that has V1 in its first column, V2 189 00:11:42,260 --> 00:11:46,560 in its second column, V3 in its third column. 190 00:11:46,560 --> 00:11:48,560 So it's got three columns. 191 00:11:48,560 --> 00:11:49,600 And V1 -- 192 00:11:49,600 --> 00:11:51,950 I don't know, that looks like about two one 193 00:11:51,950 --> 00:11:52,880 to me. 194 00:11:52,880 --> 00:11:56,430 V2 looks like it might be one two. 195 00:11:56,430 --> 00:12:01,990 V3 looks like it might be maybe two, maybe two and a half, 196 00:12:01,990 --> 00:12:03,570 minus one. 197 00:12:07,520 --> 00:12:08,230 OK. 198 00:12:08,230 --> 00:12:13,900 Those are my three vectors, and I put them in the columns of A. 199 00:12:13,900 --> 00:12:18,530 Now that matrix A is two by three. 200 00:12:18,530 --> 00:12:22,960 It fits this pattern, that where we know we've got extra 201 00:12:22,960 --> 00:12:25,900 variables, we know we have some free variables, 202 00:12:25,900 --> 00:12:28,640 we know that there's some combination -- 203 00:12:28,640 --> 00:12:34,650 and let me instead of x-s, let me call them c1, c2, and c3 -- 204 00:12:34,650 --> 00:12:39,380 that gives the zero vector. 205 00:12:39,380 --> 00:12:41,830 Sorry that my little bit of art got in the way. 206 00:12:41,830 --> 00:12:44,470 Do you see the point? 207 00:12:44,470 --> 00:12:48,600 When I have a matrix, I'm interested 208 00:12:48,600 --> 00:12:53,710 in whether its columns are dependent or independent. 209 00:12:53,710 --> 00:12:56,470 The columns are dependent if there 210 00:12:56,470 --> 00:12:58,480 is something in the null space. 211 00:12:58,480 --> 00:13:00,860 The columns are dependent because this, 212 00:13:00,860 --> 00:13:03,090 this thing in the null space says 213 00:13:03,090 --> 00:13:09,190 that c1 of that plus c2 of that plus c3 of this is zero. 214 00:13:09,190 --> 00:13:13,670 So in other words, I can go out some V1, out some more V2, 215 00:13:13,670 --> 00:13:16,260 back on V3, and end up zero. 216 00:13:19,010 --> 00:13:19,850 OK. 217 00:13:19,850 --> 00:13:25,290 So let -- here I've give the general, abstract definition, 218 00:13:25,290 --> 00:13:28,360 but let me repeat that definition -- 219 00:13:28,360 --> 00:13:31,840 this is like repeat -- 220 00:13:31,840 --> 00:13:39,270 let me call them Vs now. 221 00:13:39,270 --> 00:13:47,450 V1 up to Vn are the columns of a matrix A. 222 00:13:47,450 --> 00:13:50,120 In other words, this is telling me 223 00:13:50,120 --> 00:13:57,510 that if I'm in m dimensional space, 224 00:13:57,510 --> 00:14:00,630 like two dimensional space in the example, 225 00:14:00,630 --> 00:14:03,940 I can answer the dependence-independence 226 00:14:03,940 --> 00:14:10,180 question directly by putting those vectors 227 00:14:10,180 --> 00:14:11,580 in the columns of a matrix. 228 00:14:14,330 --> 00:14:30,380 They are independent if the null space of A, of A, is what? 229 00:14:33,230 --> 00:14:36,370 If I have a bunch of columns in a matrix, 230 00:14:36,370 --> 00:14:38,870 I'm looking at their combinations, 231 00:14:38,870 --> 00:14:44,850 but that's just A times the vector of c-s. 232 00:14:44,850 --> 00:14:48,020 And these columns will be independent 233 00:14:48,020 --> 00:14:56,390 if the null space of A is the zero vector. 234 00:15:01,270 --> 00:15:11,960 They are dependent if there's something else in there. 235 00:15:11,960 --> 00:15:17,380 If there's something else in the null space, if A times c 236 00:15:17,380 --> 00:15:25,240 gives the zero vector for some non-zero vector 237 00:15:25,240 --> 00:15:26,575 c in the null space. 238 00:15:29,160 --> 00:15:30,770 Then they're dependent, because that's 239 00:15:30,770 --> 00:15:34,380 telling me a combination of the columns gives the zero column. 240 00:15:34,380 --> 00:15:38,040 I think you're with be, because we've seen, 241 00:15:38,040 --> 00:15:40,340 like, lecture after lecture, we're 242 00:15:40,340 --> 00:15:43,500 looking at the combinations of the columns and asking, 243 00:15:43,500 --> 00:15:45,550 do we get zero or don't we? 244 00:15:45,550 --> 00:15:49,030 And now we're giving the official name, 245 00:15:49,030 --> 00:15:54,010 dependent if we do, independent if we don't. 246 00:15:54,010 --> 00:15:57,890 So I could express this in other words now. 247 00:15:57,890 --> 00:16:01,420 I could say the rank -- what's the rank in this independent 248 00:16:01,420 --> 00:16:02,450 case? 249 00:16:02,450 --> 00:16:05,020 The rank r of the, of the matrix, 250 00:16:05,020 --> 00:16:08,235 in the case of independent columns, is? 251 00:16:11,570 --> 00:16:14,740 So the columns are independent. 252 00:16:14,740 --> 00:16:18,560 So how many pivot columns have I got. 253 00:16:18,560 --> 00:16:19,850 All n. 254 00:16:19,850 --> 00:16:23,310 All the columns would be pivot columns, 255 00:16:23,310 --> 00:16:25,980 because free columns are telling me 256 00:16:25,980 --> 00:16:29,820 that they're a combination of earlier columns. 257 00:16:29,820 --> 00:16:33,080 So this would be the case where the rank is n. 258 00:16:33,080 --> 00:16:36,910 This would be the case where the rank is smaller than n. 259 00:16:39,650 --> 00:16:44,770 So in this case the rank is n and the null space of A 260 00:16:44,770 --> 00:16:48,750 is only the zero vector. 261 00:16:48,750 --> 00:16:50,800 And no free variables. 262 00:16:50,800 --> 00:16:52,400 No free variables. 263 00:16:56,520 --> 00:16:58,975 And this is the case yes free variables. 264 00:17:04,300 --> 00:17:09,589 If you'll allow me to stretch the English language that far. 265 00:17:09,589 --> 00:17:16,290 That's the case where we have, a combination 266 00:17:16,290 --> 00:17:19,040 that gives the zero column. 267 00:17:19,040 --> 00:17:23,089 I'm often interested in the case when my vectors are 268 00:17:23,089 --> 00:17:25,560 popped into a matrix. 269 00:17:25,560 --> 00:17:28,349 So the, the definition over there of independence 270 00:17:28,349 --> 00:17:31,270 didn't talk about any matrix. 271 00:17:31,270 --> 00:17:36,130 The vectors didn't have to be vectors in N dimensional space. 272 00:17:36,130 --> 00:17:38,530 And I want to give you some examples 273 00:17:38,530 --> 00:17:41,710 of vectors that aren't what you think 274 00:17:41,710 --> 00:17:43,970 of immediately as vectors. 275 00:17:43,970 --> 00:17:49,420 But most of the time, this is -- the vectors we think of are 276 00:17:49,420 --> 00:17:51,200 columns. 277 00:17:51,200 --> 00:17:54,570 And we can put them in a matrix. 278 00:17:54,570 --> 00:17:57,330 And then independence or dependence 279 00:17:57,330 --> 00:18:01,650 comes back to the null space. 280 00:18:01,650 --> 00:18:02,162 OK. 281 00:18:02,162 --> 00:18:03,620 So that's the idea of independence. 282 00:18:06,410 --> 00:18:13,770 Can I just, yeah, let me go on to spanning a 283 00:18:13,770 --> 00:18:19,480 What does it mean for a bunch of vectors to span a space? 284 00:18:19,480 --> 00:18:20,580 space. 285 00:18:20,580 --> 00:18:23,800 Well, actually, we've seen it already. 286 00:18:23,800 --> 00:18:28,040 You remember, if we had a columns in a matrix, 287 00:18:28,040 --> 00:18:33,130 we took all their combinations and that gave us 288 00:18:33,130 --> 00:18:36,640 the column space. 289 00:18:36,640 --> 00:18:41,250 Those vectors that we started with span that column space. 290 00:18:41,250 --> 00:18:44,490 So spanning a space means -- 291 00:18:44,490 --> 00:18:48,440 so let me move that important stuff right up. 292 00:18:53,130 --> 00:18:54,050 OK. 293 00:18:54,050 --> 00:19:02,400 So vectors -- let me call them, say, V1 up to -- 294 00:19:02,400 --> 00:19:06,360 call you some different letter, say Vl -- 295 00:19:06,360 --> 00:19:15,680 span a space, a subspace, or just a vector space 296 00:19:15,680 --> 00:19:24,790 I could say, span a space means, means 297 00:19:24,790 --> 00:19:40,270 the space consists of all combinations of those vectors. 298 00:19:46,464 --> 00:19:48,505 That's exactly what we did with the column space. 299 00:19:51,040 --> 00:19:54,740 So now I could say in shorthand the columns of a matrix 300 00:19:54,740 --> 00:19:57,090 span the column space. 301 00:19:57,090 --> 00:20:00,750 So you remember it's a bunch of vectors that have this property 302 00:20:00,750 --> 00:20:05,450 that they span a space, and actually if I give you a bunch 303 00:20:05,450 --> 00:20:06,480 of vectors and say -- 304 00:20:06,480 --> 00:20:10,720 OK, let S be the space that they span, 305 00:20:10,720 --> 00:20:14,870 in other words let S contain all their combinations, 306 00:20:14,870 --> 00:20:17,750 that space S will be the smallest 307 00:20:17,750 --> 00:20:21,830 space with those vectors in it, right? 308 00:20:21,830 --> 00:20:24,300 Because any space with those vectors in it 309 00:20:24,300 --> 00:20:28,730 must have all the combinations of those vectors in it. 310 00:20:28,730 --> 00:20:34,540 And if I stop there, then I've got the smallest space, 311 00:20:34,540 --> 00:20:37,300 and that's the space that they span. 312 00:20:37,300 --> 00:20:37,800 OK. 313 00:20:37,800 --> 00:20:39,150 So I'm just -- 314 00:20:39,150 --> 00:20:44,150 rather than, needing to say, take all linear combinations 315 00:20:44,150 --> 00:20:48,950 and put them in a space, I'm compressing that 316 00:20:48,950 --> 00:20:50,740 into the word span. 317 00:20:53,169 --> 00:20:53,835 Straightforward. 318 00:20:57,520 --> 00:21:00,015 So if I think of a, of the column space of a 319 00:21:00,015 --> 00:21:00,515 OK. matrix. 320 00:21:03,890 --> 00:21:07,380 I've got their -- so I start with the columns. 321 00:21:07,380 --> 00:21:08,930 I take all their combinations. 322 00:21:08,930 --> 00:21:10,760 That gives me the columns space. 323 00:21:10,760 --> 00:21:13,440 They span the column space. 324 00:21:13,440 --> 00:21:16,230 Now are they independent? 325 00:21:16,230 --> 00:21:19,620 Maybe yes, maybe no. 326 00:21:19,620 --> 00:21:23,060 It depends on the particular columns that went into that 327 00:21:23,060 --> 00:21:24,470 matrix. 328 00:21:24,470 --> 00:21:29,700 But obviously I'm highly interested in a set 329 00:21:29,700 --> 00:21:36,870 of vectors that spans a space and is independent. 330 00:21:36,870 --> 00:21:41,630 That's, that means like I've got the right number of vectors. 331 00:21:41,630 --> 00:21:47,580 If I didn't have all of them, I wouldn't have my whole space. 332 00:21:47,580 --> 00:21:50,290 If I had more than that, they probably wouldn't -- 333 00:21:50,290 --> 00:21:52,450 they wouldn't be independent. 334 00:21:52,450 --> 00:21:56,860 So, like, basis -- and that's the word that's coming -- 335 00:21:56,860 --> 00:21:58,600 is just right. 336 00:21:58,600 --> 00:22:01,310 So here let me put what that word means. 337 00:22:01,310 --> 00:22:14,270 A basis for a vector space is, is a, is a sequence of vectors 338 00:22:14,270 --> 00:22:14,770 -- 339 00:22:20,050 --> 00:22:27,040 shall I call them V1, V2, up to let me say Vd now, 340 00:22:27,040 --> 00:22:33,600 I'll stop with that letters -- that has two properties. 341 00:22:33,600 --> 00:22:36,230 I've got enough vectors and not too many. 342 00:22:36,230 --> 00:22:39,340 It's a natural idea of a basis. 343 00:22:39,340 --> 00:22:42,570 So a basis is a bunch of vectors in the space 344 00:22:42,570 --> 00:22:47,200 and it's a so it's a sequence of vectors with two properties, 345 00:22:47,200 --> 00:22:50,050 with two properties. 346 00:22:54,760 --> 00:22:57,675 One, they are independent. 347 00:23:05,020 --> 00:23:07,990 And two -- you know what's coming? 348 00:23:07,990 --> 00:23:10,040 -- they span the space. 349 00:23:20,560 --> 00:23:21,880 OK. 350 00:23:21,880 --> 00:23:25,050 Let me take -- 351 00:23:25,050 --> 00:23:28,510 so time for examples, of course. 352 00:23:28,510 --> 00:23:32,320 So I'm asking you now to put definition one, 353 00:23:32,320 --> 00:23:38,030 the definition of independence, together with definition two, 354 00:23:38,030 --> 00:23:41,800 and let's look at examples, because this is -- 355 00:23:41,800 --> 00:23:44,840 this combination means the set I've -- 356 00:23:44,840 --> 00:23:47,690 of vectors I have is just right, and the -- 357 00:23:47,690 --> 00:23:51,210 so that this idea of a basis will be central. 358 00:23:51,210 --> 00:23:54,480 I'll always be asking you now for a basis. 359 00:23:54,480 --> 00:23:58,760 Whenever I look at a subspace, if I ask you for -- 360 00:23:58,760 --> 00:24:00,760 if you give me a basis for that subspace, 361 00:24:00,760 --> 00:24:02,920 you've told me what it is. 362 00:24:02,920 --> 00:24:07,430 You've told me everything I need to know about that subspace. 363 00:24:07,430 --> 00:24:10,520 Those -- I take their combinations and I know that I 364 00:24:10,520 --> 00:24:12,460 need all the combinations. 365 00:24:12,460 --> 00:24:13,210 OK. 366 00:24:13,210 --> 00:24:13,790 Examples. 367 00:24:13,790 --> 00:24:16,640 OK, so examples of a basis. 368 00:24:16,640 --> 00:24:20,060 Let me start with two dimensional space. 369 00:24:20,060 --> 00:24:22,350 Suppose the space -- say example. 370 00:24:26,520 --> 00:24:31,150 The space is, oh, let's make it R^3. 371 00:24:31,150 --> 00:24:35,980 Real three dimensional space. 372 00:24:35,980 --> 00:24:37,330 Give me one basis. 373 00:24:37,330 --> 00:24:38,705 One basis is? 374 00:24:43,970 --> 00:24:49,010 So I want some vectors, because if I ask you for a basis, 375 00:24:49,010 --> 00:24:53,390 I'm asking you for vectors, a little list of vectors. 376 00:24:53,390 --> 00:24:57,610 And it should be just right. 377 00:24:57,610 --> 00:25:02,880 So what would be a basis for three dimensional space? 378 00:25:02,880 --> 00:25:05,540 Well, the first basis that comes to mind, why don't we 379 00:25:05,540 --> 00:25:07,240 write that down. 380 00:25:07,240 --> 00:25:09,220 The first basis that comes to mind 381 00:25:09,220 --> 00:25:18,850 is this vector, this vector, and this vector. 382 00:25:18,850 --> 00:25:19,790 OK. 383 00:25:19,790 --> 00:25:23,540 That's one basis. 384 00:25:23,540 --> 00:25:27,190 Not the only basis, that's going to be my point. 385 00:25:27,190 --> 00:25:30,930 But let's just see -- yes, that's a basis. 386 00:25:30,930 --> 00:25:33,500 Are, are those vectors independent? 387 00:25:36,110 --> 00:25:40,120 So that's the like the x, y, z axes, so if those are not 388 00:25:40,120 --> 00:25:41,780 independent, we're in trouble. 389 00:25:41,780 --> 00:25:43,580 Certainly, they are. 390 00:25:43,580 --> 00:25:48,630 Take a combination c1 of this vector plus c2 of this vector 391 00:25:48,630 --> 00:25:51,340 plus c3 of that vector and try to make 392 00:25:51,340 --> 00:25:55,290 it give the zero vector. 393 00:25:55,290 --> 00:25:57,110 What are the c-s? 394 00:25:57,110 --> 00:26:03,080 If c1 of that plus c2 of that plus c3 of that gives me 0 0 0, 395 00:26:03,080 --> 00:26:05,600 then the c-s are all -- 396 00:26:05,600 --> 00:26:06,920 0, right. 397 00:26:06,920 --> 00:26:09,980 So that's the test for independence. 398 00:26:09,980 --> 00:26:16,210 In the language of matrices, which was under that board, 399 00:26:16,210 --> 00:26:19,700 I could make those the columns of a matrix. 400 00:26:19,700 --> 00:26:22,880 Well, it would be the identity matrix. 401 00:26:22,880 --> 00:26:25,590 Then I would ask, what's the null space of the identity 402 00:26:25,590 --> 00:26:26,760 matrix? 403 00:26:26,760 --> 00:26:30,560 And you would say it's only the zero vector. 404 00:26:30,560 --> 00:26:34,740 And I would say, fine, then the columns are independent. 405 00:26:34,740 --> 00:26:38,370 The only thing -- the identity times a vector giving zero, 406 00:26:38,370 --> 00:26:40,810 the only vector that does that is zero. 407 00:26:40,810 --> 00:26:41,980 OK. 408 00:26:41,980 --> 00:26:45,710 Now that's not the only basis. 409 00:26:45,710 --> 00:26:46,660 Far from it. 410 00:26:46,660 --> 00:26:50,680 Tell me another basis, a second basis, another basis. 411 00:26:57,560 --> 00:27:03,030 So, give me -- well, I'll just start it out. 412 00:27:03,030 --> 00:27:04,090 One one two. 413 00:27:06,950 --> 00:27:10,780 Two two five. 414 00:27:10,780 --> 00:27:14,420 Suppose I stopped there. 415 00:27:14,420 --> 00:27:20,980 Has that little bunch of vectors got the properties that 416 00:27:20,980 --> 00:27:24,350 I'm asking for in a basis for R^3? 417 00:27:24,350 --> 00:27:26,880 We're looking for a basis for R^3. 418 00:27:26,880 --> 00:27:30,430 Are they independent, those two column vectors? 419 00:27:30,430 --> 00:27:31,330 Yes. 420 00:27:31,330 --> 00:27:33,610 Do they span R^3? 421 00:27:33,610 --> 00:27:34,340 No. 422 00:27:34,340 --> 00:27:36,440 Our feeling is no. 423 00:27:36,440 --> 00:27:37,550 Our feeling is no. 424 00:27:37,550 --> 00:27:41,640 Our feeling is that there're some vectors in R3 that 425 00:27:41,640 --> 00:27:44,120 are not combinations of those. 426 00:27:44,120 --> 00:27:44,930 OK. 427 00:27:44,930 --> 00:27:47,830 So suppose I add in -- 428 00:27:47,830 --> 00:27:50,390 I need another vector then, because these two 429 00:27:50,390 --> 00:27:52,021 don't span the space. 430 00:27:52,021 --> 00:27:52,520 OK. 431 00:27:52,520 --> 00:27:56,360 Now it would be foolish for me to put in three three seven, 432 00:27:56,360 --> 00:27:58,810 right, as the third vector. 433 00:27:58,810 --> 00:27:59,950 That would be a goof. 434 00:27:59,950 --> 00:28:03,380 Because that, if I put in three three seven, 435 00:28:03,380 --> 00:28:07,680 those vectors would be dependent, right? 436 00:28:07,680 --> 00:28:09,440 If I put in three three seven, it 437 00:28:09,440 --> 00:28:12,170 would be the sum of those two, it 438 00:28:12,170 --> 00:28:15,020 would lie in the same plane as those. 439 00:28:15,020 --> 00:28:18,590 It wouldn't be independent. 440 00:28:18,590 --> 00:28:21,930 My attempt to create a basis would be dead. 441 00:28:21,930 --> 00:28:26,420 But if I take -- so what vector can I take? 442 00:28:26,420 --> 00:28:30,340 I can take any vector that's not in that plane. 443 00:28:30,340 --> 00:28:31,430 Let me try -- 444 00:28:31,430 --> 00:28:33,590 I hope that 3 3 8 would do it. 445 00:28:37,340 --> 00:28:40,470 At least it's not the sum of those two vectors. 446 00:28:40,470 --> 00:28:44,160 But I believe that's a basis. 447 00:28:44,160 --> 00:28:49,310 And what's the test then, for that to be a basis? 448 00:28:49,310 --> 00:28:53,160 Because I just picked those numbers, and if I had picked, 449 00:28:53,160 --> 00:29:02,300 5 7 -14 how would we know do we have a basis or don't we? 450 00:29:02,300 --> 00:29:05,430 You would put them in the columns of a matrix, 451 00:29:05,430 --> 00:29:09,280 and you would do elimination, row reduction -- 452 00:29:09,280 --> 00:29:15,340 and you would see do you get any free variables 453 00:29:15,340 --> 00:29:18,500 or are all the columns pivot columns. 454 00:29:18,500 --> 00:29:20,590 Well now actually we have a square -- 455 00:29:20,590 --> 00:29:22,760 the matrix would be three by three. 456 00:29:22,760 --> 00:29:26,260 So, what's the test on the matrix then? 457 00:29:29,280 --> 00:29:34,790 The matrix -- so in this case, when my space is R^3 and I have 458 00:29:34,790 --> 00:29:44,740 three vectors, my matrix is square and what I asking about 459 00:29:44,740 --> 00:29:49,780 that matrix in order for those columns to be a basis? 460 00:29:49,780 --> 00:29:51,190 So in this -- 461 00:29:51,190 --> 00:30:06,380 for R^n, if I have -- n vectors give a basis if the n by n 462 00:30:06,380 --> 00:30:19,850 matrix with those columns, with those columns, is what? 463 00:30:19,850 --> 00:30:24,070 What's the requirement on that matrix? 464 00:30:24,070 --> 00:30:27,010 Invertible, right, right. 465 00:30:27,010 --> 00:30:28,590 The matrix should be invertible. 466 00:30:28,590 --> 00:30:33,187 For a square matrix, that's the, that's the perfect answer. 467 00:30:33,187 --> 00:30:33,770 Is invertible. 468 00:30:38,260 --> 00:30:43,380 So that's when, that's when the space is the whole space R^n. 469 00:30:46,140 --> 00:30:50,360 Let me, let me be sure you're with me here. 470 00:30:50,360 --> 00:30:53,540 Let me remove that. 471 00:30:53,540 --> 00:31:01,580 Are those two vectors a basis for any space at all? 472 00:31:01,580 --> 00:31:04,020 Is there a vector space that those really 473 00:31:04,020 --> 00:31:08,540 are a basis for, those, that pair of vectors, this guy 474 00:31:08,540 --> 00:31:11,580 and this 1, 1 1 2 and 2 2 5? 475 00:31:11,580 --> 00:31:15,360 Is there a space for which that's a basis? 476 00:31:15,360 --> 00:31:16,220 Sure. 477 00:31:16,220 --> 00:31:21,970 They're independent, so they satisfy the first requirement, 478 00:31:21,970 --> 00:31:24,740 so what space shall I take for them to be a basis 479 00:31:24,740 --> 00:31:25,240 of? 480 00:31:25,240 --> 00:31:29,080 What spaces will they be a basis for? 481 00:31:29,080 --> 00:31:31,180 The one they span. 482 00:31:31,180 --> 00:31:32,350 Their combinations. 483 00:31:32,350 --> 00:31:33,920 It's a plane, right? 484 00:31:33,920 --> 00:31:36,340 It'll be a plane inside R^3. 485 00:31:36,340 --> 00:31:40,930 So if I take this vector 1 1 2, say it goes there, 486 00:31:40,930 --> 00:31:44,240 and this vector 2 2 5, say it goes there, 487 00:31:44,240 --> 00:31:50,797 those are a basis for -- because they span a plane. 488 00:31:50,797 --> 00:31:52,880 And they're a basis for the plane, because they're 489 00:31:52,880 --> 00:31:53,550 independent. 490 00:31:53,550 --> 00:31:56,800 If I stick in some third guy, like 3 3 7, 491 00:31:56,800 --> 00:32:01,050 which is in the plane -- suppose I put in, try to put in 3 3 7, 492 00:32:01,050 --> 00:32:05,740 then the three vectors would still span the plane, 493 00:32:05,740 --> 00:32:09,180 but they wouldn't be a basis anymore because they're not 494 00:32:09,180 --> 00:32:12,260 independent anymore. 495 00:32:12,260 --> 00:32:19,420 So, we're looking at the question of -- 496 00:32:19,420 --> 00:32:21,360 again, 497 00:32:21,360 --> 00:32:25,390 OK. the case with independent columns 498 00:32:25,390 --> 00:32:32,090 is the case where the column vectors span the column space. 499 00:32:32,090 --> 00:32:35,830 They're independent, so they're a basis for the column space. 500 00:32:35,830 --> 00:32:36,520 OK. 501 00:32:36,520 --> 00:32:42,490 So now there's one bit of intuition. 502 00:32:42,490 --> 00:32:46,420 Let me go back to all of R^n. 503 00:32:46,420 --> 00:32:48,150 So I -- where I put 3 3 8. 504 00:32:51,110 --> 00:32:51,740 OK. 505 00:32:51,740 --> 00:32:56,510 The first message is that the basis is not unique, right. 506 00:32:56,510 --> 00:32:58,090 There's zillions of bases. 507 00:32:58,090 --> 00:33:02,790 I take any invertible three by three matrix, 508 00:33:02,790 --> 00:33:07,800 its columns are a basis for R^3. 509 00:33:07,800 --> 00:33:11,200 The column space is R^3, and if those, 510 00:33:11,200 --> 00:33:15,180 if that matrix is invertible, those columns are independent, 511 00:33:15,180 --> 00:33:17,100 I've got a basis for R^3. 512 00:33:17,100 --> 00:33:19,860 So there're many, many bases. 513 00:33:19,860 --> 00:33:27,730 But there is something in common for all those bases. 514 00:33:27,730 --> 00:33:32,810 There's something that this basis shares with that basis 515 00:33:32,810 --> 00:33:35,880 and every other basis for R^3. 516 00:33:35,880 --> 00:33:37,515 And what's that? 517 00:33:40,040 --> 00:33:47,410 Well, you saw it coming, because when I stopped here and asked 518 00:33:47,410 --> 00:33:51,180 if that was a basis for R^3, you said no. 519 00:33:51,180 --> 00:33:54,730 And I know that you said no because you knew there 520 00:33:54,730 --> 00:33:57,410 weren't enough vectors there. 521 00:33:57,410 --> 00:34:03,670 And the great fact is that there're many, many bases, 522 00:34:03,670 --> 00:34:12,139 but -- let me put in somebody else, just for variety. 523 00:34:12,139 --> 00:34:14,850 There are many, many bases, but they all 524 00:34:14,850 --> 00:34:18,639 have the same number of vectors. 525 00:34:18,639 --> 00:34:21,350 If we're talking about the space R^3, 526 00:34:21,350 --> 00:34:25,040 then that number of vectors is three. 527 00:34:25,040 --> 00:34:27,650 If we're talking about the space R^n, 528 00:34:27,650 --> 00:34:31,320 then that number of vectors is n. 529 00:34:31,320 --> 00:34:36,389 If we're talking about some other space, 530 00:34:36,389 --> 00:34:41,570 the column space of some matrix, or the null space of some 531 00:34:41,570 --> 00:34:45,420 matrix, or some other space that we haven't even thought 532 00:34:45,420 --> 00:34:52,820 of, then that still is true that every basis -- 533 00:34:52,820 --> 00:34:57,270 that there're lots of bases but every basis has the same number 534 00:34:57,270 --> 00:34:57,950 of vectors. 535 00:34:57,950 --> 00:35:02,000 Let me write that great fact down. 536 00:35:02,000 --> 00:35:07,940 Every basis -- we're given a space. 537 00:35:07,940 --> 00:35:10,680 Given a space. 538 00:35:13,950 --> 00:35:18,840 R^3 or R^n or some other column space of a matrix or the null 539 00:35:18,840 --> 00:35:21,790 space of a matrix or some other vector space. 540 00:35:21,790 --> 00:35:25,200 Then the great fact is that every basis 541 00:35:25,200 --> 00:35:40,365 for this, for the space has the same number of vectors. 542 00:35:47,750 --> 00:35:50,770 If one basis has six vectors, then every other basis 543 00:35:50,770 --> 00:35:52,770 has six vectors. 544 00:35:52,770 --> 00:35:56,050 So that number six is telling me like 545 00:35:56,050 --> 00:35:59,250 it's telling me how big is the space. 546 00:35:59,250 --> 00:36:01,640 It's telling me how many vectors do 547 00:36:01,640 --> 00:36:04,650 I have to have to have a basis. 548 00:36:04,650 --> 00:36:08,440 And of course we're seeing it this way. 549 00:36:08,440 --> 00:36:10,900 That number six, if we had seven vectors, 550 00:36:10,900 --> 00:36:13,150 then we've got too many. 551 00:36:13,150 --> 00:36:17,070 If we have five vectors we haven't got enough. 552 00:36:17,070 --> 00:36:21,860 Sixes are like just right for whatever space that is. 553 00:36:21,860 --> 00:36:24,600 And what do we call that number? 554 00:36:24,600 --> 00:36:29,550 That number is -- now I'm ready for the last definition today. 555 00:36:29,550 --> 00:36:33,650 It's the dimension of that space. 556 00:36:33,650 --> 00:36:37,410 So every basis for a space has the same number of vectors in 557 00:36:37,410 --> 00:36:37,910 it. 558 00:36:37,910 --> 00:36:41,860 Not the same vectors, all sorts of bases -- 559 00:36:41,860 --> 00:36:44,850 but the same number of vectors is always the same, 560 00:36:44,850 --> 00:36:47,010 and that number is the dimension. 561 00:36:47,010 --> 00:36:47,900 This is definitional. 562 00:36:50,780 --> 00:36:58,245 This number is the dimension of the space. 563 00:37:03,390 --> 00:37:03,890 OK. 564 00:37:06,670 --> 00:37:08,417 OK. 565 00:37:08,417 --> 00:37:09,375 Let's do some examples. 566 00:37:12,930 --> 00:37:14,410 Because now we've got definitions. 567 00:37:14,410 --> 00:37:17,550 Let me repeat the four things, the four words that 568 00:37:17,550 --> 00:37:19,270 have now got defined. 569 00:37:19,270 --> 00:37:23,320 Independence, that looks at combinations not 570 00:37:23,320 --> 00:37:24,560 being zero. 571 00:37:24,560 --> 00:37:27,700 Spanning, that looks at all the combinations. 572 00:37:27,700 --> 00:37:30,580 Basis, that's the one that combines 573 00:37:30,580 --> 00:37:32,360 independence and spanning. 574 00:37:32,360 --> 00:37:36,460 And now we've got the idea of the dimension of a space. 575 00:37:36,460 --> 00:37:40,470 It's the number of vectors in any basis, 576 00:37:40,470 --> 00:37:43,160 because all bases have the same number. 577 00:37:43,160 --> 00:37:44,580 OK. 578 00:37:44,580 --> 00:37:47,750 Let's take examples. 579 00:37:47,750 --> 00:37:53,790 Suppose I take, my space is -- examples now -- 580 00:37:53,790 --> 00:37:58,170 space is the, say, the column space of this matrix. 581 00:37:58,170 --> 00:38:00,240 Let me write down a matrix. 582 00:38:00,240 --> 00:38:06,370 1 1 1, 2 1 2, and I'll -- just to make it clear, 583 00:38:06,370 --> 00:38:12,430 I'll take the sum there, 3 2 3, and let me take the sum of all 584 00:38:12,430 --> 00:38:15,050 -- oh, let me put in one -- yeah, 585 00:38:15,050 --> 00:38:18,530 I'll put in one one one again. 586 00:38:18,530 --> 00:38:20,300 OK. 587 00:38:20,300 --> 00:38:21,260 So that's four vectors. 588 00:38:24,710 --> 00:38:29,310 OK, do they span the column space of that matrix? 589 00:38:29,310 --> 00:38:35,290 Let me repeat, do they span the column space of that matrix? 590 00:38:35,290 --> 00:38:37,700 By definition, that's what the column space -- 591 00:38:37,700 --> 00:38:39,280 Yes. where it comes from. 592 00:38:39,280 --> 00:38:41,470 Are they a basis for the column space? 593 00:38:41,470 --> 00:38:43,920 Are they independent? 594 00:38:43,920 --> 00:38:45,860 No, they're not independent. 595 00:38:45,860 --> 00:38:49,460 There's something in that null space. 596 00:38:49,460 --> 00:38:55,140 Maybe we can -- so let's look at the null space of the matrix. 597 00:38:55,140 --> 00:38:58,490 Tell me a vector that's in the null space of that matrix. 598 00:39:01,090 --> 00:39:05,320 So I'm looking for some vector that combines those columns 599 00:39:05,320 --> 00:39:08,350 and produces the zero column. 600 00:39:08,350 --> 00:39:10,680 Or in other words, I'm looking for solutions 601 00:39:10,680 --> 00:39:12,280 to A X equals zero. 602 00:39:12,280 --> 00:39:16,930 So tell me a vector in the null space. 603 00:39:16,930 --> 00:39:21,070 Maybe -- well, this was, this column was that one plus that 604 00:39:21,070 --> 00:39:25,030 one, so maybe if I have one of those and minus one of those 605 00:39:25,030 --> 00:39:26,560 that would be a vector in the null 606 00:39:26,560 --> 00:39:27,060 space. 607 00:39:29,450 --> 00:39:33,440 So, you've already told me now, are those vectors independent, 608 00:39:33,440 --> 00:39:38,140 the answer is -- those column vectors, the answer is -- 609 00:39:38,140 --> 00:39:38,670 no. 610 00:39:38,670 --> 00:39:39,170 Right? 611 00:39:39,170 --> 00:39:40,770 They're not independent. 612 00:39:40,770 --> 00:39:44,060 Because -- you knew they weren't independent. 613 00:39:44,060 --> 00:39:47,060 Anyway, minus one of this minus one 614 00:39:47,060 --> 00:39:50,530 of this plus one of this zero of that is the zero vector. 615 00:39:54,120 --> 00:39:55,660 OK, so they're not independent. 616 00:39:55,660 --> 00:39:55,720 OK. 617 00:39:55,720 --> 00:39:57,500 They span, but they're not independent. 618 00:39:57,500 --> 00:40:04,560 Tell me a basis for that column space. 619 00:40:04,560 --> 00:40:06,300 What's a basis for the column space? 620 00:40:06,300 --> 00:40:09,220 These are all the questions that the homework asks, the quizzes 621 00:40:09,220 --> 00:40:11,800 ask, the final exam will ask. 622 00:40:11,800 --> 00:40:17,710 Find a basis for the column space of this matrix. 623 00:40:17,710 --> 00:40:20,280 OK. 624 00:40:20,280 --> 00:40:22,500 Now there's many answers, but give me 625 00:40:22,500 --> 00:40:25,590 the most natural answer. 626 00:40:25,590 --> 00:40:29,670 Columns one and two. 627 00:40:29,670 --> 00:40:31,170 Columns one and two. 628 00:40:31,170 --> 00:40:32,410 That's the natural answer. 629 00:40:32,410 --> 00:40:35,320 Those are the pivot columns, because, I mean, 630 00:40:35,320 --> 00:40:37,140 we s- we begin systematically. 631 00:40:37,140 --> 00:40:39,300 We look at the first column, it's OK. 632 00:40:39,300 --> 00:40:41,330 We can put that in the basis. 633 00:40:41,330 --> 00:40:43,550 We look at the second column, it's OK. 634 00:40:43,550 --> 00:40:46,300 We can put that in the basis. 635 00:40:46,300 --> 00:40:48,960 The third column we can't put in the basis. 636 00:40:48,960 --> 00:40:54,070 The fourth column we can't, again. 637 00:40:54,070 --> 00:40:57,570 So the rank of the matrix is -- 638 00:40:57,570 --> 00:40:59,850 what's the rank of our matrix? 639 00:40:59,850 --> 00:41:01,210 Two. 640 00:41:01,210 --> 00:41:02,000 Two. 641 00:41:02,000 --> 00:41:06,610 And, and now that rank is also -- we also have another word. 642 00:41:06,610 --> 00:41:08,710 We, we have a great theorem here. 643 00:41:08,710 --> 00:41:23,060 The rank of A, that rank r, is the number of pivot columns 644 00:41:23,060 --> 00:41:25,340 and it's also -- 645 00:41:25,340 --> 00:41:27,890 well, so now please use my new word. 646 00:41:32,640 --> 00:41:34,700 This, it's the number two, of course, 647 00:41:34,700 --> 00:41:41,790 two is the rank of my matrix, it's 648 00:41:41,790 --> 00:41:44,440 the number of pivot columns, those pivot columns form 649 00:41:44,440 --> 00:41:49,104 a basis, of course, so what's two? 650 00:41:49,104 --> 00:41:49,895 It's the dimension. 651 00:41:52,820 --> 00:41:55,580 The rank of A, the number of pivot columns, 652 00:41:55,580 --> 00:42:02,555 is the dimension of the column space. 653 00:42:07,450 --> 00:42:08,840 Of course, you say. 654 00:42:08,840 --> 00:42:10,670 It had to be. 655 00:42:10,670 --> 00:42:12,000 Right. 656 00:42:12,000 --> 00:42:17,440 But just watch, look for one moment at the, 657 00:42:17,440 --> 00:42:20,240 the language, the way the English words 658 00:42:20,240 --> 00:42:21,810 get involved here. 659 00:42:21,810 --> 00:42:28,610 I take the rank of a matrix, the rank of a matrix. 660 00:42:28,610 --> 00:42:34,580 It's a number of columns and it's the dimension of -- 661 00:42:34,580 --> 00:42:37,810 not the dimension of the matrix, that's what I want to say. 662 00:42:37,810 --> 00:42:43,890 It's the dimension of a space, a subspace, the column space. 663 00:42:43,890 --> 00:42:46,800 Do you see, I don't take the dimension of A. 664 00:42:46,800 --> 00:42:49,600 That's not what I want. 665 00:42:49,600 --> 00:42:53,190 I'm looking for the dimension of the column space of A. 666 00:42:53,190 --> 00:42:56,050 If you use those words right, it shows you've got the idea 667 00:42:56,050 --> 00:42:57,400 right. 668 00:42:57,400 --> 00:42:59,170 Similarly here. 669 00:42:59,170 --> 00:43:03,430 I don't talk about the rank of a subspace. 670 00:43:03,430 --> 00:43:05,440 It's a matrix that has a rank. 671 00:43:05,440 --> 00:43:08,040 I talk about the rank of a matrix. 672 00:43:08,040 --> 00:43:11,750 And the beauty is that these definitions just 673 00:43:11,750 --> 00:43:14,160 merge so that the rank of a matrix 674 00:43:14,160 --> 00:43:16,170 is the dimension of its column space. 675 00:43:16,170 --> 00:43:18,840 And in this example it's two. 676 00:43:18,840 --> 00:43:22,570 And then the further question is, what's a basis? 677 00:43:22,570 --> 00:43:25,450 And the first two columns are a basis. 678 00:43:25,450 --> 00:43:27,500 Tell me another basis. 679 00:43:27,500 --> 00:43:29,710 Another basis for the columns space. 680 00:43:29,710 --> 00:43:31,390 You see I just keep hammering away. 681 00:43:31,390 --> 00:43:35,140 I apologize, but it's, I have to be sure you 682 00:43:35,140 --> 00:43:36,660 have the idea of basis. 683 00:43:36,660 --> 00:43:40,940 Tell me another basis for the column space. 684 00:43:40,940 --> 00:43:47,410 Well, you could take columns one and three. 685 00:43:47,410 --> 00:43:49,930 That would be a basis for the column space. 686 00:43:49,930 --> 00:43:53,220 Or columns two and three would be a basis. 687 00:43:53,220 --> 00:43:55,660 Or columns two and four. 688 00:43:55,660 --> 00:43:58,871 Or tell me another basis that's not made out of those columns 689 00:43:58,871 --> 00:43:59,370 at all? 690 00:44:03,290 --> 00:44:07,180 So -- I guess I'm giving you infinitely many possibilities, 691 00:44:07,180 --> 00:44:11,150 so I can't expect a unanimous answer here. 692 00:44:11,150 --> 00:44:14,770 I'll tell you -- but let's look at another basis, though. 693 00:44:14,770 --> 00:44:17,836 I'll just -- because it's only one out of zillions, 694 00:44:17,836 --> 00:44:19,960 I'm going to put it down and I'm going to erase it. 695 00:44:19,960 --> 00:44:26,680 Another basis for the column space would be -- 696 00:44:26,680 --> 00:44:27,879 let's see. 697 00:44:27,879 --> 00:44:29,670 I'll put in some things that are not there. 698 00:44:29,670 --> 00:44:33,830 Say, oh well, just to make it -- my life easy, 2 2 2. 699 00:44:33,830 --> 00:44:37,040 That's in the column space. 700 00:44:37,040 --> 00:44:40,380 And, that was sort of obvious. 701 00:44:40,380 --> 00:44:43,458 Let me take the sum of those, say 6 4 6. 702 00:44:46,810 --> 00:44:52,040 Or the sum of all of the columns, 7 5 7, why not. 703 00:44:52,040 --> 00:44:54,910 That's in the column space. 704 00:44:54,910 --> 00:44:59,540 Those are independent and I've got the number right, 705 00:44:59,540 --> 00:45:01,290 I've got two. 706 00:45:01,290 --> 00:45:02,940 Actually, this is a key point. 707 00:45:05,890 --> 00:45:09,170 If you know the dimension of the space you're working with, 708 00:45:09,170 --> 00:45:14,480 and we know that this column -- we know that the dimension, 709 00:45:14,480 --> 00:45:19,430 DIM, the dimension of the column space is two. 710 00:45:23,730 --> 00:45:30,120 If you know the dimension, then -- 711 00:45:30,120 --> 00:45:33,670 and we have a couple of vectors that are independent, 712 00:45:33,670 --> 00:45:35,710 they'll automatically be a 713 00:45:35,710 --> 00:45:36,550 basis. 714 00:45:36,550 --> 00:45:38,890 If we've got the number of vectors right, 715 00:45:38,890 --> 00:45:43,760 two vectors in this case, then if they're independent, 716 00:45:43,760 --> 00:45:47,037 they can't help but span the space. 717 00:45:47,037 --> 00:45:48,620 Because if they didn't span the space, 718 00:45:48,620 --> 00:45:52,130 there'd be a third guy to help span the space, 719 00:45:52,130 --> 00:45:54,160 but it couldn't be independent. 720 00:45:54,160 --> 00:45:58,190 So, it just has to be independent 721 00:45:58,190 --> 00:46:01,640 if we've got the numbers right. 722 00:46:01,640 --> 00:46:02,590 And they span. 723 00:46:02,590 --> 00:46:03,280 OK. 724 00:46:03,280 --> 00:46:04,100 Very good. 725 00:46:04,100 --> 00:46:06,060 So you got the dimension of a space. 726 00:46:06,060 --> 00:46:08,870 So this was another basis that I just invented. 727 00:46:08,870 --> 00:46:09,470 OK. 728 00:46:09,470 --> 00:46:16,270 Now, now I get to ask about the null space. 729 00:46:16,270 --> 00:46:18,320 What's the dimension of the null space? 730 00:46:18,320 --> 00:46:20,700 So we, we got a great fact there, 731 00:46:20,700 --> 00:46:28,560 the dimension of the column space is the rank. 732 00:46:28,560 --> 00:46:30,760 Now I want to ask you about the null space. 733 00:46:30,760 --> 00:46:34,190 That's the other part of the lecture, 734 00:46:34,190 --> 00:46:38,320 and it'll go on to the next lecture. 735 00:46:38,320 --> 00:46:40,320 OK. 736 00:46:40,320 --> 00:46:44,230 So we know the dimension of the column space is two, the rank. 737 00:46:44,230 --> 00:46:46,280 What about the null space? 738 00:46:46,280 --> 00:46:47,870 This is a vector in the null space. 739 00:46:47,870 --> 00:46:49,770 Are there other vectors in the null space? 740 00:46:52,280 --> 00:46:53,530 Yes or no? 741 00:46:53,530 --> 00:46:54,870 Yes. 742 00:46:54,870 --> 00:46:58,820 So this isn't a basis because it's doesn't span, right? 743 00:46:58,820 --> 00:47:01,970 There's more in the null space than we've got so far. 744 00:47:01,970 --> 00:47:04,570 I need another vector at least. 745 00:47:04,570 --> 00:47:08,790 So tell me another vector in the null space. 746 00:47:08,790 --> 00:47:12,290 Well, the natural choice, the choice you naturally think of 747 00:47:12,290 --> 00:47:16,140 is I'm going on to the fourth column, 748 00:47:16,140 --> 00:47:20,210 I'm letting that free variable be a one, 749 00:47:20,210 --> 00:47:23,750 and that free variable be a zero, and I'm asking 750 00:47:23,750 --> 00:47:26,130 is that fourth column a combination 751 00:47:26,130 --> 00:47:27,410 of my pivot columns? 752 00:47:27,410 --> 00:47:28,670 Yes, it is. 753 00:47:28,670 --> 00:47:31,650 And it's -- that will do. 754 00:47:35,730 --> 00:47:37,820 So what I've written there are actually the two 755 00:47:37,820 --> 00:47:39,860 special solutions, right? 756 00:47:39,860 --> 00:47:44,520 I took the two free variables, free and free. 757 00:47:44,520 --> 00:47:50,410 I gave them the values 1 0 or 0 I figured out the rest. 758 00:47:50,410 --> 00:47:53,640 So do you see, let me just say it in words. 759 00:47:53,640 --> 00:47:58,000 This vector, these vectors in the null space are telling me, 760 00:47:58,000 --> 00:48:00,190 they're telling me the combinations 761 00:48:00,190 --> 00:48:02,610 of the columns that give zero. 762 00:48:02,610 --> 00:48:08,770 They're telling me in what way the, the columns are dependent. 763 00:48:08,770 --> 00:48:10,770 That's what the null space is doing. 764 00:48:10,770 --> 00:48:13,890 Have I got enough now? 765 00:48:13,890 --> 00:48:15,780 And what's the null space now? 766 00:48:15,780 --> 00:48:18,540 We have to think about the null space. 767 00:48:18,540 --> 00:48:20,740 These are two vectors in the null space. 768 00:48:20,740 --> 00:48:21,790 They're independent. 769 00:48:21,790 --> 00:48:24,620 Are they a basis for the null space? 770 00:48:24,620 --> 00:48:27,650 What's the dimension of the null space? 771 00:48:27,650 --> 00:48:30,100 You see that those questions just keep coming up all the 772 00:48:30,100 --> 00:48:31,120 time. 773 00:48:31,120 --> 00:48:34,500 Are they a basis for the null space? 774 00:48:34,500 --> 00:48:36,940 You can tell me the answer even though we haven't 775 00:48:36,940 --> 00:48:39,460 written out a proof of that. 776 00:48:39,460 --> 00:48:40,190 Can you? 777 00:48:40,190 --> 00:48:41,050 Yes or no? 778 00:48:41,050 --> 00:48:44,510 Do these two special solutions form 779 00:48:44,510 --> 00:48:46,400 a basis for the null space? 780 00:48:46,400 --> 00:48:48,790 In other words, does the null space 781 00:48:48,790 --> 00:48:52,390 consist of all combinations of those two guys? 782 00:48:52,390 --> 00:48:54,110 Yes or no? 783 00:48:54,110 --> 00:48:55,180 Yes. 784 00:48:55,180 --> 00:48:56,880 Yes. 785 00:48:56,880 --> 00:48:58,970 The null space is two dimensional. 786 00:48:58,970 --> 00:49:01,500 The null space, the dimension of the null space, 787 00:49:01,500 --> 00:49:03,530 is the number of free variables. 788 00:49:03,530 --> 00:49:09,330 So the dimension of the null space 789 00:49:09,330 --> 00:49:12,065 is the number of free variables. 790 00:49:17,260 --> 00:49:21,420 And at the last second, give me the formula. 791 00:49:21,420 --> 00:49:24,490 This is then the key formula that we know. 792 00:49:24,490 --> 00:49:29,740 How many free variables are there in terms of R, the rank, 793 00:49:29,740 --> 00:49:35,020 m -- the number of rows, n, the number of columns? 794 00:49:35,020 --> 00:49:36,740 What do we get? 795 00:49:36,740 --> 00:49:42,360 We have n columns, r of them are pivot columns, 796 00:49:42,360 --> 00:49:48,200 so n-r is the number of free columns, free variables. 797 00:49:48,200 --> 00:49:51,950 And now it's the dimension of the null space. 798 00:49:51,950 --> 00:49:53,000 OK. 799 00:49:53,000 --> 00:49:53,950 That's great. 800 00:49:53,950 --> 00:49:57,280 That's the key spaces, their bases, and their dimensions. 801 00:49:57,280 --> 00:49:58,830 Thanks.