1 00:00:09,070 --> 00:00:09,660 Okay. 2 00:00:09,660 --> 00:00:13,950 This is lecture six in linear algebra, 3 00:00:13,950 --> 00:00:19,790 and we're at the start of this new chapter, chapter three 4 00:00:19,790 --> 00:00:22,500 in the text, which is really getting 5 00:00:22,500 --> 00:00:24,930 to the center of linear algebra. 6 00:00:24,930 --> 00:00:31,460 And I had time to make a first start on it 7 00:00:31,460 --> 00:00:33,690 at the end of lecture five. 8 00:00:33,690 --> 00:00:38,430 But now is lecture six is officially 9 00:00:38,430 --> 00:00:42,610 the lecture on vector spaces and subspaces. 10 00:00:42,610 --> 00:00:47,070 And then especially -- there are two subspaces that 11 00:00:47,070 --> 00:00:49,840 we're specially interested in. 12 00:00:49,840 --> 00:00:52,680 One is the column space of a matrix, 13 00:00:52,680 --> 00:00:55,630 the other is the null space of the matrix. 14 00:00:55,630 --> 00:00:59,050 So, I got to tell you what those are. 15 00:00:59,050 --> 00:01:00,140 Okay. 16 00:01:00,140 --> 00:01:05,140 So, first to remember from lecture five, 17 00:01:05,140 --> 00:01:08,350 what is a vector space? 18 00:01:08,350 --> 00:01:14,410 It's a bunch of vectors that -- where I'm allowed -- 19 00:01:14,410 --> 00:01:15,970 where I can add -- 20 00:01:15,970 --> 00:01:19,690 I can add any two vectors in the space 21 00:01:19,690 --> 00:01:23,300 and the answer stays in the space. 22 00:01:23,300 --> 00:01:27,610 Or I can multiply any vector in the space by any constant 23 00:01:27,610 --> 00:01:30,870 and the result stays in the space. 24 00:01:30,870 --> 00:01:34,950 So that's -- in fact if I combine those two into one, 25 00:01:34,950 --> 00:01:40,710 you can see that -- if I can add and I can multiply by numbers, 26 00:01:40,710 --> 00:01:44,750 that really means that I can take linear combinations. 27 00:01:44,750 --> 00:01:50,220 So the quick way to say it is that all linear combinations, 28 00:01:50,220 --> 00:01:54,250 C -- any multiple of V plus any multiple of W stay 29 00:01:54,250 --> 00:01:56,220 in the space. 30 00:01:56,220 --> 00:02:01,070 So, can I give you examples that are vector spaces and also 31 00:02:01,070 --> 00:02:04,850 some examples that are not, to make that point clear? 32 00:02:04,850 --> 00:02:10,220 So, suppose I'm in three dimensions. 33 00:02:10,220 --> 00:02:16,950 Then one way to get us one space is the whole three dimensional 34 00:02:16,950 --> 00:02:18,200 space. 35 00:02:18,200 --> 00:02:22,770 So the whole space R^3, three dimensional space, 36 00:02:22,770 --> 00:02:27,100 would be a vector space, because if I have a couple of vectors I 37 00:02:27,100 --> 00:02:30,040 can add them and I'm certainly okay and they follow all 38 00:02:30,040 --> 00:02:32,240 the rules. 39 00:02:32,240 --> 00:02:34,600 So R^3 is easy. 40 00:02:34,600 --> 00:02:39,320 Now I'm interested also in subspaces. 41 00:02:39,320 --> 00:02:42,370 So there's this key word, subspaces. 42 00:02:42,370 --> 00:02:48,530 That's a space -- that's some vectors inside the given space, 43 00:02:48,530 --> 00:02:53,940 inside R three that still make up a vector space of their own. 44 00:02:53,940 --> 00:02:56,990 It's a vector space inside a vector space. 45 00:02:56,990 --> 00:03:00,030 And the simplest example was a plane. 46 00:03:00,030 --> 00:03:05,070 So, like, can I just sketch it -- there is a plane. 47 00:03:05,070 --> 00:03:07,280 It's got to go through the origin, 48 00:03:07,280 --> 00:03:10,390 and of course it goes infinitely far. 49 00:03:10,390 --> 00:03:12,620 That's of that's a subspace now. 50 00:03:12,620 --> 00:03:18,190 Do you see that if I have two vectors on the plane 51 00:03:18,190 --> 00:03:22,920 and I add them, the result stays in the plane. 52 00:03:22,920 --> 00:03:26,770 If I take a vector in the plane and I multiply by minus two, 53 00:03:26,770 --> 00:03:28,520 I'm still in the plane. 54 00:03:28,520 --> 00:03:31,240 So that plane is a subspace. 55 00:03:31,240 --> 00:03:33,120 So let me just make that point. 56 00:03:33,120 --> 00:03:44,120 Plane through zero, through that zero zero zero is a subspace. 57 00:03:44,120 --> 00:03:45,060 Okay. 58 00:03:45,060 --> 00:03:51,710 And also, another subspace would be a line. 59 00:03:51,710 --> 00:03:54,230 A line through zero zero zero -- yeah, 60 00:03:54,230 --> 00:03:56,240 the line has to go through the origin. 61 00:03:56,240 --> 00:03:59,390 All subspaces have got to contain the origin, 62 00:03:59,390 --> 00:04:01,910 contain zero -- the zero vector. 63 00:04:01,910 --> 00:04:07,015 So this line is a subspace. 64 00:04:09,900 --> 00:04:13,910 Really, if I want to say it really correctly, 65 00:04:13,910 --> 00:04:18,899 I should say a subspace of R^3. 66 00:04:18,899 --> 00:04:23,710 That of R^3 was, like, understood there. 67 00:04:23,710 --> 00:04:30,730 Now -- so let me call this plane P. And let me call this line L. 68 00:04:30,730 --> 00:04:40,540 And let me ask you about other sets of vectors. 69 00:04:40,540 --> 00:04:43,210 Suppose I took -- 70 00:04:43,210 --> 00:04:45,310 yeah -- so here's a first question. 71 00:04:45,310 --> 00:04:50,030 Suppose I take two subspaces, like P and L. 72 00:04:50,030 --> 00:04:54,610 And I just put them together, take their union, 73 00:04:54,610 --> 00:04:56,890 take all the vectors -- 74 00:04:56,890 --> 00:05:00,880 so now you've got P and L in mind, here. 75 00:05:00,880 --> 00:05:03,870 So I have two subspaces. 76 00:05:03,870 --> 00:05:09,960 I have two subspaces and, for example, P -- 77 00:05:09,960 --> 00:05:12,070 a plane and L a line. 78 00:05:12,070 --> 00:05:13,410 Okay. 79 00:05:13,410 --> 00:05:17,970 Now I want to ask you about the union of those. 80 00:05:17,970 --> 00:05:20,190 So P union L. 81 00:05:20,190 --> 00:05:29,990 This is all vectors in P or L or both. 82 00:05:33,120 --> 00:05:36,240 Is that a subspace? 83 00:05:36,240 --> 00:05:38,040 Is this a subspace? 84 00:05:38,040 --> 00:05:48,380 This is or is not a subspace? 85 00:05:48,380 --> 00:05:51,690 Because we're -- I just want to be sure that I've got 86 00:05:51,690 --> 00:05:54,340 the central idea. 87 00:05:54,340 --> 00:05:57,200 Suppose I take the vectors in the plane 88 00:05:57,200 --> 00:06:03,230 and also the vectors on that line, 89 00:06:03,230 --> 00:06:05,930 put them together, so I've got a bunch of vectors, 90 00:06:05,930 --> 00:06:07,030 is it a subspace? 91 00:06:07,030 --> 00:06:09,210 Can you give me, like, so the camera 92 00:06:09,210 --> 00:06:11,700 can hear it or maybe the tape. 93 00:06:11,700 --> 00:06:14,650 Can you say yes or no? 94 00:06:14,650 --> 00:06:20,770 Do I have a subspace if I put -- if I take all the vectors 95 00:06:20,770 --> 00:06:25,660 on the plane plus all -- and all the ones on the line and just 96 00:06:25,660 --> 00:06:30,520 join them together -- but I'm not taking this guy that's -- 97 00:06:30,520 --> 00:06:32,240 actually, I'm not taking most of them, 98 00:06:32,240 --> 00:06:35,680 because most vectors are not on the line or the plane, 99 00:06:35,680 --> 00:06:37,590 they're off somewhere else. 100 00:06:37,590 --> 00:06:39,970 Do I have a subspace? 101 00:06:39,970 --> 00:06:40,697 STUDENTS: No. 102 00:06:40,697 --> 00:06:41,280 STRANG: Right. 103 00:06:41,280 --> 00:06:41,790 Thank you. 104 00:06:41,790 --> 00:06:42,510 No. 105 00:06:42,510 --> 00:06:44,980 Because -- why not? 106 00:06:44,980 --> 00:06:47,130 Because I can't add. 107 00:06:47,130 --> 00:06:52,090 Because if I that requirement isn't satisfied. 108 00:06:52,090 --> 00:06:56,050 If I take one vector like this guy and another vector 109 00:06:56,050 --> 00:07:00,100 that happens to come from L and add, I'm off somewhere else. 110 00:07:00,100 --> 00:07:06,870 You see that I've gone outside the union if I just add 111 00:07:06,870 --> 00:07:09,070 something from P and something from L, 112 00:07:09,070 --> 00:07:15,550 then normally what'll happen is I'm outside the union -- 113 00:07:15,550 --> 00:07:16,770 and I don't have a subspace. 114 00:07:16,770 --> 00:07:19,120 So the correct answer is -- 115 00:07:19,120 --> 00:07:21,270 is not. 116 00:07:21,270 --> 00:07:21,850 Okay. 117 00:07:21,850 --> 00:07:25,630 Now let me ask you about -- the other thing we do is take 118 00:07:25,630 --> 00:07:28,240 the intersection. 119 00:07:28,240 --> 00:07:30,410 So what does intersection mean? 120 00:07:30,410 --> 00:07:43,830 Intersection means all vectors that are in both P and L. 121 00:07:43,830 --> 00:07:45,970 Is this a subspace. 122 00:07:45,970 --> 00:07:49,730 Yeah, so I guess I want to go back up to the same question. 123 00:07:49,730 --> 00:07:52,360 This is or is not a subspace? 124 00:07:52,360 --> 00:07:56,910 And you can answer me -- answer the question first for this 125 00:07:56,910 --> 00:08:00,640 particular example, this picture I drew. 126 00:08:00,640 --> 00:08:03,780 What is P intersect L for this case? 127 00:08:03,780 --> 00:08:05,280 STUDENT: It's only zero. 128 00:08:05,280 --> 00:08:07,810 STRANG: It's only zero. 129 00:08:07,810 --> 00:08:11,240 At least, sort of this was the artist's idea as he drew it 130 00:08:11,240 --> 00:08:17,250 that, that that line L was not in the plane and, went off 131 00:08:17,250 --> 00:08:21,120 somewhere else -- and then the only point that was in common 132 00:08:21,120 --> 00:08:22,770 was the zero vector. 133 00:08:22,770 --> 00:08:25,430 Is the zero vector by itself a subspace? 134 00:08:25,430 --> 00:08:26,000 STUDENT: Yes. 135 00:08:26,000 --> 00:08:27,740 STRANG: Yes, absolutely. 136 00:08:27,740 --> 00:08:34,049 And what about, if I don't have this plane and this line 137 00:08:34,049 --> 00:08:38,789 but any subspace and any other subspace? 138 00:08:38,789 --> 00:08:43,740 So now -- can I ask that question for any two subspaces? 139 00:08:43,740 --> 00:08:46,800 So maybe I'll write it up here. 140 00:08:46,800 --> 00:08:50,240 If I'm strong enough. 141 00:08:50,240 --> 00:08:51,000 Okay. 142 00:08:51,000 --> 00:08:52,510 So this is the general question. 143 00:08:52,510 --> 00:09:00,350 I have subspaces, say S and T. 144 00:09:00,350 --> 00:09:08,640 And I want to ask you about their intersection S intersect 145 00:09:08,640 --> 00:09:11,300 T and I want -- 146 00:09:11,300 --> 00:09:13,220 it is a subspace. 147 00:09:17,700 --> 00:09:19,430 Do you see why? 148 00:09:19,430 --> 00:09:24,610 Do you see why if I take the vectors that are in both one- 149 00:09:24,610 --> 00:09:27,930 th- that are in both of the subspaces -- 150 00:09:27,930 --> 00:09:30,490 so that's like a smaller set of vectors, probably, 151 00:09:30,490 --> 00:09:32,540 because it's -- we've added requirements. 152 00:09:32,540 --> 00:09:36,370 It has to be in S and in T. 153 00:09:36,370 --> 00:09:38,100 How do I know that's a subspace? 154 00:09:38,100 --> 00:09:40,560 Can we just think through that abstract stuff 155 00:09:40,560 --> 00:09:43,640 and then I get to the examples. 156 00:09:43,640 --> 00:09:44,870 Okay. 157 00:09:44,870 --> 00:09:45,940 So why? 158 00:09:45,940 --> 00:09:49,490 Suppose I take a couple of vectors 159 00:09:49,490 --> 00:09:51,040 that are in the intersection. 160 00:09:51,040 --> 00:09:56,500 Why is the sum also in the intersection? 161 00:09:56,500 --> 00:10:00,900 Okay, so let me give a name to these vectors, say V and W. 162 00:10:00,900 --> 00:10:02,280 They're in the intersection. 163 00:10:02,280 --> 00:10:05,310 So that means they're both in S. 164 00:10:05,310 --> 00:10:08,370 Also means they're both in T. 165 00:10:08,370 --> 00:10:11,960 So what can I say about V plus W? 166 00:10:11,960 --> 00:10:14,350 Is it in S? 167 00:10:14,350 --> 00:10:15,040 Yes. 168 00:10:15,040 --> 00:10:16,480 Right? 169 00:10:16,480 --> 00:10:21,300 If I take two vectors, V and W that are both in S, 170 00:10:21,300 --> 00:10:24,900 then the sum is in S, because S was a subspace. 171 00:10:24,900 --> 00:10:27,830 And if they're both in T and I add them, 172 00:10:27,830 --> 00:10:31,990 then the result is also in T, because T was a subspace. 173 00:10:31,990 --> 00:10:37,100 So the result V plus W is in the intersection. 174 00:10:37,100 --> 00:10:42,690 It's in both and requirement one is satisfied. 175 00:10:42,690 --> 00:10:44,480 Requirement two's the same. 176 00:10:44,480 --> 00:10:49,080 If I take a vector that's in both, I multiply by seven. 177 00:10:49,080 --> 00:10:53,470 Seven times that vector is in S, because the vector was in S. 178 00:10:53,470 --> 00:10:57,290 Seven times that vector's in T because the original one was in 179 00:10:57,290 --> 00:11:01,080 T. So seven times that vector is in the intersection. 180 00:11:01,080 --> 00:11:05,740 In other words, when you take the intersection of two 181 00:11:05,740 --> 00:11:08,830 subspaces, you get probably a smaller subspace, 182 00:11:08,830 --> 00:11:10,440 but it is a subspace. 183 00:11:10,440 --> 00:11:11,780 Okay. 184 00:11:11,780 --> 00:11:17,270 So that's like sort of just emphasizing what 185 00:11:17,270 --> 00:11:20,080 these two requirements mean. 186 00:11:20,080 --> 00:11:24,700 Again -- Let me circle those, because those are so important. 187 00:11:24,700 --> 00:11:30,490 The sum and the scale of multiplication which combines 188 00:11:30,490 --> 00:11:33,850 into linear combinations. 189 00:11:33,850 --> 00:11:37,130 That's what you have to do inside the subspace. 190 00:11:37,130 --> 00:11:37,980 Okay. 191 00:11:37,980 --> 00:11:41,530 On to the column space. 192 00:11:41,530 --> 00:11:42,030 Okay. 193 00:11:42,030 --> 00:11:46,690 So my lecture last time started that and I want to continue it. 194 00:11:46,690 --> 00:11:47,190 Okay. 195 00:11:47,190 --> 00:11:48,680 Column space of a matrix. 196 00:11:53,690 --> 00:11:54,190 Of A. 197 00:11:54,190 --> 00:11:56,060 Okay. 198 00:11:56,060 --> 00:11:59,240 Can I take an example? 199 00:11:59,240 --> 00:12:03,030 Say one two three four. 200 00:12:03,030 --> 00:12:06,410 One one one one. 201 00:12:06,410 --> 00:12:07,570 Two three four five. 202 00:12:11,760 --> 00:12:14,370 Okay. 203 00:12:14,370 --> 00:12:19,610 That's my matrix A. 204 00:12:19,610 --> 00:12:25,900 So, it's got columns, three columns. 205 00:12:25,900 --> 00:12:29,700 Those columns are vectors, so the column space of this A, 206 00:12:29,700 --> 00:12:31,700 of this A -- 207 00:12:31,700 --> 00:12:33,970 let's stay with this example for a while. 208 00:12:33,970 --> 00:12:40,520 The column space of this matrix is a subspace of R -- 209 00:12:40,520 --> 00:12:41,600 R what? 210 00:12:41,600 --> 00:12:43,990 So what space are we in if I'm looking 211 00:12:43,990 --> 00:12:46,730 at the columns of this matrix? 212 00:12:46,730 --> 00:12:48,770 R^4 , right? 213 00:12:48,770 --> 00:12:52,680 These are vectors in R^4, they're four dimensional 214 00:12:52,680 --> 00:12:53,800 vectors. 215 00:12:53,800 --> 00:13:04,010 So it's this column space of A is a subspace of R^4 here, 216 00:13:04,010 --> 00:13:07,030 because A was four by -- 217 00:13:07,030 --> 00:13:09,190 A is a four by three matrix. 218 00:13:09,190 --> 00:13:12,080 This tells me how many rows there are, 219 00:13:12,080 --> 00:13:16,020 how many components in a column, and so we're in R^4. 220 00:13:16,020 --> 00:13:20,030 Okay, now what's in that subspace? 221 00:13:20,030 --> 00:13:24,300 So the column space of A, it's a subspace of R^4. 222 00:13:24,300 --> 00:13:28,790 I call it the column space of A, like that. 223 00:13:28,790 --> 00:13:34,670 So that's my little symbol for some subspace of R^4. 224 00:13:34,670 --> 00:13:37,480 What's in that subspace? 225 00:13:37,480 --> 00:13:40,010 Well, that column certainly is. 226 00:13:40,010 --> 00:13:41,350 One two three four. 227 00:13:41,350 --> 00:13:42,460 This column is in. 228 00:13:42,460 --> 00:13:45,750 This column is in, and what else? 229 00:13:45,750 --> 00:13:48,480 So it's got the columns of A in it, 230 00:13:48,480 --> 00:13:51,671 but that's not enough, certainly. 231 00:13:51,671 --> 00:13:52,170 Right? 232 00:13:52,170 --> 00:13:55,700 I don't have a subspace if I just put in three vectors. 233 00:13:55,700 --> 00:13:59,210 So how do I fill that out to be a subspace? 234 00:13:59,210 --> 00:14:06,000 I take their linear combinations. 235 00:14:06,000 --> 00:14:15,110 So the column space of A is all linear combinations -- 236 00:14:15,110 --> 00:14:16,710 combinations of the columns. 237 00:14:19,480 --> 00:14:22,890 And that does give me a subspace. 238 00:14:22,890 --> 00:14:24,890 It does give me a vector space, because if I 239 00:14:24,890 --> 00:14:28,040 have one linear combination and I multiply by eleven, 240 00:14:28,040 --> 00:14:30,680 I've got another linear combination. 241 00:14:30,680 --> 00:14:32,190 If I have a linear combination, I 242 00:14:32,190 --> 00:14:33,730 add to another linear combination 243 00:14:33,730 --> 00:14:35,970 I get a third combination. 244 00:14:35,970 --> 00:14:40,260 So that -- this is like the smallest space -- 245 00:14:40,260 --> 00:14:43,040 like, it's got to have those three columns in it, 246 00:14:43,040 --> 00:14:45,270 and it has to have their combinations 247 00:14:45,270 --> 00:14:47,110 and that's where we stop. 248 00:14:47,110 --> 00:14:47,680 Okay. 249 00:14:47,680 --> 00:14:54,120 Now I'm going to be interested in that space. 250 00:14:54,120 --> 00:14:56,930 So I, like -- get some idea of what's in that space. 251 00:14:56,930 --> 00:14:58,450 How big is that space? 252 00:14:58,450 --> 00:15:02,880 Is that space the whole four dimensional space? 253 00:15:02,880 --> 00:15:05,290 Or is it a subspace inside? 254 00:15:05,290 --> 00:15:13,870 Can you -- let me just see if we can get a yes or no answer 255 00:15:13,870 --> 00:15:19,930 sometimes without being ready for the complete proof. 256 00:15:22,660 --> 00:15:23,540 What do you think? 257 00:15:23,540 --> 00:15:26,590 Is the subspace that I'm talking about here, 258 00:15:26,590 --> 00:15:28,700 the combinations of those three guys, 259 00:15:28,700 --> 00:15:32,250 does that fill the full four dimensional space? 260 00:15:32,250 --> 00:15:34,900 Maybe yes or no on that one. 261 00:15:34,900 --> 00:15:35,560 No. 262 00:15:35,560 --> 00:15:36,060 No. 263 00:15:36,060 --> 00:15:39,640 Somehow our feeling is, and it happens 264 00:15:39,640 --> 00:15:42,730 to be right, that if we start with three vectors 265 00:15:42,730 --> 00:15:45,660 and take their combinations, we can't get the whole four 266 00:15:45,660 --> 00:15:48,680 dimensional space. 267 00:15:48,680 --> 00:15:52,590 Now -- so somehow we get a smaller space. 268 00:15:52,590 --> 00:15:55,010 But how much smaller? 269 00:15:55,010 --> 00:15:57,280 That's going to come up here. 270 00:15:57,280 --> 00:16:00,970 That's not so immediate. 271 00:16:00,970 --> 00:16:07,950 Let me first make this critical connection with -- 272 00:16:07,950 --> 00:16:15,980 with, linear equations, because behind our abstract definition, 273 00:16:15,980 --> 00:16:17,360 we have a purpose. 274 00:16:17,360 --> 00:16:19,750 And that is to understand Ax=b. 275 00:16:19,750 --> 00:16:22,950 So suppose I make the connection -- 276 00:16:22,950 --> 00:16:33,330 w- w- does A x=b always have a solution for every b? 277 00:16:33,330 --> 00:16:47,850 Have a solution for every right-hand side? 278 00:16:47,850 --> 00:16:49,880 I guess that's going to be a yes or no question. 279 00:16:53,220 --> 00:16:59,580 And then I'm going to ask which right-hand sides are okay? 280 00:16:59,580 --> 00:17:02,050 That's really the question I'm after, 281 00:17:02,050 --> 00:17:08,900 is which right-hand sides (b) do make up -- 282 00:17:08,900 --> 00:17:12,210 you can see from the way I'm speaking what the question -- 283 00:17:12,210 --> 00:17:13,839 As it is. 284 00:17:13,839 --> 00:17:16,020 The answer is no. 285 00:17:16,020 --> 00:17:21,710 A x=b does not have a solution for every b. 286 00:17:21,710 --> 00:17:26,069 Why do I say no? 287 00:17:26,069 --> 00:17:34,660 Because A x=b is -- like, this is four equations, 288 00:17:34,660 --> 00:17:35,940 and only three unknowns. 289 00:17:39,590 --> 00:17:40,860 Right? 290 00:17:40,860 --> 00:17:46,057 X is -- let me right out that whole -- 291 00:17:46,057 --> 00:17:47,390 what the whole thing looks like. 292 00:17:49,940 --> 00:17:50,600 Yeah. 293 00:17:50,600 --> 00:17:54,680 Let me write out A x=b. 294 00:17:54,680 --> 00:17:59,310 A x is -- 295 00:17:59,310 --> 00:18:02,260 these columns are one two three four. 296 00:18:02,260 --> 00:18:07,050 One one one one and two three four five. 297 00:18:07,050 --> 00:18:12,410 Then x, of course, has three components, x1, x2, x3. 298 00:18:12,410 --> 00:18:15,610 And I'm trying to get the -- 299 00:18:15,610 --> 00:18:19,257 hit the right-hand side, b1,b2,b3 and b4. 300 00:18:25,430 --> 00:18:27,850 So my first point is, I can't always do it. 301 00:18:30,560 --> 00:18:34,140 In a way, that just says again what you told me five minutes 302 00:18:34,140 --> 00:18:36,220 ago -- 303 00:18:36,220 --> 00:18:39,540 that the combinations of these columns 304 00:18:39,540 --> 00:18:42,630 don't fill the whole four dimensional space. 305 00:18:42,630 --> 00:18:46,190 There's going to be some vectors b, a lot of vectors b, 306 00:18:46,190 --> 00:18:50,530 that are not combinations of these three columns, 307 00:18:50,530 --> 00:18:53,200 because the combinations of those columns are, like, 308 00:18:53,200 --> 00:18:56,130 going to be just a little plane or something inside -- 309 00:18:56,130 --> 00:18:58,050 inside R^4. 310 00:18:58,050 --> 00:19:03,990 Now, so and you see that I do have four equations and only 311 00:19:03,990 --> 00:19:05,210 three unknowns. 312 00:19:05,210 --> 00:19:09,110 So, like anybody is going to say, no you dope, 313 00:19:09,110 --> 00:19:11,650 you can't usually solve four equations 314 00:19:11,650 --> 00:19:13,330 with only three unknowns. 315 00:19:13,330 --> 00:19:17,710 But now I want to say sometimes you can. 316 00:19:17,710 --> 00:19:22,330 For some right-hand sides, I can solve this. 317 00:19:22,330 --> 00:19:25,900 So that's the bunch of right-hand sides 318 00:19:25,900 --> 00:19:28,710 that I'm interested in right now. 319 00:19:28,710 --> 00:19:34,860 Which right-hand sides allow me to solve this? 320 00:19:34,860 --> 00:19:36,740 This is the question for today. 321 00:19:36,740 --> 00:19:40,240 It's going to have, like, a nice clear answer. 322 00:19:40,240 --> 00:19:50,800 So my question is -- is which bs, which vectors b, 323 00:19:50,800 --> 00:19:55,200 allow this system to be solved? 324 00:19:59,640 --> 00:20:03,150 And I want to ask you -- 325 00:20:03,150 --> 00:20:08,090 so that's, like, gets two question marks to indicate 326 00:20:08,090 --> 00:20:10,350 that's -- this is the important question. 327 00:20:10,350 --> 00:20:15,030 Okay, first, before we give a total answer, 328 00:20:15,030 --> 00:20:18,000 give me just a partial answer. 329 00:20:18,000 --> 00:20:20,910 Tell me one right-hand side that I know 330 00:20:20,910 --> 00:20:23,160 I can solve this thing for. 331 00:20:23,160 --> 00:20:24,910 So -- all zeroes. 332 00:20:24,910 --> 00:20:25,410 Okay. 333 00:20:25,410 --> 00:20:28,000 That's the, like, guaranteed. 334 00:20:28,000 --> 00:20:31,990 If these were all zero, then I know I can solve it, 335 00:20:31,990 --> 00:20:35,550 let the x-s all be zero, no problem. 336 00:20:35,550 --> 00:20:38,890 So that's always a -- okay. 337 00:20:38,890 --> 00:20:39,660 Okay. 338 00:20:39,660 --> 00:20:42,460 A x=0 I can always solve. 339 00:20:42,460 --> 00:20:45,040 Now tell me another right-hand side, 340 00:20:45,040 --> 00:20:52,740 just a specific set of numbers for which I can solve these 341 00:20:52,740 --> 00:20:55,700 three -- these four equations with only three unknowns, 342 00:20:55,700 --> 00:21:00,130 but if you give me a good right-hand side, I can do it. 343 00:21:00,130 --> 00:21:00,842 So tell me one? 344 00:21:00,842 --> 00:21:01,550 STUDENT: 1 2 3 4. 345 00:21:01,550 --> 00:21:04,740 STRANG: 1 2 3 4? 346 00:21:04,740 --> 00:21:09,575 If I -- can I solve -- is that a good right-hand side? 347 00:21:12,340 --> 00:21:15,340 Can you solve -- can you find a solution that -- 348 00:21:15,340 --> 00:21:18,630 X one plus X two plus two X three is one, 349 00:21:18,630 --> 00:21:22,630 two X one plus X two plus three X three is two and two more 350 00:21:22,630 --> 00:21:23,858 equations -- 351 00:21:27,210 --> 00:21:30,220 so I'm asking you to solve in your head in -- 352 00:21:30,220 --> 00:21:35,750 within five seconds, four equations and three unknowns, 353 00:21:35,750 --> 00:21:41,310 but you can do it, because the right-hand side is, like, 354 00:21:41,310 --> 00:21:44,610 showing up here is -- it's one of the columns. 355 00:21:44,610 --> 00:21:48,330 So tell me what's the X that does solve it? 356 00:21:48,330 --> 00:21:50,140 One zero zero. 357 00:21:50,140 --> 00:21:55,960 One zero zero solves it, because -- 358 00:21:55,960 --> 00:22:01,450 well, so you can multiply this out by rows, but oh God, 359 00:22:01,450 --> 00:22:06,360 it's much nicer to say -- okay, this is one of this column, 360 00:22:06,360 --> 00:22:09,190 zero of this, zero of this, so it's one of that column, 361 00:22:09,190 --> 00:22:11,730 which is exactly what we wanted. 362 00:22:11,730 --> 00:22:13,290 Okay. 363 00:22:13,290 --> 00:22:17,280 So there is a b that's okay. 364 00:22:17,280 --> 00:22:19,250 Now tell me another B that's okay, 365 00:22:19,250 --> 00:22:23,030 another right-hand side that would be all right? 366 00:22:23,030 --> 00:22:25,055 Well -- all ones? 367 00:22:29,480 --> 00:22:33,450 Actually -- and then what's the solution in that case? 368 00:22:33,450 --> 00:22:36,080 0 1 0, thanks. 369 00:22:36,080 --> 00:22:40,190 And, in fact, it's much e- like, one way 370 00:22:40,190 --> 00:22:44,230 to do it is think of a solution first, right, 371 00:22:44,230 --> 00:22:50,380 and then just see what b turns out to be. 372 00:22:50,380 --> 00:22:53,030 What b turns out to be, right. 373 00:22:53,030 --> 00:22:54,790 Okay. 374 00:22:54,790 --> 00:22:58,220 So I think of a solution -- so I think of an x, 375 00:22:58,220 --> 00:23:00,380 I think of any -- 376 00:23:00,380 --> 00:23:03,810 x1, x2, x3, I do this multiplication 377 00:23:03,810 --> 00:23:04,980 and what have I got? 378 00:23:09,640 --> 00:23:13,920 Now I'm ready to answer the big question. 379 00:23:13,920 --> 00:23:20,830 I can solve A x=b exactly when the right-hand side B is 380 00:23:20,830 --> 00:23:24,920 a vector in the column space. 381 00:23:24,920 --> 00:23:25,900 Good. 382 00:23:25,900 --> 00:23:33,140 I can solve A x=b when b is a combination of the columns, 383 00:23:33,140 --> 00:23:36,130 when it's in the column space -- 384 00:23:36,130 --> 00:23:39,730 so let me write that answer down. 385 00:23:39,730 --> 00:23:57,910 I can solve Ax=b exactly when B is in the column space. 386 00:23:57,910 --> 00:24:02,210 Let me just say again why that is. 387 00:24:02,210 --> 00:24:07,030 Because it -- the column space by its definition contains all 388 00:24:07,030 --> 00:24:07,870 the combinations. 389 00:24:07,870 --> 00:24:10,710 It contains all the Ax-s. 390 00:24:10,710 --> 00:24:17,360 The column space really consists of all vectors A times any X. 391 00:24:17,360 --> 00:24:22,900 So those are the bs that I can deal with. 392 00:24:22,900 --> 00:24:26,130 If b is a combination of the columns, 393 00:24:26,130 --> 00:24:32,080 then that combination tells me what X should be. 394 00:24:32,080 --> 00:24:35,630 If b is not a combination of the columns, then there is no x. 395 00:24:35,630 --> 00:24:38,650 There's no way to solve A x equal b. 396 00:24:38,650 --> 00:24:40,420 Okay. 397 00:24:40,420 --> 00:24:42,700 So the column space -- 398 00:24:42,700 --> 00:24:45,510 that's really why we're interested in this column 399 00:24:45,510 --> 00:24:48,170 space, because it's the central guy. 400 00:24:48,170 --> 00:24:56,390 It says when we can solve, and that -- 401 00:24:56,390 --> 00:24:58,880 we got to understand this column space better. 402 00:25:02,420 --> 00:25:03,000 Let's see. 403 00:25:03,000 --> 00:25:07,260 Do I want to think -- 404 00:25:07,260 --> 00:25:10,600 yeah, somehow -- oh, well, let's just -- 405 00:25:10,600 --> 00:25:13,360 as long as we've got it here, what do I get for this 406 00:25:13,360 --> 00:25:14,550 particular example? 407 00:25:14,550 --> 00:25:24,670 If I take combinations of this and this and this, 408 00:25:24,670 --> 00:25:28,230 I'll tell you the question that's in my mind. 409 00:25:28,230 --> 00:25:30,930 It's not even proper to use this word yet, 410 00:25:30,930 --> 00:25:33,280 but you'll know what it means. 411 00:25:33,280 --> 00:25:38,060 Are those three columns independent? 412 00:25:38,060 --> 00:25:44,370 If I take the combinations of the three columns -- 413 00:25:44,370 --> 00:25:50,550 does each column contribute something new or now? 414 00:25:50,550 --> 00:25:53,360 So that if I take the combinations of those three 415 00:25:53,360 --> 00:25:57,160 columns, do I, like, get some three dimensional subspace -- 416 00:25:57,160 --> 00:26:00,630 do I have three vectors that are, like, you know, 417 00:26:00,630 --> 00:26:07,440 independent, whatever that means? 418 00:26:07,440 --> 00:26:10,080 Or do I -- is one of those columns, like, 419 00:26:10,080 --> 00:26:12,370 contributing nothing new -- 420 00:26:12,370 --> 00:26:15,630 So that actually only two of the columns 421 00:26:15,630 --> 00:26:19,040 would have given the same column space? 422 00:26:19,040 --> 00:26:21,080 Yeah -- that's a good way to ask the question. 423 00:26:21,080 --> 00:26:22,550 Finally I think of it. 424 00:26:22,550 --> 00:26:25,340 Can I throw away any columns -- 425 00:26:25,340 --> 00:26:27,339 and have the same column space? 426 00:26:27,339 --> 00:26:27,880 STUDENT: Yes. 427 00:26:27,880 --> 00:26:29,100 STRANG: Yes. 428 00:26:29,100 --> 00:26:31,290 And which one do you suggest I throw away? 429 00:26:31,290 --> 00:26:32,670 STUDENT: Column three -- three. 430 00:26:32,670 --> 00:26:37,350 STRANG: Well, three is the natural, like, guy to target. 431 00:26:37,350 --> 00:26:39,490 So if I -- and why? 432 00:26:39,490 --> 00:26:44,750 Because -- what's so bad about three here? 433 00:26:44,750 --> 00:26:46,240 Column three? 434 00:26:46,240 --> 00:26:48,310 It's the sum of these, right? 435 00:26:48,310 --> 00:26:52,370 So it's not -- if I'm taking -- if I have combinations of these 436 00:26:52,370 --> 00:26:54,510 two and I put in this one, actually, 437 00:26:54,510 --> 00:26:56,780 I don't get anything more. 438 00:26:56,780 --> 00:27:02,460 So later on I will call these pivot columns. 439 00:27:02,460 --> 00:27:06,780 And the third guy will not be a pivot column in this -- 440 00:27:06,780 --> 00:27:08,760 with those numbers. 441 00:27:08,760 --> 00:27:12,940 Now actually -- honesty makes me ask you this question. 442 00:27:12,940 --> 00:27:16,350 Could I have thrown away column one? 443 00:27:16,350 --> 00:27:17,940 Yes, I could. 444 00:27:17,940 --> 00:27:19,920 I could. 445 00:27:19,920 --> 00:27:22,910 So when I say pivot columns, my convention 446 00:27:22,910 --> 00:27:26,710 is, okay, I'll keep the first ones as long as they're not 447 00:27:26,710 --> 00:27:27,590 dependent. 448 00:27:27,590 --> 00:27:30,930 So I keep this guy, he's fine, he's a line. 449 00:27:30,930 --> 00:27:32,100 I keep the second guy. 450 00:27:32,100 --> 00:27:33,810 It's in a second direction. 451 00:27:33,810 --> 00:27:39,220 But the third one, which is in the same plane as the first two 452 00:27:39,220 --> 00:27:40,790 gives me nothing new. 453 00:27:40,790 --> 00:27:44,560 It's dependent in the language that we will use 454 00:27:44,560 --> 00:27:46,990 and I don't need it. 455 00:27:46,990 --> 00:27:48,160 Okay. 456 00:27:48,160 --> 00:27:53,570 So I would describe the column space of this matrix as a two 457 00:27:53,570 --> 00:27:56,040 dimensional subspace of R^4. 458 00:27:58,680 --> 00:28:01,070 A two dimensional subspace of R^4. 459 00:28:01,070 --> 00:28:01,990 Okay. 460 00:28:01,990 --> 00:28:05,430 So you're seeing how these vector spaces work and you -- 461 00:28:05,430 --> 00:28:08,760 you're seeing that we -- some idea of dependence 462 00:28:08,760 --> 00:28:11,780 or independence is in our future. 463 00:28:11,780 --> 00:28:12,400 Okay. 464 00:28:12,400 --> 00:28:16,590 Now I want to speak about another vector 465 00:28:16,590 --> 00:28:20,460 space, the null space. 466 00:28:20,460 --> 00:28:23,890 So again I'm getting a little ahead 467 00:28:23,890 --> 00:28:28,191 because it's in section three point two, but that's okay. 468 00:28:28,191 --> 00:28:28,690 All right. 469 00:28:28,690 --> 00:28:31,870 Now I'm ready for the null space. 470 00:28:31,870 --> 00:28:33,340 Let me keep the same matrix. 471 00:28:36,830 --> 00:28:39,370 And this is going to be a different -- 472 00:28:39,370 --> 00:28:41,500 totally different subspace. 473 00:28:41,500 --> 00:28:43,270 Totally different. 474 00:28:43,270 --> 00:28:44,930 Okay. 475 00:28:44,930 --> 00:28:47,370 Now -- so let me make space for it. 476 00:28:47,370 --> 00:28:50,190 Now -- here comes a completely different subspace, 477 00:28:50,190 --> 00:29:00,120 the null space of A. 478 00:29:00,120 --> 00:29:00,935 What's in it? 479 00:29:04,240 --> 00:29:10,070 It contains not right-hand sides b. 480 00:29:10,070 --> 00:29:12,750 It contains x-s. 481 00:29:12,750 --> 00:29:16,570 It contains all x-s that solve -- 482 00:29:16,570 --> 00:29:18,040 this word null is going to -- 483 00:29:18,040 --> 00:29:22,580 I mean, that's the key word here, meaning zero. 484 00:29:22,580 --> 00:29:30,670 So this contains -- this is all solutions x, 485 00:29:30,670 --> 00:29:36,450 and of course x is our vectors, x1, x2 and x3, 486 00:29:36,450 --> 00:29:41,220 to the equation A x=0. 487 00:29:43,800 --> 00:29:47,800 Well, four equations, because we've got -- 488 00:29:47,800 --> 00:29:50,230 so, do you see what I'm doing? 489 00:29:50,230 --> 00:29:53,890 I'm now saying, okay, columns were great, 490 00:29:53,890 --> 00:29:56,010 the column space we understood. 491 00:29:56,010 --> 00:29:59,260 Now I'm interested in x-s. 492 00:29:59,260 --> 00:30:02,860 I'm not -- the only b I'm interested in now is the b 493 00:30:02,860 --> 00:30:03,670 of all zeroes. 494 00:30:03,670 --> 00:30:06,300 The right-hand side is now zeroes. 495 00:30:06,300 --> 00:30:08,550 And I'm interested in solutions. 496 00:30:12,140 --> 00:30:12,810 x-s. 497 00:30:12,810 --> 00:30:18,740 So t- where is this null space for this example? 498 00:30:18,740 --> 00:30:23,600 These x-s are -- have three components. 499 00:30:23,600 --> 00:30:26,310 So the null space is a subspace -- 500 00:30:26,310 --> 00:30:31,390 we still have to show it is a subspace -- of R^3. 501 00:30:31,390 --> 00:30:35,590 So this is -- and we will show -- 502 00:30:35,590 --> 00:30:46,220 these vectors x, this is in R^3, where the column space was 503 00:30:46,220 --> 00:30:50,910 in R^4 in our example. 504 00:30:50,910 --> 00:30:58,340 For an m by n matrix, this is m and this is n, 505 00:30:58,340 --> 00:31:00,570 because the number of columns, n, 506 00:31:00,570 --> 00:31:03,000 tells me how many unknowns, how many x-s 507 00:31:03,000 --> 00:31:05,950 multiply those columns, so it tells me 508 00:31:05,950 --> 00:31:10,480 the big space, in this case R three that I'm in. 509 00:31:10,480 --> 00:31:14,300 Now tell me -- why don't we figure out what the null space 510 00:31:14,300 --> 00:31:20,310 is for this example, just by looking at it. 511 00:31:20,310 --> 00:31:24,740 I mean, that's the beauty of small examples, 512 00:31:24,740 --> 00:31:30,500 that my official way to find null spaces and column spaces 513 00:31:30,500 --> 00:31:36,200 and get all the facts straight would be elimination, 514 00:31:36,200 --> 00:31:37,340 and we'll do that. 515 00:31:37,340 --> 00:31:40,120 But with a small example, we can see that -- 516 00:31:40,120 --> 00:31:43,150 see what's going on without going through the mechanics 517 00:31:43,150 --> 00:31:43,950 of elimination. 518 00:31:43,950 --> 00:31:48,670 So this null space -- 519 00:31:48,670 --> 00:31:51,540 so I'm talking about -- again, the null space, 520 00:31:51,540 --> 00:31:53,520 and let me copy again the matrix. 521 00:31:56,530 --> 00:32:04,190 One two three four, one one one one and two three four five. 522 00:32:04,190 --> 00:32:05,330 What's in the null space? 523 00:32:05,330 --> 00:32:11,410 So I'm taking A times x, so let me right it again, 524 00:32:11,410 --> 00:32:16,015 and I want you to solve those four equations. 525 00:32:19,500 --> 00:32:21,490 In fact, I want you to find all solutions 526 00:32:21,490 --> 00:32:24,350 to those four equations. 527 00:32:24,350 --> 00:32:26,870 Well, actually, just first of all find one. 528 00:32:26,870 --> 00:32:28,420 Why should I ask you for all of them? 529 00:32:28,420 --> 00:32:31,020 Tell me one -- well, tell me one solution that y- 530 00:32:31,020 --> 00:32:35,660 you don't even have to look at the matrix to know one solution 531 00:32:35,660 --> 00:32:38,030 to this set of equations. 532 00:32:38,030 --> 00:32:41,430 It is zero vector. 533 00:32:41,430 --> 00:32:46,810 Whatever that matrix is, its null space contains zero -- 534 00:32:46,810 --> 00:32:51,020 because A times the zero vector sure gives the zero right-hand 535 00:32:51,020 --> 00:32:51,660 side. 536 00:32:51,660 --> 00:32:55,700 So the null space certainly contains zero. 537 00:32:55,700 --> 00:32:58,220 A- so it's got a chance to be a vector space now, 538 00:32:58,220 --> 00:33:00,200 and it will turn out it is. 539 00:33:00,200 --> 00:33:00,880 Okay. 540 00:33:00,880 --> 00:33:04,180 Tell me another solution. 541 00:33:04,180 --> 00:33:07,800 So this particular null space -- and of course I'm going to call 542 00:33:07,800 --> 00:33:11,110 it N(A) for null space -- 543 00:33:11,110 --> 00:33:17,160 this contains-- well we've already located the zero 544 00:33:17,160 --> 00:33:20,970 vector, and now you're going to tell me another vector 545 00:33:20,970 --> 00:33:24,550 that's in the null space, another solution, another x, 546 00:33:24,550 --> 00:33:26,340 another -- 547 00:33:26,340 --> 00:33:28,140 you see what I'm asking you for is 548 00:33:28,140 --> 00:33:29,950 a combination of those columns. 549 00:33:29,950 --> 00:33:33,150 That's what I'm always looking at combinations of columns, 550 00:33:33,150 --> 00:33:40,800 but now I'm looking at the weights, the coefficients 551 00:33:40,800 --> 00:33:41,710 in the combination. 552 00:33:41,710 --> 00:33:46,620 So tell me a good set of numbers to put in there. 553 00:33:46,620 --> 00:33:50,420 One one -- STUDENTS: Minus one. 554 00:33:50,420 --> 00:33:51,530 STRANG: One one minus one. 555 00:33:51,530 --> 00:33:53,340 Thanks. 556 00:33:53,340 --> 00:33:55,280 One one minus one. 557 00:33:55,280 --> 00:33:56,730 So there's a vector that's in it. 558 00:33:59,260 --> 00:34:00,250 Okay. 559 00:34:00,250 --> 00:34:02,600 But have I got a subspace at this point? 560 00:34:02,600 --> 00:34:04,690 Certainly not, right? 561 00:34:04,690 --> 00:34:06,850 I've got just a couple of vectors. 562 00:34:06,850 --> 00:34:08,860 No way they make a subspace. 563 00:34:08,860 --> 00:34:12,940 Tell me -- actually, why don't I jump the whole way now? 564 00:34:12,940 --> 00:34:16,840 Tell me -- well, tell me one more solution, 565 00:34:16,840 --> 00:34:19,460 one more X that would work. 566 00:34:19,460 --> 00:34:21,199 Student: 2 2 -2. 567 00:34:21,199 --> 00:34:22,520 STRANG: 2 2 -2? 568 00:34:22,520 --> 00:34:27,750 Oh, well, tell me all of them, that would have been easier. 569 00:34:27,750 --> 00:34:30,370 Tell me the whole lot, now. 570 00:34:30,370 --> 00:34:34,389 What is the null space for this matrix? 571 00:34:34,389 --> 00:34:38,670 It's all vectors of the form -- what could this be? 572 00:34:38,670 --> 00:34:44,900 It could be one one minus one, it could be it could be any 573 00:34:44,900 --> 00:34:50,880 number C, any number -- the same number again and -- 574 00:34:50,880 --> 00:34:52,010 STUDENTS: Minus. 575 00:34:52,010 --> 00:34:53,420 STRANG: Minus C. 576 00:34:53,420 --> 00:34:55,010 In other words -- 577 00:34:55,010 --> 00:34:59,910 actually, any multiple of this guy. 578 00:34:59,910 --> 00:35:02,270 Oh, that's the perfect description, 579 00:35:02,270 --> 00:35:08,050 because now the zero vector's automatically included 580 00:35:08,050 --> 00:35:10,040 because C could be zero. 581 00:35:10,040 --> 00:35:12,810 The vector I had is included, because C could be one. 582 00:35:12,810 --> 00:35:14,750 But now any vector. 583 00:35:14,750 --> 00:35:16,860 And that's actually it. 584 00:35:20,830 --> 00:35:24,450 And do I have a subspace? 585 00:35:24,450 --> 00:35:26,060 And what does it look like? 586 00:35:26,060 --> 00:35:30,620 It's in -- how would you describe this, the null space, 587 00:35:30,620 --> 00:35:35,810 this -- all these vectors of this form C C minus C, like, 588 00:35:35,810 --> 00:35:38,730 seven seven minus seven. 589 00:35:38,730 --> 00:35:41,080 Minus eleven minus eleven plus eleven. 590 00:35:41,080 --> 00:35:43,810 What have I got here? 591 00:35:43,810 --> 00:35:46,120 If -- describe that whole null space of -- what -- 592 00:35:46,120 --> 00:35:50,550 if I drew it, what do I draw? 593 00:35:50,550 --> 00:35:51,840 A line, right? 594 00:35:51,840 --> 00:35:53,350 The null space is a line. 595 00:35:53,350 --> 00:36:02,724 It's the line through -- in R^3 and the vector one one negative 596 00:36:02,724 --> 00:36:05,140 one maybe goes down here, I don't know where it goes, say, 597 00:36:05,140 --> 00:36:06,530 down here. 598 00:36:06,530 --> 00:36:12,660 There's the vector one one negative one that you gave me. 599 00:36:12,660 --> 00:36:15,610 And where is the vector C C negative C? 600 00:36:15,610 --> 00:36:17,050 It's on this line. 601 00:36:17,050 --> 00:36:19,800 Of course, there's zero zero zero that we had. 602 00:36:19,800 --> 00:36:23,720 And what we've got is that whole -- oops -- that whole line, 603 00:36:23,720 --> 00:36:28,640 going both ways, through the origin. 604 00:36:28,640 --> 00:36:31,030 The null space is a line in R^3. 605 00:36:35,970 --> 00:36:37,930 Okay. 606 00:36:37,930 --> 00:36:43,040 For that example, we could find all the combinations 607 00:36:43,040 --> 00:36:46,590 of the columns that gave zero at sight. 608 00:36:46,590 --> 00:36:51,600 Now, can I just take one more time, 609 00:36:51,600 --> 00:36:57,860 to go back to the definition of subspace, vector space, 610 00:36:57,860 --> 00:37:01,060 and ask you -- 611 00:37:01,060 --> 00:37:05,400 how do I know that the null space is a vector space? 612 00:37:05,400 --> 00:37:08,610 How I entitled to use this word space? 613 00:37:08,610 --> 00:37:13,070 I'll never use that word space without meaning that the two 614 00:37:13,070 --> 00:37:15,800 requirements are satisfied. 615 00:37:15,800 --> 00:37:18,250 Can we just check that they are? 616 00:37:18,250 --> 00:37:20,810 So I'm going to check that -- 617 00:37:20,810 --> 00:37:22,660 can I just continue here? 618 00:37:22,660 --> 00:37:39,240 Check that -- that the solutions to A x=0 always give 619 00:37:39,240 --> 00:37:42,740 a subspace. 620 00:37:42,740 --> 00:37:47,700 And, of course, the key word is that= "Space." 621 00:37:47,700 --> 00:37:49,840 So what do I have to check? 622 00:37:53,410 --> 00:37:57,340 I have to show that if I have one solution, call it x, 623 00:37:57,340 --> 00:38:01,240 and another solution, call it x*, 624 00:38:01,240 --> 00:38:05,970 that their sum is also a solution, right? 625 00:38:05,970 --> 00:38:07,290 That's a requirement. 626 00:38:07,290 --> 00:38:10,010 To use that word space, I have to say -- 627 00:38:10,010 --> 00:38:18,650 I have to convince myself that if A x is zero and also -- 628 00:38:18,650 --> 00:38:25,140 and A x* is zero, or maybe I should have said if A v is zero 629 00:38:25,140 --> 00:38:30,130 and A w is zero, then what about v plus w? 630 00:38:30,130 --> 00:38:32,630 Shall I -- let me use those letters. 631 00:38:32,630 --> 00:38:47,300 If A v is zero and A w is zero, then what -- if that and that, 632 00:38:47,300 --> 00:38:49,240 then what's my point here? 633 00:38:49,240 --> 00:38:55,170 That A times (v+w) must be zero. 634 00:38:55,170 --> 00:38:59,590 That says that if v is in the null space and w's in the null 635 00:38:59,590 --> 00:39:03,680 space, then their sum v+w is in the null space. 636 00:39:03,680 --> 00:39:06,250 And of course, now that I've written it down, 637 00:39:06,250 --> 00:39:11,460 it's totally absurd, ridiculously simple -- 638 00:39:11,460 --> 00:39:17,290 because matrix multiplication allows me to separate that out 639 00:39:17,290 --> 00:39:19,460 into A v plus A w. 640 00:39:22,050 --> 00:39:23,560 I shouldn't say absurdly simple. 641 00:39:23,560 --> 00:39:24,840 That was a dumb thing to say. 642 00:39:24,840 --> 00:39:28,960 Could -- we've used, here, a basic law of matrix 643 00:39:28,960 --> 00:39:30,580 multiplication. 644 00:39:30,580 --> 00:39:34,070 Actually, we've used it without proving it, but that's okay. 645 00:39:34,070 --> 00:39:39,340 We only live so long, we just skip that proof. 646 00:39:39,340 --> 00:39:42,820 I think it's called the distributive law that I can 647 00:39:42,820 --> 00:39:45,770 split these -- split this into two pieces. 648 00:39:45,770 --> 00:39:52,150 But now you see the point, that A v is zero and A w is zero 649 00:39:52,150 --> 00:39:54,520 so I have zero plus zero and I do get zero. 650 00:39:54,520 --> 00:39:56,430 It checks. 651 00:39:56,430 --> 00:40:02,070 And, similarly, I have to show that if A v is zero, 652 00:40:02,070 --> 00:40:10,130 then A times any multiple, say 12v is also zero. 653 00:40:10,130 --> 00:40:11,590 And how do I know that? 654 00:40:11,590 --> 00:40:14,900 Because I'm allowed to s- bring that twelve outside. 655 00:40:14,900 --> 00:40:20,640 A number, a scaler can move outside, so I have twelve A vs, 656 00:40:20,640 --> 00:40:21,740 twelve zeroes -- 657 00:40:21,740 --> 00:40:23,960 I have zero. 658 00:40:23,960 --> 00:40:25,550 Okay. 659 00:40:25,550 --> 00:40:33,070 Just to -- it's really critical to understand the -- 660 00:40:33,070 --> 00:40:34,390 oh yeah. 661 00:40:34,390 --> 00:40:38,220 Here -- I was going to say, understand what's the point 662 00:40:38,220 --> 00:40:39,470 of a vector space? 663 00:40:39,470 --> 00:40:44,790 Let me make that point by changing the right-hand side. 664 00:40:44,790 --> 00:40:45,920 Oops. 665 00:40:45,920 --> 00:40:46,420 Okay. 666 00:40:46,420 --> 00:40:49,740 Let me change the right-hand side to one two three four. 667 00:40:49,740 --> 00:40:51,632 Oh, okay. 668 00:40:51,632 --> 00:40:53,840 Why don't we do all of linear algebra in one lecture, 669 00:40:53,840 --> 00:40:54,890 then we -- 670 00:40:54,890 --> 00:40:55,900 okay. 671 00:40:55,900 --> 00:41:00,500 I would like to know the solutions to this equation. 672 00:41:00,500 --> 00:41:01,670 For those four equations. 673 00:41:05,060 --> 00:41:06,660 So I have four equations. 674 00:41:06,660 --> 00:41:09,039 I have only three unknowns, so if I 675 00:41:09,039 --> 00:41:10,830 don't have a pretty special right-hand side 676 00:41:10,830 --> 00:41:12,640 there won't be any solution at all. 677 00:41:12,640 --> 00:41:16,070 But that is a very special right-hand side. 678 00:41:16,070 --> 00:41:21,450 And we know that there is a solution, one zero zero. 679 00:41:21,450 --> 00:41:23,730 Were there any more solutions? 680 00:41:23,730 --> 00:41:27,320 And did they form a vector space? 681 00:41:27,320 --> 00:41:28,600 Okay. 682 00:41:28,600 --> 00:41:31,550 So I'm asking two questions there. 683 00:41:31,550 --> 00:41:35,530 One is, do -- so my right-hand side now is not zero anymore. 684 00:41:35,530 --> 00:41:37,100 I'm not looking at the null space 685 00:41:37,100 --> 00:41:40,070 because I changed from zeroes. 686 00:41:40,070 --> 00:41:45,930 So my first question is, do the solutions, if there are any 687 00:41:45,930 --> 00:41:50,610 and there are, do they form a subspace? 688 00:41:50,610 --> 00:41:53,270 Let's answer that question first. 689 00:41:53,270 --> 00:41:54,210 Yes or no. 690 00:41:54,210 --> 00:41:59,640 Do I get a subspace if I look at the solutions to -- 691 00:41:59,640 --> 00:42:03,300 let me go back to x1 x2 x3. 692 00:42:03,300 --> 00:42:08,910 I'm looking at all the x-s, at all those vectors in R^3 that 693 00:42:08,910 --> 00:42:10,380 solve A x -b. 694 00:42:10,380 --> 00:42:13,400 The only thing I've changed is b isn't zero anymore. 695 00:42:17,360 --> 00:42:21,045 Do the x-s, the solutions, form a vector space? 696 00:42:25,860 --> 00:42:32,670 The solutions to this do not form a subspace. 697 00:42:32,670 --> 00:42:36,150 The solutions don't, because -- 698 00:42:36,150 --> 00:42:38,460 how shall I see that? 699 00:42:38,460 --> 00:42:42,950 The zero vector is not a solution, so I never even got 700 00:42:42,950 --> 00:42:43,700 started. 701 00:42:43,700 --> 00:42:46,090 The zero vector doesn't solve this system. 702 00:42:46,090 --> 00:42:51,940 I can't -- solutions can't be a vector space. 703 00:42:51,940 --> 00:42:55,480 Now what are they like? 704 00:42:55,480 --> 00:42:58,810 Well, we'll see this, but let's do it for this example. 705 00:42:58,810 --> 00:43:02,200 So one zero zero was a solution. 706 00:43:02,200 --> 00:43:03,690 You saw that right away. 707 00:43:03,690 --> 00:43:05,310 Are there any other solutions? 708 00:43:05,310 --> 00:43:08,730 Can you tell me a second solution 709 00:43:08,730 --> 00:43:10,430 to this system of equations? 710 00:43:10,430 --> 00:43:14,970 STUDENTS: 0 -1 1 STRANG: 0 -1 1. 711 00:43:14,970 --> 00:43:19,560 Boy, that's -- 0 -1 1. 712 00:43:19,560 --> 00:43:20,680 Yes. 713 00:43:20,680 --> 00:43:24,090 Because that says I take minus this column plus this one 714 00:43:24,090 --> 00:43:24,790 and sure enough. 715 00:43:24,790 --> 00:43:27,730 That's right. 716 00:43:27,730 --> 00:43:30,750 So there are -- there's a bunch of solutions here. 717 00:43:35,160 --> 00:43:37,300 But they're not a subspace. 718 00:43:37,300 --> 00:43:38,540 I'll tell you what it's like. 719 00:43:38,540 --> 00:43:41,460 It's like a plane that doesn't go through the origin, 720 00:43:41,460 --> 00:43:43,930 or a line that doesn't go through the origin. 721 00:43:43,930 --> 00:43:45,870 Maybe in this case it's a line that 722 00:43:45,870 --> 00:43:47,870 doesn't go through the origin, if I graft 723 00:43:47,870 --> 00:43:50,570 the solutions to A x equal B. 724 00:43:50,570 --> 00:43:53,550 So you -- I think you've got the idea. 725 00:43:53,550 --> 00:43:56,780 Subspaces have to go through the origin. 726 00:43:56,780 --> 00:44:01,590 If I'm looking at x-s, then they'd better solve Ax=0. 727 00:44:01,590 --> 00:44:04,690 In a way I've got -- 728 00:44:04,690 --> 00:44:10,800 my two subspaces that I -- talking about today are kind 729 00:44:10,800 --> 00:44:17,340 of the two ways I can tell you what a -- about subspace. 730 00:44:17,340 --> 00:44:20,100 If I want to tell you about the column space, 731 00:44:20,100 --> 00:44:24,780 I tell you a few columns and I say take their combinations. 732 00:44:24,780 --> 00:44:27,330 Like I build up this subspace. 733 00:44:27,330 --> 00:44:31,440 I put in a few vectors, their combinations make a subspace. 734 00:44:31,440 --> 00:44:35,010 Now, when I went to -- let me come back to the one that is 735 00:44:35,010 --> 00:44:36,180 a subspace here. 736 00:44:40,980 --> 00:44:44,430 Here, when I talked about the null space, 737 00:44:44,430 --> 00:44:46,960 I didn't tell you what's in it. 738 00:44:46,960 --> 00:44:49,350 We had to figure out what was in it. 739 00:44:49,350 --> 00:44:53,290 What I told you was the equations that I'm -- 740 00:44:53,290 --> 00:44:55,180 that has to be satisfied. 741 00:44:55,180 --> 00:44:56,680 You see those -- 742 00:44:56,680 --> 00:44:59,890 like, those are the two natural ways to tell you 743 00:44:59,890 --> 00:45:02,460 what's in a subspace. 744 00:45:02,460 --> 00:45:06,730 I can either give you a few vectors and say fill it out, 745 00:45:06,730 --> 00:45:08,660 take combinations -- 746 00:45:08,660 --> 00:45:12,800 or I can give you a system of equations, the requirements 747 00:45:12,800 --> 00:45:17,390 that the x-s have to satisfy. 748 00:45:17,390 --> 00:45:20,400 And both of those ways produce subspaces 749 00:45:20,400 --> 00:45:25,080 and they're the important ways to construct subspaces. 750 00:45:25,080 --> 00:45:29,730 Okay, so today's lecture actually got, 751 00:45:29,730 --> 00:45:33,260 the essentials of three point two, 752 00:45:33,260 --> 00:45:35,250 the idea of the null space. 753 00:45:35,250 --> 00:45:37,740 Now we have to tackle, Wednesday, 754 00:45:37,740 --> 00:45:40,490 the job of how do we actually get hold 755 00:45:40,490 --> 00:45:43,530 of that subspace in an example that's bigger 756 00:45:43,530 --> 00:45:45,960 and we can't see it just by eye. 757 00:45:45,960 --> 00:45:48,770 Okay. 758 00:45:48,770 --> 00:45:57,190 See you Wednesday. 759 00:45:57,190 --> 00:45:58,740 Thanks.