WEBVTT 00:00:09.070 --> 00:00:09.660 Okay. 00:00:09.660 --> 00:00:13.950 This is lecture six in linear algebra, 00:00:13.950 --> 00:00:19.790 and we're at the start of this new chapter, chapter three 00:00:19.790 --> 00:00:22.500 in the text, which is really getting 00:00:22.500 --> 00:00:24.930 to the center of linear algebra. 00:00:24.930 --> 00:00:31.460 And I had time to make a first start on it 00:00:31.460 --> 00:00:33.690 at the end of lecture five. 00:00:33.690 --> 00:00:38.430 But now is lecture six is officially 00:00:38.430 --> 00:00:42.610 the lecture on vector spaces and subspaces. 00:00:42.610 --> 00:00:47.070 And then especially -- there are two subspaces that 00:00:47.070 --> 00:00:49.840 we're specially interested in. 00:00:49.840 --> 00:00:52.680 One is the column space of a matrix, 00:00:52.680 --> 00:00:55.630 the other is the null space of the matrix. 00:00:55.630 --> 00:00:59.050 So, I got to tell you what those are. 00:00:59.050 --> 00:01:00.140 Okay. 00:01:00.140 --> 00:01:05.140 So, first to remember from lecture five, 00:01:05.140 --> 00:01:08.350 what is a vector space? 00:01:08.350 --> 00:01:14.410 It's a bunch of vectors that -- where I'm allowed -- 00:01:14.410 --> 00:01:15.970 where I can add -- 00:01:15.970 --> 00:01:19.690 I can add any two vectors in the space 00:01:19.690 --> 00:01:23.300 and the answer stays in the space. 00:01:23.300 --> 00:01:27.610 Or I can multiply any vector in the space by any constant 00:01:27.610 --> 00:01:30.870 and the result stays in the space. 00:01:30.870 --> 00:01:34.950 So that's -- in fact if I combine those two into one, 00:01:34.950 --> 00:01:40.710 you can see that -- if I can add and I can multiply by numbers, 00:01:40.710 --> 00:01:44.750 that really means that I can take linear combinations. 00:01:44.750 --> 00:01:50.220 So the quick way to say it is that all linear combinations, 00:01:50.220 --> 00:01:54.250 C -- any multiple of V plus any multiple of W stay 00:01:54.250 --> 00:01:56.220 in the space. 00:01:56.220 --> 00:02:01.070 So, can I give you examples that are vector spaces and also 00:02:01.070 --> 00:02:04.850 some examples that are not, to make that point clear? 00:02:04.850 --> 00:02:10.220 So, suppose I'm in three dimensions. 00:02:10.220 --> 00:02:16.950 Then one way to get us one space is the whole three dimensional 00:02:16.950 --> 00:02:18.200 space. 00:02:18.200 --> 00:02:22.770 So the whole space R^3, three dimensional space, 00:02:22.770 --> 00:02:27.100 would be a vector space, because if I have a couple of vectors I 00:02:27.100 --> 00:02:30.040 can add them and I'm certainly okay and they follow all 00:02:30.040 --> 00:02:32.240 the rules. 00:02:32.240 --> 00:02:34.600 So R^3 is easy. 00:02:34.600 --> 00:02:39.320 Now I'm interested also in subspaces. 00:02:39.320 --> 00:02:42.370 So there's this key word, subspaces. 00:02:42.370 --> 00:02:48.530 That's a space -- that's some vectors inside the given space, 00:02:48.530 --> 00:02:53.940 inside R three that still make up a vector space of their own. 00:02:53.940 --> 00:02:56.990 It's a vector space inside a vector space. 00:02:56.990 --> 00:03:00.030 And the simplest example was a plane. 00:03:00.030 --> 00:03:05.070 So, like, can I just sketch it -- there is a plane. 00:03:05.070 --> 00:03:07.280 It's got to go through the origin, 00:03:07.280 --> 00:03:10.390 and of course it goes infinitely far. 00:03:10.390 --> 00:03:12.620 That's of that's a subspace now. 00:03:12.620 --> 00:03:18.190 Do you see that if I have two vectors on the plane 00:03:18.190 --> 00:03:22.920 and I add them, the result stays in the plane. 00:03:22.920 --> 00:03:26.770 If I take a vector in the plane and I multiply by minus two, 00:03:26.770 --> 00:03:28.520 I'm still in the plane. 00:03:28.520 --> 00:03:31.240 So that plane is a subspace. 00:03:31.240 --> 00:03:33.120 So let me just make that point. 00:03:33.120 --> 00:03:44.120 Plane through zero, through that zero zero zero is a subspace. 00:03:44.120 --> 00:03:45.060 Okay. 00:03:45.060 --> 00:03:51.710 And also, another subspace would be a line. 00:03:51.710 --> 00:03:54.230 A line through zero zero zero -- yeah, 00:03:54.230 --> 00:03:56.240 the line has to go through the origin. 00:03:56.240 --> 00:03:59.390 All subspaces have got to contain the origin, 00:03:59.390 --> 00:04:01.910 contain zero -- the zero vector. 00:04:01.910 --> 00:04:07.015 So this line is a subspace. 00:04:09.900 --> 00:04:13.910 Really, if I want to say it really correctly, 00:04:13.910 --> 00:04:18.899 I should say a subspace of R^3. 00:04:18.899 --> 00:04:23.710 That of R^3 was, like, understood there. 00:04:23.710 --> 00:04:30.730 Now -- so let me call this plane P. And let me call this line L. 00:04:30.730 --> 00:04:40.540 And let me ask you about other sets of vectors. 00:04:40.540 --> 00:04:43.210 Suppose I took -- 00:04:43.210 --> 00:04:45.310 yeah -- so here's a first question. 00:04:45.310 --> 00:04:50.030 Suppose I take two subspaces, like P and L. 00:04:50.030 --> 00:04:54.610 And I just put them together, take their union, 00:04:54.610 --> 00:04:56.890 take all the vectors -- 00:04:56.890 --> 00:05:00.880 so now you've got P and L in mind, here. 00:05:00.880 --> 00:05:03.870 So I have two subspaces. 00:05:03.870 --> 00:05:09.960 I have two subspaces and, for example, P -- 00:05:09.960 --> 00:05:12.070 a plane and L a line. 00:05:12.070 --> 00:05:13.410 Okay. 00:05:13.410 --> 00:05:17.970 Now I want to ask you about the union of those. 00:05:17.970 --> 00:05:20.190 So P union L. 00:05:20.190 --> 00:05:29.990 This is all vectors in P or L or both. 00:05:33.120 --> 00:05:36.240 Is that a subspace? 00:05:36.240 --> 00:05:38.040 Is this a subspace? 00:05:38.040 --> 00:05:48.380 This is or is not a subspace? 00:05:48.380 --> 00:05:51.690 Because we're -- I just want to be sure that I've got 00:05:51.690 --> 00:05:54.340 the central idea. 00:05:54.340 --> 00:05:57.200 Suppose I take the vectors in the plane 00:05:57.200 --> 00:06:03.230 and also the vectors on that line, 00:06:03.230 --> 00:06:05.930 put them together, so I've got a bunch of vectors, 00:06:05.930 --> 00:06:07.030 is it a subspace? 00:06:07.030 --> 00:06:09.210 Can you give me, like, so the camera 00:06:09.210 --> 00:06:11.700 can hear it or maybe the tape. 00:06:11.700 --> 00:06:14.650 Can you say yes or no? 00:06:14.650 --> 00:06:20.770 Do I have a subspace if I put -- if I take all the vectors 00:06:20.770 --> 00:06:25.660 on the plane plus all -- and all the ones on the line and just 00:06:25.660 --> 00:06:30.520 join them together -- but I'm not taking this guy that's -- 00:06:30.520 --> 00:06:32.240 actually, I'm not taking most of them, 00:06:32.240 --> 00:06:35.680 because most vectors are not on the line or the plane, 00:06:35.680 --> 00:06:37.590 they're off somewhere else. 00:06:37.590 --> 00:06:39.970 Do I have a subspace? 00:06:39.970 --> 00:06:40.697 STUDENTS: No. 00:06:40.697 --> 00:06:41.280 STRANG: Right. 00:06:41.280 --> 00:06:41.790 Thank you. 00:06:41.790 --> 00:06:42.510 No. 00:06:42.510 --> 00:06:44.980 Because -- why not? 00:06:44.980 --> 00:06:47.130 Because I can't add. 00:06:47.130 --> 00:06:52.090 Because if I that requirement isn't satisfied. 00:06:52.090 --> 00:06:56.050 If I take one vector like this guy and another vector 00:06:56.050 --> 00:07:00.100 that happens to come from L and add, I'm off somewhere else. 00:07:00.100 --> 00:07:06.870 You see that I've gone outside the union if I just add 00:07:06.870 --> 00:07:09.070 something from P and something from L, 00:07:09.070 --> 00:07:15.550 then normally what'll happen is I'm outside the union -- 00:07:15.550 --> 00:07:16.770 and I don't have a subspace. 00:07:16.770 --> 00:07:19.120 So the correct answer is -- 00:07:19.120 --> 00:07:21.270 is not. 00:07:21.270 --> 00:07:21.850 Okay. 00:07:21.850 --> 00:07:25.630 Now let me ask you about -- the other thing we do is take 00:07:25.630 --> 00:07:28.240 the intersection. 00:07:28.240 --> 00:07:30.410 So what does intersection mean? 00:07:30.410 --> 00:07:43.830 Intersection means all vectors that are in both P and L. 00:07:43.830 --> 00:07:45.970 Is this a subspace. 00:07:45.970 --> 00:07:49.730 Yeah, so I guess I want to go back up to the same question. 00:07:49.730 --> 00:07:52.360 This is or is not a subspace? 00:07:52.360 --> 00:07:56.910 And you can answer me -- answer the question first for this 00:07:56.910 --> 00:08:00.640 particular example, this picture I drew. 00:08:00.640 --> 00:08:03.780 What is P intersect L for this case? 00:08:03.780 --> 00:08:05.280 STUDENT: It's only zero. 00:08:05.280 --> 00:08:07.810 STRANG: It's only zero. 00:08:07.810 --> 00:08:11.240 At least, sort of this was the artist's idea as he drew it 00:08:11.240 --> 00:08:17.250 that, that that line L was not in the plane and, went off 00:08:17.250 --> 00:08:21.120 somewhere else -- and then the only point that was in common 00:08:21.120 --> 00:08:22.770 was the zero vector. 00:08:22.770 --> 00:08:25.430 Is the zero vector by itself a subspace? 00:08:25.430 --> 00:08:26.000 STUDENT: Yes. 00:08:26.000 --> 00:08:27.740 STRANG: Yes, absolutely. 00:08:27.740 --> 00:08:34.049 And what about, if I don't have this plane and this line 00:08:34.049 --> 00:08:38.789 but any subspace and any other subspace? 00:08:38.789 --> 00:08:43.740 So now -- can I ask that question for any two subspaces? 00:08:43.740 --> 00:08:46.800 So maybe I'll write it up here. 00:08:46.800 --> 00:08:50.240 If I'm strong enough. 00:08:50.240 --> 00:08:51.000 Okay. 00:08:51.000 --> 00:08:52.510 So this is the general question. 00:08:52.510 --> 00:09:00.350 I have subspaces, say S and T. 00:09:00.350 --> 00:09:08.640 And I want to ask you about their intersection S intersect 00:09:08.640 --> 00:09:11.300 T and I want -- 00:09:11.300 --> 00:09:13.220 it is a subspace. 00:09:17.700 --> 00:09:19.430 Do you see why? 00:09:19.430 --> 00:09:24.610 Do you see why if I take the vectors that are in both one- 00:09:24.610 --> 00:09:27.930 th- that are in both of the subspaces -- 00:09:27.930 --> 00:09:30.490 so that's like a smaller set of vectors, probably, 00:09:30.490 --> 00:09:32.540 because it's -- we've added requirements. 00:09:32.540 --> 00:09:36.370 It has to be in S and in T. 00:09:36.370 --> 00:09:38.100 How do I know that's a subspace? 00:09:38.100 --> 00:09:40.560 Can we just think through that abstract stuff 00:09:40.560 --> 00:09:43.640 and then I get to the examples. 00:09:43.640 --> 00:09:44.870 Okay. 00:09:44.870 --> 00:09:45.940 So why? 00:09:45.940 --> 00:09:49.490 Suppose I take a couple of vectors 00:09:49.490 --> 00:09:51.040 that are in the intersection. 00:09:51.040 --> 00:09:56.500 Why is the sum also in the intersection? 00:09:56.500 --> 00:10:00.900 Okay, so let me give a name to these vectors, say V and W. 00:10:00.900 --> 00:10:02.280 They're in the intersection. 00:10:02.280 --> 00:10:05.310 So that means they're both in S. 00:10:05.310 --> 00:10:08.370 Also means they're both in T. 00:10:08.370 --> 00:10:11.960 So what can I say about V plus W? 00:10:11.960 --> 00:10:14.350 Is it in S? 00:10:14.350 --> 00:10:15.040 Yes. 00:10:15.040 --> 00:10:16.480 Right? 00:10:16.480 --> 00:10:21.300 If I take two vectors, V and W that are both in S, 00:10:21.300 --> 00:10:24.900 then the sum is in S, because S was a subspace. 00:10:24.900 --> 00:10:27.830 And if they're both in T and I add them, 00:10:27.830 --> 00:10:31.990 then the result is also in T, because T was a subspace. 00:10:31.990 --> 00:10:37.100 So the result V plus W is in the intersection. 00:10:37.100 --> 00:10:42.690 It's in both and requirement one is satisfied. 00:10:42.690 --> 00:10:44.480 Requirement two's the same. 00:10:44.480 --> 00:10:49.080 If I take a vector that's in both, I multiply by seven. 00:10:49.080 --> 00:10:53.470 Seven times that vector is in S, because the vector was in S. 00:10:53.470 --> 00:10:57.290 Seven times that vector's in T because the original one was in 00:10:57.290 --> 00:11:01.080 T. So seven times that vector is in the intersection. 00:11:01.080 --> 00:11:05.740 In other words, when you take the intersection of two 00:11:05.740 --> 00:11:08.830 subspaces, you get probably a smaller subspace, 00:11:08.830 --> 00:11:10.440 but it is a subspace. 00:11:10.440 --> 00:11:11.780 Okay. 00:11:11.780 --> 00:11:17.270 So that's like sort of just emphasizing what 00:11:17.270 --> 00:11:20.080 these two requirements mean. 00:11:20.080 --> 00:11:24.700 Again -- Let me circle those, because those are so important. 00:11:24.700 --> 00:11:30.490 The sum and the scale of multiplication which combines 00:11:30.490 --> 00:11:33.850 into linear combinations. 00:11:33.850 --> 00:11:37.130 That's what you have to do inside the subspace. 00:11:37.130 --> 00:11:37.980 Okay. 00:11:37.980 --> 00:11:41.530 On to the column space. 00:11:41.530 --> 00:11:42.030 Okay. 00:11:42.030 --> 00:11:46.690 So my lecture last time started that and I want to continue it. 00:11:46.690 --> 00:11:47.190 Okay. 00:11:47.190 --> 00:11:48.680 Column space of a matrix. 00:11:53.690 --> 00:11:54.190 Of A. 00:11:54.190 --> 00:11:56.060 Okay. 00:11:56.060 --> 00:11:59.240 Can I take an example? 00:11:59.240 --> 00:12:03.030 Say one two three four. 00:12:03.030 --> 00:12:06.410 One one one one. 00:12:06.410 --> 00:12:07.570 Two three four five. 00:12:11.760 --> 00:12:14.370 Okay. 00:12:14.370 --> 00:12:19.610 That's my matrix A. 00:12:19.610 --> 00:12:25.900 So, it's got columns, three columns. 00:12:25.900 --> 00:12:29.700 Those columns are vectors, so the column space of this A, 00:12:29.700 --> 00:12:31.700 of this A -- 00:12:31.700 --> 00:12:33.970 let's stay with this example for a while. 00:12:33.970 --> 00:12:40.520 The column space of this matrix is a subspace of R -- 00:12:40.520 --> 00:12:41.600 R what? 00:12:41.600 --> 00:12:43.990 So what space are we in if I'm looking 00:12:43.990 --> 00:12:46.730 at the columns of this matrix? 00:12:46.730 --> 00:12:48.770 R^4 , right? 00:12:48.770 --> 00:12:52.680 These are vectors in R^4, they're four dimensional 00:12:52.680 --> 00:12:53.800 vectors. 00:12:53.800 --> 00:13:04.010 So it's this column space of A is a subspace of R^4 here, 00:13:04.010 --> 00:13:07.030 because A was four by -- 00:13:07.030 --> 00:13:09.190 A is a four by three matrix. 00:13:09.190 --> 00:13:12.080 This tells me how many rows there are, 00:13:12.080 --> 00:13:16.020 how many components in a column, and so we're in R^4. 00:13:16.020 --> 00:13:20.030 Okay, now what's in that subspace? 00:13:20.030 --> 00:13:24.300 So the column space of A, it's a subspace of R^4. 00:13:24.300 --> 00:13:28.790 I call it the column space of A, like that. 00:13:28.790 --> 00:13:34.670 So that's my little symbol for some subspace of R^4. 00:13:34.670 --> 00:13:37.480 What's in that subspace? 00:13:37.480 --> 00:13:40.010 Well, that column certainly is. 00:13:40.010 --> 00:13:41.350 One two three four. 00:13:41.350 --> 00:13:42.460 This column is in. 00:13:42.460 --> 00:13:45.750 This column is in, and what else? 00:13:45.750 --> 00:13:48.480 So it's got the columns of A in it, 00:13:48.480 --> 00:13:51.671 but that's not enough, certainly. 00:13:51.671 --> 00:13:52.170 Right? 00:13:52.170 --> 00:13:55.700 I don't have a subspace if I just put in three vectors. 00:13:55.700 --> 00:13:59.210 So how do I fill that out to be a subspace? 00:13:59.210 --> 00:14:06.000 I take their linear combinations. 00:14:06.000 --> 00:14:15.110 So the column space of A is all linear combinations -- 00:14:15.110 --> 00:14:16.710 combinations of the columns. 00:14:19.480 --> 00:14:22.890 And that does give me a subspace. 00:14:22.890 --> 00:14:24.890 It does give me a vector space, because if I 00:14:24.890 --> 00:14:28.040 have one linear combination and I multiply by eleven, 00:14:28.040 --> 00:14:30.680 I've got another linear combination. 00:14:30.680 --> 00:14:32.190 If I have a linear combination, I 00:14:32.190 --> 00:14:33.730 add to another linear combination 00:14:33.730 --> 00:14:35.970 I get a third combination. 00:14:35.970 --> 00:14:40.260 So that -- this is like the smallest space -- 00:14:40.260 --> 00:14:43.040 like, it's got to have those three columns in it, 00:14:43.040 --> 00:14:45.270 and it has to have their combinations 00:14:45.270 --> 00:14:47.110 and that's where we stop. 00:14:47.110 --> 00:14:47.680 Okay. 00:14:47.680 --> 00:14:54.120 Now I'm going to be interested in that space. 00:14:54.120 --> 00:14:56.930 So I, like -- get some idea of what's in that space. 00:14:56.930 --> 00:14:58.450 How big is that space? 00:14:58.450 --> 00:15:02.880 Is that space the whole four dimensional space? 00:15:02.880 --> 00:15:05.290 Or is it a subspace inside? 00:15:05.290 --> 00:15:13.870 Can you -- let me just see if we can get a yes or no answer 00:15:13.870 --> 00:15:19.930 sometimes without being ready for the complete proof. 00:15:22.660 --> 00:15:23.540 What do you think? 00:15:23.540 --> 00:15:26.590 Is the subspace that I'm talking about here, 00:15:26.590 --> 00:15:28.700 the combinations of those three guys, 00:15:28.700 --> 00:15:32.250 does that fill the full four dimensional space? 00:15:32.250 --> 00:15:34.900 Maybe yes or no on that one. 00:15:34.900 --> 00:15:35.560 No. 00:15:35.560 --> 00:15:36.060 No. 00:15:36.060 --> 00:15:39.640 Somehow our feeling is, and it happens 00:15:39.640 --> 00:15:42.730 to be right, that if we start with three vectors 00:15:42.730 --> 00:15:45.660 and take their combinations, we can't get the whole four 00:15:45.660 --> 00:15:48.680 dimensional space. 00:15:48.680 --> 00:15:52.590 Now -- so somehow we get a smaller space. 00:15:52.590 --> 00:15:55.010 But how much smaller? 00:15:55.010 --> 00:15:57.280 That's going to come up here. 00:15:57.280 --> 00:16:00.970 That's not so immediate. 00:16:00.970 --> 00:16:07.950 Let me first make this critical connection with -- 00:16:07.950 --> 00:16:15.980 with, linear equations, because behind our abstract definition, 00:16:15.980 --> 00:16:17.360 we have a purpose. 00:16:17.360 --> 00:16:19.750 And that is to understand Ax=b. 00:16:19.750 --> 00:16:22.950 So suppose I make the connection -- 00:16:22.950 --> 00:16:33.330 w- w- does A x=b always have a solution for every b? 00:16:33.330 --> 00:16:47.850 Have a solution for every right-hand side? 00:16:47.850 --> 00:16:49.880 I guess that's going to be a yes or no question. 00:16:53.220 --> 00:16:59.580 And then I'm going to ask which right-hand sides are okay? 00:16:59.580 --> 00:17:02.050 That's really the question I'm after, 00:17:02.050 --> 00:17:08.900 is which right-hand sides (b) do make up -- 00:17:08.900 --> 00:17:12.210 you can see from the way I'm speaking what the question -- 00:17:12.210 --> 00:17:13.839 As it is. 00:17:13.839 --> 00:17:16.020 The answer is no. 00:17:16.020 --> 00:17:21.710 A x=b does not have a solution for every b. 00:17:21.710 --> 00:17:26.069 Why do I say no? 00:17:26.069 --> 00:17:34.660 Because A x=b is -- like, this is four equations, 00:17:34.660 --> 00:17:35.940 and only three unknowns. 00:17:39.590 --> 00:17:40.860 Right? 00:17:40.860 --> 00:17:46.057 X is -- let me right out that whole -- 00:17:46.057 --> 00:17:47.390 what the whole thing looks like. 00:17:49.940 --> 00:17:50.600 Yeah. 00:17:50.600 --> 00:17:54.680 Let me write out A x=b. 00:17:54.680 --> 00:17:59.310 A x is -- 00:17:59.310 --> 00:18:02.260 these columns are one two three four. 00:18:02.260 --> 00:18:07.050 One one one one and two three four five. 00:18:07.050 --> 00:18:12.410 Then x, of course, has three components, x1, x2, x3. 00:18:12.410 --> 00:18:15.610 And I'm trying to get the -- 00:18:15.610 --> 00:18:19.257 hit the right-hand side, b1,b2,b3 and b4. 00:18:25.430 --> 00:18:27.850 So my first point is, I can't always do it. 00:18:30.560 --> 00:18:34.140 In a way, that just says again what you told me five minutes 00:18:34.140 --> 00:18:36.220 ago -- 00:18:36.220 --> 00:18:39.540 that the combinations of these columns 00:18:39.540 --> 00:18:42.630 don't fill the whole four dimensional space. 00:18:42.630 --> 00:18:46.190 There's going to be some vectors b, a lot of vectors b, 00:18:46.190 --> 00:18:50.530 that are not combinations of these three columns, 00:18:50.530 --> 00:18:53.200 because the combinations of those columns are, like, 00:18:53.200 --> 00:18:56.130 going to be just a little plane or something inside -- 00:18:56.130 --> 00:18:58.050 inside R^4. 00:18:58.050 --> 00:19:03.990 Now, so and you see that I do have four equations and only 00:19:03.990 --> 00:19:05.210 three unknowns. 00:19:05.210 --> 00:19:09.110 So, like anybody is going to say, no you dope, 00:19:09.110 --> 00:19:11.650 you can't usually solve four equations 00:19:11.650 --> 00:19:13.330 with only three unknowns. 00:19:13.330 --> 00:19:17.710 But now I want to say sometimes you can. 00:19:17.710 --> 00:19:22.330 For some right-hand sides, I can solve this. 00:19:22.330 --> 00:19:25.900 So that's the bunch of right-hand sides 00:19:25.900 --> 00:19:28.710 that I'm interested in right now. 00:19:28.710 --> 00:19:34.860 Which right-hand sides allow me to solve this? 00:19:34.860 --> 00:19:36.740 This is the question for today. 00:19:36.740 --> 00:19:40.240 It's going to have, like, a nice clear answer. 00:19:40.240 --> 00:19:50.800 So my question is -- is which bs, which vectors b, 00:19:50.800 --> 00:19:55.200 allow this system to be solved? 00:19:59.640 --> 00:20:03.150 And I want to ask you -- 00:20:03.150 --> 00:20:08.090 so that's, like, gets two question marks to indicate 00:20:08.090 --> 00:20:10.350 that's -- this is the important question. 00:20:10.350 --> 00:20:15.030 Okay, first, before we give a total answer, 00:20:15.030 --> 00:20:18.000 give me just a partial answer. 00:20:18.000 --> 00:20:20.910 Tell me one right-hand side that I know 00:20:20.910 --> 00:20:23.160 I can solve this thing for. 00:20:23.160 --> 00:20:24.910 So -- all zeroes. 00:20:24.910 --> 00:20:25.410 Okay. 00:20:25.410 --> 00:20:28.000 That's the, like, guaranteed. 00:20:28.000 --> 00:20:31.990 If these were all zero, then I know I can solve it, 00:20:31.990 --> 00:20:35.550 let the x-s all be zero, no problem. 00:20:35.550 --> 00:20:38.890 So that's always a -- okay. 00:20:38.890 --> 00:20:39.660 Okay. 00:20:39.660 --> 00:20:42.460 A x=0 I can always solve. 00:20:42.460 --> 00:20:45.040 Now tell me another right-hand side, 00:20:45.040 --> 00:20:52.740 just a specific set of numbers for which I can solve these 00:20:52.740 --> 00:20:55.700 three -- these four equations with only three unknowns, 00:20:55.700 --> 00:21:00.130 but if you give me a good right-hand side, I can do it. 00:21:00.130 --> 00:21:00.842 So tell me one? 00:21:00.842 --> 00:21:01.550 STUDENT: 1 2 3 4. 00:21:01.550 --> 00:21:04.740 STRANG: 1 2 3 4? 00:21:04.740 --> 00:21:09.575 If I -- can I solve -- is that a good right-hand side? 00:21:12.340 --> 00:21:15.340 Can you solve -- can you find a solution that -- 00:21:15.340 --> 00:21:18.630 X one plus X two plus two X three is one, 00:21:18.630 --> 00:21:22.630 two X one plus X two plus three X three is two and two more 00:21:22.630 --> 00:21:23.858 equations -- 00:21:27.210 --> 00:21:30.220 so I'm asking you to solve in your head in -- 00:21:30.220 --> 00:21:35.750 within five seconds, four equations and three unknowns, 00:21:35.750 --> 00:21:41.310 but you can do it, because the right-hand side is, like, 00:21:41.310 --> 00:21:44.610 showing up here is -- it's one of the columns. 00:21:44.610 --> 00:21:48.330 So tell me what's the X that does solve it? 00:21:48.330 --> 00:21:50.140 One zero zero. 00:21:50.140 --> 00:21:55.960 One zero zero solves it, because -- 00:21:55.960 --> 00:22:01.450 well, so you can multiply this out by rows, but oh God, 00:22:01.450 --> 00:22:06.360 it's much nicer to say -- okay, this is one of this column, 00:22:06.360 --> 00:22:09.190 zero of this, zero of this, so it's one of that column, 00:22:09.190 --> 00:22:11.730 which is exactly what we wanted. 00:22:11.730 --> 00:22:13.290 Okay. 00:22:13.290 --> 00:22:17.280 So there is a b that's okay. 00:22:17.280 --> 00:22:19.250 Now tell me another B that's okay, 00:22:19.250 --> 00:22:23.030 another right-hand side that would be all right? 00:22:23.030 --> 00:22:25.055 Well -- all ones? 00:22:29.480 --> 00:22:33.450 Actually -- and then what's the solution in that case? 00:22:33.450 --> 00:22:36.080 0 1 0, thanks. 00:22:36.080 --> 00:22:40.190 And, in fact, it's much e- like, one way 00:22:40.190 --> 00:22:44.230 to do it is think of a solution first, right, 00:22:44.230 --> 00:22:50.380 and then just see what b turns out to be. 00:22:50.380 --> 00:22:53.030 What b turns out to be, right. 00:22:53.030 --> 00:22:54.790 Okay. 00:22:54.790 --> 00:22:58.220 So I think of a solution -- so I think of an x, 00:22:58.220 --> 00:23:00.380 I think of any -- 00:23:00.380 --> 00:23:03.810 x1, x2, x3, I do this multiplication 00:23:03.810 --> 00:23:04.980 and what have I got? 00:23:09.640 --> 00:23:13.920 Now I'm ready to answer the big question. 00:23:13.920 --> 00:23:20.830 I can solve A x=b exactly when the right-hand side B is 00:23:20.830 --> 00:23:24.920 a vector in the column space. 00:23:24.920 --> 00:23:25.900 Good. 00:23:25.900 --> 00:23:33.140 I can solve A x=b when b is a combination of the columns, 00:23:33.140 --> 00:23:36.130 when it's in the column space -- 00:23:36.130 --> 00:23:39.730 so let me write that answer down. 00:23:39.730 --> 00:23:57.910 I can solve Ax=b exactly when B is in the column space. 00:23:57.910 --> 00:24:02.210 Let me just say again why that is. 00:24:02.210 --> 00:24:07.030 Because it -- the column space by its definition contains all 00:24:07.030 --> 00:24:07.870 the combinations. 00:24:07.870 --> 00:24:10.710 It contains all the Ax-s. 00:24:10.710 --> 00:24:17.360 The column space really consists of all vectors A times any X. 00:24:17.360 --> 00:24:22.900 So those are the bs that I can deal with. 00:24:22.900 --> 00:24:26.130 If b is a combination of the columns, 00:24:26.130 --> 00:24:32.080 then that combination tells me what X should be. 00:24:32.080 --> 00:24:35.630 If b is not a combination of the columns, then there is no x. 00:24:35.630 --> 00:24:38.650 There's no way to solve A x equal b. 00:24:38.650 --> 00:24:40.420 Okay. 00:24:40.420 --> 00:24:42.700 So the column space -- 00:24:42.700 --> 00:24:45.510 that's really why we're interested in this column 00:24:45.510 --> 00:24:48.170 space, because it's the central guy. 00:24:48.170 --> 00:24:56.390 It says when we can solve, and that -- 00:24:56.390 --> 00:24:58.880 we got to understand this column space better. 00:25:02.420 --> 00:25:03.000 Let's see. 00:25:03.000 --> 00:25:07.260 Do I want to think -- 00:25:07.260 --> 00:25:10.600 yeah, somehow -- oh, well, let's just -- 00:25:10.600 --> 00:25:13.360 as long as we've got it here, what do I get for this 00:25:13.360 --> 00:25:14.550 particular example? 00:25:14.550 --> 00:25:24.670 If I take combinations of this and this and this, 00:25:24.670 --> 00:25:28.230 I'll tell you the question that's in my mind. 00:25:28.230 --> 00:25:30.930 It's not even proper to use this word yet, 00:25:30.930 --> 00:25:33.280 but you'll know what it means. 00:25:33.280 --> 00:25:38.060 Are those three columns independent? 00:25:38.060 --> 00:25:44.370 If I take the combinations of the three columns -- 00:25:44.370 --> 00:25:50.550 does each column contribute something new or now? 00:25:50.550 --> 00:25:53.360 So that if I take the combinations of those three 00:25:53.360 --> 00:25:57.160 columns, do I, like, get some three dimensional subspace -- 00:25:57.160 --> 00:26:00.630 do I have three vectors that are, like, you know, 00:26:00.630 --> 00:26:07.440 independent, whatever that means? 00:26:07.440 --> 00:26:10.080 Or do I -- is one of those columns, like, 00:26:10.080 --> 00:26:12.370 contributing nothing new -- 00:26:12.370 --> 00:26:15.630 So that actually only two of the columns 00:26:15.630 --> 00:26:19.040 would have given the same column space? 00:26:19.040 --> 00:26:21.080 Yeah -- that's a good way to ask the question. 00:26:21.080 --> 00:26:22.550 Finally I think of it. 00:26:22.550 --> 00:26:25.340 Can I throw away any columns -- 00:26:25.340 --> 00:26:27.339 and have the same column space? 00:26:27.339 --> 00:26:27.880 STUDENT: Yes. 00:26:27.880 --> 00:26:29.100 STRANG: Yes. 00:26:29.100 --> 00:26:31.290 And which one do you suggest I throw away? 00:26:31.290 --> 00:26:32.670 STUDENT: Column three -- three. 00:26:32.670 --> 00:26:37.350 STRANG: Well, three is the natural, like, guy to target. 00:26:37.350 --> 00:26:39.490 So if I -- and why? 00:26:39.490 --> 00:26:44.750 Because -- what's so bad about three here? 00:26:44.750 --> 00:26:46.240 Column three? 00:26:46.240 --> 00:26:48.310 It's the sum of these, right? 00:26:48.310 --> 00:26:52.370 So it's not -- if I'm taking -- if I have combinations of these 00:26:52.370 --> 00:26:54.510 two and I put in this one, actually, 00:26:54.510 --> 00:26:56.780 I don't get anything more. 00:26:56.780 --> 00:27:02.460 So later on I will call these pivot columns. 00:27:02.460 --> 00:27:06.780 And the third guy will not be a pivot column in this -- 00:27:06.780 --> 00:27:08.760 with those numbers. 00:27:08.760 --> 00:27:12.940 Now actually -- honesty makes me ask you this question. 00:27:12.940 --> 00:27:16.350 Could I have thrown away column one? 00:27:16.350 --> 00:27:17.940 Yes, I could. 00:27:17.940 --> 00:27:19.920 I could. 00:27:19.920 --> 00:27:22.910 So when I say pivot columns, my convention 00:27:22.910 --> 00:27:26.710 is, okay, I'll keep the first ones as long as they're not 00:27:26.710 --> 00:27:27.590 dependent. 00:27:27.590 --> 00:27:30.930 So I keep this guy, he's fine, he's a line. 00:27:30.930 --> 00:27:32.100 I keep the second guy. 00:27:32.100 --> 00:27:33.810 It's in a second direction. 00:27:33.810 --> 00:27:39.220 But the third one, which is in the same plane as the first two 00:27:39.220 --> 00:27:40.790 gives me nothing new. 00:27:40.790 --> 00:27:44.560 It's dependent in the language that we will use 00:27:44.560 --> 00:27:46.990 and I don't need it. 00:27:46.990 --> 00:27:48.160 Okay. 00:27:48.160 --> 00:27:53.570 So I would describe the column space of this matrix as a two 00:27:53.570 --> 00:27:56.040 dimensional subspace of R^4. 00:27:58.680 --> 00:28:01.070 A two dimensional subspace of R^4. 00:28:01.070 --> 00:28:01.990 Okay. 00:28:01.990 --> 00:28:05.430 So you're seeing how these vector spaces work and you -- 00:28:05.430 --> 00:28:08.760 you're seeing that we -- some idea of dependence 00:28:08.760 --> 00:28:11.780 or independence is in our future. 00:28:11.780 --> 00:28:12.400 Okay. 00:28:12.400 --> 00:28:16.590 Now I want to speak about another vector 00:28:16.590 --> 00:28:20.460 space, the null space. 00:28:20.460 --> 00:28:23.890 So again I'm getting a little ahead 00:28:23.890 --> 00:28:28.191 because it's in section three point two, but that's okay. 00:28:28.191 --> 00:28:28.690 All right. 00:28:28.690 --> 00:28:31.870 Now I'm ready for the null space. 00:28:31.870 --> 00:28:33.340 Let me keep the same matrix. 00:28:36.830 --> 00:28:39.370 And this is going to be a different -- 00:28:39.370 --> 00:28:41.500 totally different subspace. 00:28:41.500 --> 00:28:43.270 Totally different. 00:28:43.270 --> 00:28:44.930 Okay. 00:28:44.930 --> 00:28:47.370 Now -- so let me make space for it. 00:28:47.370 --> 00:28:50.190 Now -- here comes a completely different subspace, 00:28:50.190 --> 00:29:00.120 the null space of A. 00:29:00.120 --> 00:29:00.935 What's in it? 00:29:04.240 --> 00:29:10.070 It contains not right-hand sides b. 00:29:10.070 --> 00:29:12.750 It contains x-s. 00:29:12.750 --> 00:29:16.570 It contains all x-s that solve -- 00:29:16.570 --> 00:29:18.040 this word null is going to -- 00:29:18.040 --> 00:29:22.580 I mean, that's the key word here, meaning zero. 00:29:22.580 --> 00:29:30.670 So this contains -- this is all solutions x, 00:29:30.670 --> 00:29:36.450 and of course x is our vectors, x1, x2 and x3, 00:29:36.450 --> 00:29:41.220 to the equation A x=0. 00:29:43.800 --> 00:29:47.800 Well, four equations, because we've got -- 00:29:47.800 --> 00:29:50.230 so, do you see what I'm doing? 00:29:50.230 --> 00:29:53.890 I'm now saying, okay, columns were great, 00:29:53.890 --> 00:29:56.010 the column space we understood. 00:29:56.010 --> 00:29:59.260 Now I'm interested in x-s. 00:29:59.260 --> 00:30:02.860 I'm not -- the only b I'm interested in now is the b 00:30:02.860 --> 00:30:03.670 of all zeroes. 00:30:03.670 --> 00:30:06.300 The right-hand side is now zeroes. 00:30:06.300 --> 00:30:08.550 And I'm interested in solutions. 00:30:12.140 --> 00:30:12.810 x-s. 00:30:12.810 --> 00:30:18.740 So t- where is this null space for this example? 00:30:18.740 --> 00:30:23.600 These x-s are -- have three components. 00:30:23.600 --> 00:30:26.310 So the null space is a subspace -- 00:30:26.310 --> 00:30:31.390 we still have to show it is a subspace -- of R^3. 00:30:31.390 --> 00:30:35.590 So this is -- and we will show -- 00:30:35.590 --> 00:30:46.220 these vectors x, this is in R^3, where the column space was 00:30:46.220 --> 00:30:50.910 in R^4 in our example. 00:30:50.910 --> 00:30:58.340 For an m by n matrix, this is m and this is n, 00:30:58.340 --> 00:31:00.570 because the number of columns, n, 00:31:00.570 --> 00:31:03.000 tells me how many unknowns, how many x-s 00:31:03.000 --> 00:31:05.950 multiply those columns, so it tells me 00:31:05.950 --> 00:31:10.480 the big space, in this case R three that I'm in. 00:31:10.480 --> 00:31:14.300 Now tell me -- why don't we figure out what the null space 00:31:14.300 --> 00:31:20.310 is for this example, just by looking at it. 00:31:20.310 --> 00:31:24.740 I mean, that's the beauty of small examples, 00:31:24.740 --> 00:31:30.500 that my official way to find null spaces and column spaces 00:31:30.500 --> 00:31:36.200 and get all the facts straight would be elimination, 00:31:36.200 --> 00:31:37.340 and we'll do that. 00:31:37.340 --> 00:31:40.120 But with a small example, we can see that -- 00:31:40.120 --> 00:31:43.150 see what's going on without going through the mechanics 00:31:43.150 --> 00:31:43.950 of elimination. 00:31:43.950 --> 00:31:48.670 So this null space -- 00:31:48.670 --> 00:31:51.540 so I'm talking about -- again, the null space, 00:31:51.540 --> 00:31:53.520 and let me copy again the matrix. 00:31:56.530 --> 00:32:04.190 One two three four, one one one one and two three four five. 00:32:04.190 --> 00:32:05.330 What's in the null space? 00:32:05.330 --> 00:32:11.410 So I'm taking A times x, so let me right it again, 00:32:11.410 --> 00:32:16.015 and I want you to solve those four equations. 00:32:19.500 --> 00:32:21.490 In fact, I want you to find all solutions 00:32:21.490 --> 00:32:24.350 to those four equations. 00:32:24.350 --> 00:32:26.870 Well, actually, just first of all find one. 00:32:26.870 --> 00:32:28.420 Why should I ask you for all of them? 00:32:28.420 --> 00:32:31.020 Tell me one -- well, tell me one solution that y- 00:32:31.020 --> 00:32:35.660 you don't even have to look at the matrix to know one solution 00:32:35.660 --> 00:32:38.030 to this set of equations. 00:32:38.030 --> 00:32:41.430 It is zero vector. 00:32:41.430 --> 00:32:46.810 Whatever that matrix is, its null space contains zero -- 00:32:46.810 --> 00:32:51.020 because A times the zero vector sure gives the zero right-hand 00:32:51.020 --> 00:32:51.660 side. 00:32:51.660 --> 00:32:55.700 So the null space certainly contains zero. 00:32:55.700 --> 00:32:58.220 A- so it's got a chance to be a vector space now, 00:32:58.220 --> 00:33:00.200 and it will turn out it is. 00:33:00.200 --> 00:33:00.880 Okay. 00:33:00.880 --> 00:33:04.180 Tell me another solution. 00:33:04.180 --> 00:33:07.800 So this particular null space -- and of course I'm going to call 00:33:07.800 --> 00:33:11.110 it N(A) for null space -- 00:33:11.110 --> 00:33:17.160 this contains-- well we've already located the zero 00:33:17.160 --> 00:33:20.970 vector, and now you're going to tell me another vector 00:33:20.970 --> 00:33:24.550 that's in the null space, another solution, another x, 00:33:24.550 --> 00:33:26.340 another -- 00:33:26.340 --> 00:33:28.140 you see what I'm asking you for is 00:33:28.140 --> 00:33:29.950 a combination of those columns. 00:33:29.950 --> 00:33:33.150 That's what I'm always looking at combinations of columns, 00:33:33.150 --> 00:33:40.800 but now I'm looking at the weights, the coefficients 00:33:40.800 --> 00:33:41.710 in the combination. 00:33:41.710 --> 00:33:46.620 So tell me a good set of numbers to put in there. 00:33:46.620 --> 00:33:50.420 One one -- STUDENTS: Minus one. 00:33:50.420 --> 00:33:51.530 STRANG: One one minus one. 00:33:51.530 --> 00:33:53.340 Thanks. 00:33:53.340 --> 00:33:55.280 One one minus one. 00:33:55.280 --> 00:33:56.730 So there's a vector that's in it. 00:33:59.260 --> 00:34:00.250 Okay. 00:34:00.250 --> 00:34:02.600 But have I got a subspace at this point? 00:34:02.600 --> 00:34:04.690 Certainly not, right? 00:34:04.690 --> 00:34:06.850 I've got just a couple of vectors. 00:34:06.850 --> 00:34:08.860 No way they make a subspace. 00:34:08.860 --> 00:34:12.940 Tell me -- actually, why don't I jump the whole way now? 00:34:12.940 --> 00:34:16.840 Tell me -- well, tell me one more solution, 00:34:16.840 --> 00:34:19.460 one more X that would work. 00:34:19.460 --> 00:34:21.199 Student: 2 2 -2. 00:34:21.199 --> 00:34:22.520 STRANG: 2 2 -2? 00:34:22.520 --> 00:34:27.750 Oh, well, tell me all of them, that would have been easier. 00:34:27.750 --> 00:34:30.370 Tell me the whole lot, now. 00:34:30.370 --> 00:34:34.389 What is the null space for this matrix? 00:34:34.389 --> 00:34:38.670 It's all vectors of the form -- what could this be? 00:34:38.670 --> 00:34:44.900 It could be one one minus one, it could be it could be any 00:34:44.900 --> 00:34:50.880 number C, any number -- the same number again and -- 00:34:50.880 --> 00:34:52.010 STUDENTS: Minus. 00:34:52.010 --> 00:34:53.420 STRANG: Minus C. 00:34:53.420 --> 00:34:55.010 In other words -- 00:34:55.010 --> 00:34:59.910 actually, any multiple of this guy. 00:34:59.910 --> 00:35:02.270 Oh, that's the perfect description, 00:35:02.270 --> 00:35:08.050 because now the zero vector's automatically included 00:35:08.050 --> 00:35:10.040 because C could be zero. 00:35:10.040 --> 00:35:12.810 The vector I had is included, because C could be one. 00:35:12.810 --> 00:35:14.750 But now any vector. 00:35:14.750 --> 00:35:16.860 And that's actually it. 00:35:20.830 --> 00:35:24.450 And do I have a subspace? 00:35:24.450 --> 00:35:26.060 And what does it look like? 00:35:26.060 --> 00:35:30.620 It's in -- how would you describe this, the null space, 00:35:30.620 --> 00:35:35.810 this -- all these vectors of this form C C minus C, like, 00:35:35.810 --> 00:35:38.730 seven seven minus seven. 00:35:38.730 --> 00:35:41.080 Minus eleven minus eleven plus eleven. 00:35:41.080 --> 00:35:43.810 What have I got here? 00:35:43.810 --> 00:35:46.120 If -- describe that whole null space of -- what -- 00:35:46.120 --> 00:35:50.550 if I drew it, what do I draw? 00:35:50.550 --> 00:35:51.840 A line, right? 00:35:51.840 --> 00:35:53.350 The null space is a line. 00:35:53.350 --> 00:36:02.724 It's the line through -- in R^3 and the vector one one negative 00:36:02.724 --> 00:36:05.140 one maybe goes down here, I don't know where it goes, say, 00:36:05.140 --> 00:36:06.530 down here. 00:36:06.530 --> 00:36:12.660 There's the vector one one negative one that you gave me. 00:36:12.660 --> 00:36:15.610 And where is the vector C C negative C? 00:36:15.610 --> 00:36:17.050 It's on this line. 00:36:17.050 --> 00:36:19.800 Of course, there's zero zero zero that we had. 00:36:19.800 --> 00:36:23.720 And what we've got is that whole -- oops -- that whole line, 00:36:23.720 --> 00:36:28.640 going both ways, through the origin. 00:36:28.640 --> 00:36:31.030 The null space is a line in R^3. 00:36:35.970 --> 00:36:37.930 Okay. 00:36:37.930 --> 00:36:43.040 For that example, we could find all the combinations 00:36:43.040 --> 00:36:46.590 of the columns that gave zero at sight. 00:36:46.590 --> 00:36:51.600 Now, can I just take one more time, 00:36:51.600 --> 00:36:57.860 to go back to the definition of subspace, vector space, 00:36:57.860 --> 00:37:01.060 and ask you -- 00:37:01.060 --> 00:37:05.400 how do I know that the null space is a vector space? 00:37:05.400 --> 00:37:08.610 How I entitled to use this word space? 00:37:08.610 --> 00:37:13.070 I'll never use that word space without meaning that the two 00:37:13.070 --> 00:37:15.800 requirements are satisfied. 00:37:15.800 --> 00:37:18.250 Can we just check that they are? 00:37:18.250 --> 00:37:20.810 So I'm going to check that -- 00:37:20.810 --> 00:37:22.660 can I just continue here? 00:37:22.660 --> 00:37:39.240 Check that -- that the solutions to A x=0 always give 00:37:39.240 --> 00:37:42.740 a subspace. 00:37:42.740 --> 00:37:47.700 And, of course, the key word is that= "Space." 00:37:47.700 --> 00:37:49.840 So what do I have to check? 00:37:53.410 --> 00:37:57.340 I have to show that if I have one solution, call it x, 00:37:57.340 --> 00:38:01.240 and another solution, call it x*, 00:38:01.240 --> 00:38:05.970 that their sum is also a solution, right? 00:38:05.970 --> 00:38:07.290 That's a requirement. 00:38:07.290 --> 00:38:10.010 To use that word space, I have to say -- 00:38:10.010 --> 00:38:18.650 I have to convince myself that if A x is zero and also -- 00:38:18.650 --> 00:38:25.140 and A x* is zero, or maybe I should have said if A v is zero 00:38:25.140 --> 00:38:30.130 and A w is zero, then what about v plus w? 00:38:30.130 --> 00:38:32.630 Shall I -- let me use those letters. 00:38:32.630 --> 00:38:47.300 If A v is zero and A w is zero, then what -- if that and that, 00:38:47.300 --> 00:38:49.240 then what's my point here? 00:38:49.240 --> 00:38:55.170 That A times (v+w) must be zero. 00:38:55.170 --> 00:38:59.590 That says that if v is in the null space and w's in the null 00:38:59.590 --> 00:39:03.680 space, then their sum v+w is in the null space. 00:39:03.680 --> 00:39:06.250 And of course, now that I've written it down, 00:39:06.250 --> 00:39:11.460 it's totally absurd, ridiculously simple -- 00:39:11.460 --> 00:39:17.290 because matrix multiplication allows me to separate that out 00:39:17.290 --> 00:39:19.460 into A v plus A w. 00:39:22.050 --> 00:39:23.560 I shouldn't say absurdly simple. 00:39:23.560 --> 00:39:24.840 That was a dumb thing to say. 00:39:24.840 --> 00:39:28.960 Could -- we've used, here, a basic law of matrix 00:39:28.960 --> 00:39:30.580 multiplication. 00:39:30.580 --> 00:39:34.070 Actually, we've used it without proving it, but that's okay. 00:39:34.070 --> 00:39:39.340 We only live so long, we just skip that proof. 00:39:39.340 --> 00:39:42.820 I think it's called the distributive law that I can 00:39:42.820 --> 00:39:45.770 split these -- split this into two pieces. 00:39:45.770 --> 00:39:52.150 But now you see the point, that A v is zero and A w is zero 00:39:52.150 --> 00:39:54.520 so I have zero plus zero and I do get zero. 00:39:54.520 --> 00:39:56.430 It checks. 00:39:56.430 --> 00:40:02.070 And, similarly, I have to show that if A v is zero, 00:40:02.070 --> 00:40:10.130 then A times any multiple, say 12v is also zero. 00:40:10.130 --> 00:40:11.590 And how do I know that? 00:40:11.590 --> 00:40:14.900 Because I'm allowed to s- bring that twelve outside. 00:40:14.900 --> 00:40:20.640 A number, a scaler can move outside, so I have twelve A vs, 00:40:20.640 --> 00:40:21.740 twelve zeroes -- 00:40:21.740 --> 00:40:23.960 I have zero. 00:40:23.960 --> 00:40:25.550 Okay. 00:40:25.550 --> 00:40:33.070 Just to -- it's really critical to understand the -- 00:40:33.070 --> 00:40:34.390 oh yeah. 00:40:34.390 --> 00:40:38.220 Here -- I was going to say, understand what's the point 00:40:38.220 --> 00:40:39.470 of a vector space? 00:40:39.470 --> 00:40:44.790 Let me make that point by changing the right-hand side. 00:40:44.790 --> 00:40:45.920 Oops. 00:40:45.920 --> 00:40:46.420 Okay. 00:40:46.420 --> 00:40:49.740 Let me change the right-hand side to one two three four. 00:40:49.740 --> 00:40:51.632 Oh, okay. 00:40:51.632 --> 00:40:53.840 Why don't we do all of linear algebra in one lecture, 00:40:53.840 --> 00:40:54.890 then we -- 00:40:54.890 --> 00:40:55.900 okay. 00:40:55.900 --> 00:41:00.500 I would like to know the solutions to this equation. 00:41:00.500 --> 00:41:01.670 For those four equations. 00:41:05.060 --> 00:41:06.660 So I have four equations. 00:41:06.660 --> 00:41:09.039 I have only three unknowns, so if I 00:41:09.039 --> 00:41:10.830 don't have a pretty special right-hand side 00:41:10.830 --> 00:41:12.640 there won't be any solution at all. 00:41:12.640 --> 00:41:16.070 But that is a very special right-hand side. 00:41:16.070 --> 00:41:21.450 And we know that there is a solution, one zero zero. 00:41:21.450 --> 00:41:23.730 Were there any more solutions? 00:41:23.730 --> 00:41:27.320 And did they form a vector space? 00:41:27.320 --> 00:41:28.600 Okay. 00:41:28.600 --> 00:41:31.550 So I'm asking two questions there. 00:41:31.550 --> 00:41:35.530 One is, do -- so my right-hand side now is not zero anymore. 00:41:35.530 --> 00:41:37.100 I'm not looking at the null space 00:41:37.100 --> 00:41:40.070 because I changed from zeroes. 00:41:40.070 --> 00:41:45.930 So my first question is, do the solutions, if there are any 00:41:45.930 --> 00:41:50.610 and there are, do they form a subspace? 00:41:50.610 --> 00:41:53.270 Let's answer that question first. 00:41:53.270 --> 00:41:54.210 Yes or no. 00:41:54.210 --> 00:41:59.640 Do I get a subspace if I look at the solutions to -- 00:41:59.640 --> 00:42:03.300 let me go back to x1 x2 x3. 00:42:03.300 --> 00:42:08.910 I'm looking at all the x-s, at all those vectors in R^3 that 00:42:08.910 --> 00:42:10.380 solve A x -b. 00:42:10.380 --> 00:42:13.400 The only thing I've changed is b isn't zero anymore. 00:42:17.360 --> 00:42:21.045 Do the x-s, the solutions, form a vector space? 00:42:25.860 --> 00:42:32.670 The solutions to this do not form a subspace. 00:42:32.670 --> 00:42:36.150 The solutions don't, because -- 00:42:36.150 --> 00:42:38.460 how shall I see that? 00:42:38.460 --> 00:42:42.950 The zero vector is not a solution, so I never even got 00:42:42.950 --> 00:42:43.700 started. 00:42:43.700 --> 00:42:46.090 The zero vector doesn't solve this system. 00:42:46.090 --> 00:42:51.940 I can't -- solutions can't be a vector space. 00:42:51.940 --> 00:42:55.480 Now what are they like? 00:42:55.480 --> 00:42:58.810 Well, we'll see this, but let's do it for this example. 00:42:58.810 --> 00:43:02.200 So one zero zero was a solution. 00:43:02.200 --> 00:43:03.690 You saw that right away. 00:43:03.690 --> 00:43:05.310 Are there any other solutions? 00:43:05.310 --> 00:43:08.730 Can you tell me a second solution 00:43:08.730 --> 00:43:10.430 to this system of equations? 00:43:10.430 --> 00:43:14.970 STUDENTS: 0 -1 1 STRANG: 0 -1 1. 00:43:14.970 --> 00:43:19.560 Boy, that's -- 0 -1 1. 00:43:19.560 --> 00:43:20.680 Yes. 00:43:20.680 --> 00:43:24.090 Because that says I take minus this column plus this one 00:43:24.090 --> 00:43:24.790 and sure enough. 00:43:24.790 --> 00:43:27.730 That's right. 00:43:27.730 --> 00:43:30.750 So there are -- there's a bunch of solutions here. 00:43:35.160 --> 00:43:37.300 But they're not a subspace. 00:43:37.300 --> 00:43:38.540 I'll tell you what it's like. 00:43:38.540 --> 00:43:41.460 It's like a plane that doesn't go through the origin, 00:43:41.460 --> 00:43:43.930 or a line that doesn't go through the origin. 00:43:43.930 --> 00:43:45.870 Maybe in this case it's a line that 00:43:45.870 --> 00:43:47.870 doesn't go through the origin, if I graft 00:43:47.870 --> 00:43:50.570 the solutions to A x equal B. 00:43:50.570 --> 00:43:53.550 So you -- I think you've got the idea. 00:43:53.550 --> 00:43:56.780 Subspaces have to go through the origin. 00:43:56.780 --> 00:44:01.590 If I'm looking at x-s, then they'd better solve Ax=0. 00:44:01.590 --> 00:44:04.690 In a way I've got -- 00:44:04.690 --> 00:44:10.800 my two subspaces that I -- talking about today are kind 00:44:10.800 --> 00:44:17.340 of the two ways I can tell you what a -- about subspace. 00:44:17.340 --> 00:44:20.100 If I want to tell you about the column space, 00:44:20.100 --> 00:44:24.780 I tell you a few columns and I say take their combinations. 00:44:24.780 --> 00:44:27.330 Like I build up this subspace. 00:44:27.330 --> 00:44:31.440 I put in a few vectors, their combinations make a subspace. 00:44:31.440 --> 00:44:35.010 Now, when I went to -- let me come back to the one that is 00:44:35.010 --> 00:44:36.180 a subspace here. 00:44:40.980 --> 00:44:44.430 Here, when I talked about the null space, 00:44:44.430 --> 00:44:46.960 I didn't tell you what's in it. 00:44:46.960 --> 00:44:49.350 We had to figure out what was in it. 00:44:49.350 --> 00:44:53.290 What I told you was the equations that I'm -- 00:44:53.290 --> 00:44:55.180 that has to be satisfied. 00:44:55.180 --> 00:44:56.680 You see those -- 00:44:56.680 --> 00:44:59.890 like, those are the two natural ways to tell you 00:44:59.890 --> 00:45:02.460 what's in a subspace. 00:45:02.460 --> 00:45:06.730 I can either give you a few vectors and say fill it out, 00:45:06.730 --> 00:45:08.660 take combinations -- 00:45:08.660 --> 00:45:12.800 or I can give you a system of equations, the requirements 00:45:12.800 --> 00:45:17.390 that the x-s have to satisfy. 00:45:17.390 --> 00:45:20.400 And both of those ways produce subspaces 00:45:20.400 --> 00:45:25.080 and they're the important ways to construct subspaces. 00:45:25.080 --> 00:45:29.730 Okay, so today's lecture actually got, 00:45:29.730 --> 00:45:33.260 the essentials of three point two, 00:45:33.260 --> 00:45:35.250 the idea of the null space. 00:45:35.250 --> 00:45:37.740 Now we have to tackle, Wednesday, 00:45:37.740 --> 00:45:40.490 the job of how do we actually get hold 00:45:40.490 --> 00:45:43.530 of that subspace in an example that's bigger 00:45:43.530 --> 00:45:45.960 and we can't see it just by eye. 00:45:45.960 --> 00:45:48.770 Okay. 00:45:48.770 --> 00:45:57.190 See you Wednesday. 00:45:57.190 --> 00:45:58.740 Thanks.