WEBVTT 00:00:09.760 --> 00:00:13.450 OK, here is lecture ten in linear algebra. 00:00:13.450 --> 00:00:17.400 Two important things to do in this lecture. 00:00:17.400 --> 00:00:21.840 One is to correct an error from lecture nine. 00:00:21.840 --> 00:00:27.250 So the blackboard with that awful error is still with us. 00:00:27.250 --> 00:00:29.580 And the second, the big thing to do 00:00:29.580 --> 00:00:34.200 is to tell you about the four subspaces that 00:00:34.200 --> 00:00:35.930 come with a matrix. 00:00:35.930 --> 00:00:39.030 We've seen two subspaces, the column space 00:00:39.030 --> 00:00:39.900 and the null space. 00:00:39.900 --> 00:00:42.490 There's two to go. 00:00:42.490 --> 00:00:45.760 First of all, and this is a great way to 00:00:45.760 --> 00:00:50.690 OK. recap and correct the previous lecture -- 00:00:50.690 --> 00:00:54.190 so you remember I was just doing R^3. 00:00:54.190 --> 00:00:58.620 I couldn't have taken a simpler example than R^3. 00:00:58.620 --> 00:01:02.230 And I wrote down the standard basis. 00:01:02.230 --> 00:01:03.945 That's the standard basis. 00:01:06.780 --> 00:01:11.750 The basis -- the obvious basis for the whole three dimensional 00:01:11.750 --> 00:01:12.820 space. 00:01:12.820 --> 00:01:16.500 And then I wanted to make the point 00:01:16.500 --> 00:01:22.940 that there was nothing special, nothing about that basis 00:01:22.940 --> 00:01:25.130 that another basis couldn't have. 00:01:25.130 --> 00:01:26.970 It could have linear independence, 00:01:26.970 --> 00:01:28.530 it could span a space. 00:01:28.530 --> 00:01:30.350 There's lots of other bases. 00:01:30.350 --> 00:01:34.280 So I started with these vectors, one one two and two two five, 00:01:34.280 --> 00:01:36.710 and those were independent. 00:01:36.710 --> 00:01:39.200 And then I said three three seven 00:01:39.200 --> 00:01:42.990 wouldn't do, because three three seven is the sum of those. 00:01:42.990 --> 00:01:47.920 So in my innocence, I put in three three eight. 00:01:47.920 --> 00:01:53.150 I figured probably if three three seven is on the plane, 00:01:53.150 --> 00:01:56.210 is -- which I know, it's in the plane with these two, 00:01:56.210 --> 00:01:59.100 then probably three three eight sticks a little bit out 00:01:59.100 --> 00:02:02.010 of the plane and it's independent and it gives 00:02:02.010 --> 00:02:02.530 a basis. 00:02:02.530 --> 00:02:09.810 But after class, to my sorrow, a student tells me, 00:02:09.810 --> 00:02:12.620 "Wait a minute, that ba- that third vector, three three 00:02:12.620 --> 00:02:14.690 eight, is not independent." 00:02:14.690 --> 00:02:17.620 And why did she say that? 00:02:17.620 --> 00:02:22.560 She didn't actually take the time, didn't have to, 00:02:22.560 --> 00:02:27.890 to find w- w- what combination of this one and this one 00:02:27.890 --> 00:02:30.040 gives three three eight. 00:02:30.040 --> 00:02:33.170 She did something else. 00:02:33.170 --> 00:02:35.090 In other words, she looked ahead, 00:02:35.090 --> 00:02:41.080 because she said, wait a minute, if I look at that matrix, 00:02:41.080 --> 00:02:43.430 it's not invertible. 00:02:43.430 --> 00:02:47.530 That third column can't be independent of the first two, 00:02:47.530 --> 00:02:49.270 because when I look at that matrix, 00:02:49.270 --> 00:02:53.110 it's got two identical rows. 00:02:53.110 --> 00:02:56.490 I have a square matrix. 00:02:56.490 --> 00:03:01.240 Its rows are obviously dependent. 00:03:01.240 --> 00:03:06.670 And that makes the columns dependent. 00:03:06.670 --> 00:03:08.620 So there's my error. 00:03:08.620 --> 00:03:13.800 When I look at the matrix A that has those three columns, 00:03:13.800 --> 00:03:16.090 those three columns can't be independent 00:03:16.090 --> 00:03:18.170 because that matrix is not invertible 00:03:18.170 --> 00:03:21.920 because it's got two equal rows. 00:03:21.920 --> 00:03:27.260 And today's lecture will reach the conclusion, 00:03:27.260 --> 00:03:33.110 the great conclusion, that connects the column 00:03:33.110 --> 00:03:35.305 space with the row space. 00:03:38.030 --> 00:03:43.020 So those are -- the row space is now going to be another one 00:03:43.020 --> 00:03:45.510 of my fundamental subspaces. 00:03:45.510 --> 00:03:49.430 The row space of this matrix, or of this one -- well, 00:03:49.430 --> 00:03:54.880 the row space of this one is OK, but the row space of this one, 00:03:54.880 --> 00:03:58.240 I'm looking at the rows of the matrix -- oh, anyway, 00:03:58.240 --> 00:04:02.620 I'll have two equal rows and the row space will be only two 00:04:02.620 --> 00:04:04.060 dimensional. 00:04:04.060 --> 00:04:08.320 The rank of the matrix with these columns will only be two. 00:04:08.320 --> 00:04:13.730 So only two of those columns, columns can be independent too. 00:04:13.730 --> 00:04:17.730 The rows tell me something about the columns, in other words, 00:04:17.730 --> 00:04:20.560 something that I should have noticed and I didn't. 00:04:20.560 --> 00:04:21.500 OK. 00:04:21.500 --> 00:04:27.470 So now let me pin down these four fundamental subspaces. 00:04:27.470 --> 00:04:30.500 So here are the four fundamental subspaces. 00:04:36.510 --> 00:04:40.470 This is really the heart of this approach to linear algebra, 00:04:40.470 --> 00:04:44.520 to see these four subspaces, how they're related. 00:04:44.520 --> 00:04:45.830 So what are they? 00:04:45.830 --> 00:04:53.540 The column space, C of A. 00:04:53.540 --> 00:05:00.420 The null space, N of A. 00:05:00.420 --> 00:05:05.190 And now comes the row space, something new. 00:05:05.190 --> 00:05:08.700 The row space, what's in that? 00:05:08.700 --> 00:05:12.190 It's all combinations of the rows. 00:05:12.190 --> 00:05:13.620 That's natural. 00:05:13.620 --> 00:05:16.440 We want a space, so we have to take all combinations, 00:05:16.440 --> 00:05:18.240 and we start with the rows. 00:05:18.240 --> 00:05:22.450 So the rows span the row space. 00:05:22.450 --> 00:05:26.200 Are the rows a basis for the row space? 00:05:26.200 --> 00:05:29.140 Maybe so, maybe no. 00:05:29.140 --> 00:05:33.080 The rows are a basis for the row space when they're independent, 00:05:33.080 --> 00:05:39.250 but if they're dependent, as in this example, 00:05:39.250 --> 00:05:43.500 my error from last time, they're not -- 00:05:43.500 --> 00:05:45.440 those three rows are not a basis. 00:05:45.440 --> 00:05:48.280 The row space wouldn't -- would only be two dimensional. 00:05:48.280 --> 00:05:50.280 I only need two rows for a basis. 00:05:50.280 --> 00:05:52.990 So the row space, now what's in it? 00:05:52.990 --> 00:05:55.610 It's all combinations of the rows of A. 00:05:55.610 --> 00:06:01.330 All combinations of the rows of A. 00:06:01.330 --> 00:06:04.610 But I don't like working with row vectors. 00:06:04.610 --> 00:06:06.680 All my vectors have been column vectors. 00:06:06.680 --> 00:06:09.710 I'd like to stay with column vectors. 00:06:09.710 --> 00:06:14.530 How can I get to column vectors out of these rows? 00:06:14.530 --> 00:06:16.800 I transpose the matrix. 00:06:16.800 --> 00:06:20.020 So if that's OK with you, I'm going to transpose the 00:06:20.020 --> 00:06:20.590 matrix. 00:06:20.590 --> 00:06:24.450 I'm, I'm going to say all combinations 00:06:24.450 --> 00:06:31.875 of the columns of A transpose. 00:06:34.990 --> 00:06:40.750 And that allows me to use the convenient notation, the column 00:06:40.750 --> 00:06:42.755 space of A transpose. 00:06:47.310 --> 00:06:50.320 Nothing, no mathematics went on there. 00:06:50.320 --> 00:06:55.130 We just got some vectors that were lying down to stand up. 00:06:55.130 --> 00:06:59.730 But it means that we can use this column 00:06:59.730 --> 00:07:04.120 space of A transpose, that's telling me in a nice matrix 00:07:04.120 --> 00:07:08.950 notation what the row space is. 00:07:08.950 --> 00:07:09.470 OK. 00:07:09.470 --> 00:07:15.900 And finally is another null space. 00:07:15.900 --> 00:07:19.900 The fourth fundamental space will be 00:07:19.900 --> 00:07:22.120 the null space of A transpose. 00:07:22.120 --> 00:07:30.140 The fourth guy is the null space of A transpose. 00:07:30.140 --> 00:07:33.395 And of course my notation is N of A transpose. 00:07:36.230 --> 00:07:40.010 That's the null space of A transpose. 00:07:40.010 --> 00:07:46.250 Eh, we don't have a perfect name for this space as a -- 00:07:46.250 --> 00:07:51.770 connecting with A, but our usual name is the left null space, 00:07:51.770 --> 00:07:54.120 and I'll show you why in a moment. 00:07:54.120 --> 00:07:57.480 So often I call this the -- 00:07:57.480 --> 00:08:10.570 just to write that word -- the left null space of A. 00:08:10.570 --> 00:08:13.420 So just the way we have the row space of A 00:08:13.420 --> 00:08:16.810 and we switch it to the column space of A transpose, 00:08:16.810 --> 00:08:21.590 so we have this space of guys l- that I 00:08:21.590 --> 00:08:25.040 call the left null space of A, but the good notation 00:08:25.040 --> 00:08:27.910 is it's the null space of A transpose. 00:08:27.910 --> 00:08:28.410 OK. 00:08:28.410 --> 00:08:29.770 Those are four spaces. 00:08:32.980 --> 00:08:35.280 Where are those spaces? 00:08:35.280 --> 00:08:40.059 What, what big space are they in for -- when A is m by n? 00:08:47.720 --> 00:08:52.370 In that case, the null space of A, 00:08:52.370 --> 00:08:54.100 what's in the null space of A? 00:08:54.100 --> 00:08:58.850 Vectors with n components, solutions to A x equals zero. 00:08:58.850 --> 00:09:01.663 So the null space of A is in R^n. 00:09:04.800 --> 00:09:06.480 What's in the column space of A? 00:09:06.480 --> 00:09:08.220 Well, columns. 00:09:08.220 --> 00:09:11.320 How many components dothose columns have? 00:09:11.320 --> 00:09:12.350 m. 00:09:12.350 --> 00:09:15.840 So this column space is in R^m. 00:09:19.680 --> 00:09:22.010 What about the column space of A transpose, 00:09:22.010 --> 00:09:26.960 which are just a disguised way of saying the rows of A? 00:09:26.960 --> 00:09:31.040 The rows of A, in this three by six matrix, 00:09:31.040 --> 00:09:36.680 have six components, n components. 00:09:36.680 --> 00:09:38.420 The column space is in R^n. 00:09:42.790 --> 00:09:45.950 And the null space of A transpose, 00:09:45.950 --> 00:09:51.800 I see that this fourth space is already getting second, 00:09:51.800 --> 00:09:55.640 you know, second class citizen treatment 00:09:55.640 --> 00:09:57.550 and it doesn't deserve it. 00:09:57.550 --> 00:10:00.830 It's, it should be there, it is there, 00:10:00.830 --> 00:10:03.430 and shouldn't be squeezed. 00:10:03.430 --> 00:10:06.650 The null space of A transpose -- 00:10:06.650 --> 00:10:10.520 well, if the null space of A had vectors with n components, 00:10:10.520 --> 00:10:14.970 the null space of A transpose will be in R^m. 00:10:14.970 --> 00:10:19.260 I want to draw a picture of the four spaces. 00:10:19.260 --> 00:10:19.760 OK. 00:10:24.580 --> 00:10:25.130 OK. 00:10:25.130 --> 00:10:26.380 Here are the four spaces. 00:10:32.450 --> 00:10:38.580 OK, Let me put n dimensional space over on this side. 00:10:38.580 --> 00:10:42.760 Then which were the subspaces in R^n? 00:10:42.760 --> 00:10:47.490 The null space was and the row space was. 00:10:47.490 --> 00:10:53.471 So here we have the -- can I make that picture of the row 00:10:53.471 --> 00:10:53.970 space? 00:10:57.180 --> 00:11:00.880 And can I make this kind of picture of the null space? 00:11:06.140 --> 00:11:07.880 That's just meant to be a sketch, 00:11:07.880 --> 00:11:13.800 to remind you that they're in this -- which you know, how -- 00:11:13.800 --> 00:11:16.420 what type of vectors are in it? 00:11:16.420 --> 00:11:18.560 Vectors with n components. 00:11:18.560 --> 00:11:25.560 Over here, inside, consisting of vectors with m components, 00:11:25.560 --> 00:11:33.890 is the column space and what I'm calling 00:11:33.890 --> 00:11:39.600 the null space of A transpose. 00:11:39.600 --> 00:11:42.610 Those are the ones with m components. 00:11:42.610 --> 00:11:44.820 OK. 00:11:44.820 --> 00:11:48.760 To understand these spaces is our, is our job now. 00:11:48.760 --> 00:11:50.880 Because by understanding those spaces, 00:11:50.880 --> 00:11:56.730 we know everything about this half of linear algebra. 00:11:56.730 --> 00:11:59.640 What do I mean by understanding those spaces? 00:11:59.640 --> 00:12:03.250 I would like to know a basis for those spaces. 00:12:07.640 --> 00:12:10.090 For each one of those spaces, how would I create -- 00:12:10.090 --> 00:12:11.440 construct a basis? 00:12:11.440 --> 00:12:15.260 What systematic way would produce a basis? 00:12:15.260 --> 00:12:17.180 And what's their dimension? 00:12:24.611 --> 00:12:25.110 OK. 00:12:28.380 --> 00:12:30.070 So for each of the four spaces, I 00:12:30.070 --> 00:12:32.500 have to answer those questions. 00:12:32.500 --> 00:12:34.780 How do I produce a basis? 00:12:34.780 --> 00:12:38.620 And then -- which has a somewhat long answer. 00:12:38.620 --> 00:12:42.210 And what's the dimension, which is just a number, 00:12:42.210 --> 00:12:44.030 so it has a real short answer. 00:12:44.030 --> 00:12:46.250 Can I give you the short answer first? 00:12:46.250 --> 00:12:50.930 I shouldn't do it, but here it is. 00:12:50.930 --> 00:12:56.206 I can tell you the dimension of the column space. 00:12:56.206 --> 00:12:57.330 Let me start with this guy. 00:12:57.330 --> 00:12:58.590 What's its dimension? 00:12:58.590 --> 00:13:01.360 I have an m by n matrix. 00:13:01.360 --> 00:13:09.041 The dimension of the column space is the rank, 00:13:09.041 --> 00:13:09.540 r. 00:13:13.600 --> 00:13:17.210 We actually got to that at the end of the last lecture, 00:13:17.210 --> 00:13:19.570 but only for an example. 00:13:19.570 --> 00:13:24.520 So I really have to say, OK, what's going on there. 00:13:24.520 --> 00:13:28.340 I should produce a basis and then I 00:13:28.340 --> 00:13:31.260 just look to see how many vectors I needed in that basis, 00:13:31.260 --> 00:13:34.360 and the answer will be r. 00:13:34.360 --> 00:13:38.980 Actually, I'll do that, before I get on to the others. 00:13:38.980 --> 00:13:41.050 What's a basis for the columns space? 00:13:43.770 --> 00:13:46.980 We've done all the work of row reduction, 00:13:46.980 --> 00:13:50.810 identifying the pivot columns, the ones that 00:13:50.810 --> 00:13:53.940 have pivots, the ones that end up with pivots. 00:13:53.940 --> 00:13:57.500 But now I -- the pivot columns I'm interested in are columns 00:13:57.500 --> 00:14:00.530 of A, the original A. 00:14:00.530 --> 00:14:03.610 And those pivot columns, there are r of them. 00:14:03.610 --> 00:14:05.840 The rank r counts those. 00:14:05.840 --> 00:14:07.550 Those are a basis. 00:14:07.550 --> 00:14:12.520 So if I answer this question for the column space, 00:14:12.520 --> 00:14:18.100 the answer will be a basis is the pivot columns 00:14:18.100 --> 00:14:23.790 and the dimension is the rank r, and there are r pivot columns 00:14:23.790 --> 00:14:26.360 and everything great. 00:14:26.360 --> 00:14:26.950 OK. 00:14:26.950 --> 00:14:29.920 So that space we pretty well understand. 00:14:32.560 --> 00:14:36.090 I probably have a little going back to see that -- 00:14:36.090 --> 00:14:39.750 to prove that this is a right answer, 00:14:39.750 --> 00:14:42.460 but you know it's the right answer. 00:14:42.460 --> 00:14:45.255 Now let me look at the row space. 00:14:49.340 --> 00:14:51.830 OK. 00:14:51.830 --> 00:14:55.190 Shall I tell you the dimension of the row space? 00:14:55.190 --> 00:14:55.830 Yes. 00:14:55.830 --> 00:14:58.180 Before we do even an example, let me tell you 00:14:58.180 --> 00:15:00.220 the dimension of the row space. 00:15:00.220 --> 00:15:03.650 Its dimension is also r. 00:15:03.650 --> 00:15:06.830 The row space and the column space have the same dimension. 00:15:06.830 --> 00:15:08.440 That's a wonderful fact. 00:15:08.440 --> 00:15:12.950 The dimension of the column space of A transpose -- 00:15:12.950 --> 00:15:17.170 that's the row space -- is r. 00:15:17.170 --> 00:15:19.430 That, that space is r dimensional. 00:15:22.100 --> 00:15:24.790 Snd so is this one. 00:15:24.790 --> 00:15:27.570 OK. 00:15:27.570 --> 00:15:35.430 That's the sort of insight that got used in this example. 00:15:35.430 --> 00:15:40.990 If those -- are the three columns of a matrix -- 00:15:40.990 --> 00:15:44.720 let me make them the three columns of a matrix by just 00:15:44.720 --> 00:15:47.000 erasing some brackets. 00:15:47.000 --> 00:15:51.320 OK, those are the three columns of a matrix. 00:15:51.320 --> 00:15:54.130 The rank of that matrix, if I look at the columns, 00:15:54.130 --> 00:15:57.560 it wasn't obvious to me anyway. 00:15:57.560 --> 00:16:01.250 But if I look at the rows, now it's obvious. 00:16:01.250 --> 00:16:03.860 The row space of that matrix obviously 00:16:03.860 --> 00:16:07.380 is two dimensional, because I see a basis for the row 00:16:07.380 --> 00:16:10.650 space, this row and that row. 00:16:10.650 --> 00:16:12.370 And of course, strictly speaking, 00:16:12.370 --> 00:16:16.290 I'm supposed to transpose those guys, make them stand up. 00:16:16.290 --> 00:16:19.270 But the rank is two, and therefore the column space 00:16:19.270 --> 00:16:21.940 is two dimensional by this wonderful fact 00:16:21.940 --> 00:16:25.100 that the row space and column space have the same dimension. 00:16:25.100 --> 00:16:29.130 And therefore there are only two pivot columns, not three, 00:16:29.130 --> 00:16:34.310 and, those, the three columns are dependent. 00:16:34.310 --> 00:16:35.430 OK. 00:16:35.430 --> 00:16:45.470 Now let me bury that error and talk about the row space. 00:16:45.470 --> 00:16:48.690 Well, I'm going to give you the dimensions of all the spaces. 00:16:48.690 --> 00:16:52.480 Because that's such a nice answer. 00:16:52.480 --> 00:16:53.100 OK. 00:16:53.100 --> 00:16:56.460 So let me come back here. 00:16:56.460 --> 00:16:59.640 So we have this great fact to establish, 00:16:59.640 --> 00:17:06.790 that the row space, its dimension is also the rank. 00:17:06.790 --> 00:17:07.910 What about the null space? 00:17:07.910 --> 00:17:10.020 OK. 00:17:10.020 --> 00:17:13.630 What's a basis for the null space? 00:17:13.630 --> 00:17:15.970 What's the dimension of the null space? 00:17:15.970 --> 00:17:20.274 Let me, I'll put that answer up here for the null space. 00:17:24.200 --> 00:17:27.849 Well, how have we constructed the null space? 00:17:27.849 --> 00:17:32.260 We took the matrix A, we did those row operations 00:17:32.260 --> 00:17:36.170 to get it into a form U or, or even further. 00:17:36.170 --> 00:17:39.730 We got it into the reduced form R. 00:17:39.730 --> 00:17:43.490 And then we read off special solutions. 00:17:43.490 --> 00:17:44.580 Special solutions. 00:17:44.580 --> 00:17:48.110 And every special solution came from a free variable. 00:17:48.110 --> 00:17:50.720 And those special solutions are in the null space, 00:17:50.720 --> 00:17:54.360 and the great thing is they're a basis for it. 00:17:54.360 --> 00:17:59.445 So for the null space, a basis will be the special solutions. 00:18:03.280 --> 00:18:07.250 And there's one for every free variable, right? 00:18:07.250 --> 00:18:11.190 For each free variable, we give that variable the value one, 00:18:11.190 --> 00:18:13.350 the other free variables zero. 00:18:13.350 --> 00:18:17.840 We get the pivot variables, we get a vector in the -- 00:18:17.840 --> 00:18:20.680 we get a special solution. 00:18:20.680 --> 00:18:24.140 So we get altogether n-r of them, 00:18:24.140 --> 00:18:30.340 because that's the number of free variables. 00:18:30.340 --> 00:18:32.890 If we have r -- 00:18:32.890 --> 00:18:38.490 this is the dimension is r, is the number of pivot variables. 00:18:38.490 --> 00:18:40.810 This is the number of free variables. 00:18:40.810 --> 00:18:44.180 So the beauty is that those special solutions do form 00:18:44.180 --> 00:18:51.380 a basis and tell us immediately that the dimension of the null 00:18:51.380 --> 00:18:55.570 space is n -- 00:18:55.570 --> 00:19:00.590 I better write this well, because it's so nice -- n-r. 00:19:00.590 --> 00:19:04.910 And do you see the nice thing? 00:19:04.910 --> 00:19:08.320 That the two dimensions in this n dimensional space, 00:19:08.320 --> 00:19:12.300 one subspace is r dimensional -- 00:19:12.300 --> 00:19:15.400 to be proved, that's the row space. 00:19:15.400 --> 00:19:18.680 The other subspace is n-r dimensional, 00:19:18.680 --> 00:19:21.150 that's the null space. 00:19:21.150 --> 00:19:25.550 And the two dimensions like together give n. 00:19:25.550 --> 00:19:28.870 The sum of r and n-R is n. 00:19:28.870 --> 00:19:31.270 And that's just great. 00:19:31.270 --> 00:19:35.720 It's really copying the fact that we have n variables, 00:19:35.720 --> 00:19:39.750 r of them are pivot variables and n-r are free variables, 00:19:39.750 --> 00:19:41.380 and n altogether. 00:19:41.380 --> 00:19:41.880 OK. 00:19:41.880 --> 00:19:47.090 And now what's the dimension of this poor misbegotten fourth 00:19:47.090 --> 00:19:48.500 subspace? 00:19:48.500 --> 00:19:53.700 It's got to be m-r. 00:19:53.700 --> 00:20:00.110 The dimension of this left null space, left out practically, 00:20:00.110 --> 00:20:01.540 is m-r. 00:20:04.120 --> 00:20:08.690 Well, that's really just saying that this -- again, 00:20:08.690 --> 00:20:15.140 the sum of that plus that is m, and m is correct, 00:20:15.140 --> 00:20:21.180 it's the number of columns in A transpose. 00:20:21.180 --> 00:20:25.810 A transpose is just as good a matrix as A. 00:20:25.810 --> 00:20:29.590 It just happens to be n by m. 00:20:29.590 --> 00:20:38.360 It happens to have m columns, so it will have m variables 00:20:38.360 --> 00:20:41.990 when I go to A x equals 0 and m of them, 00:20:41.990 --> 00:20:46.950 and r of them will be pivot variables and m-r will 00:20:46.950 --> 00:20:49.920 be free variables. 00:20:49.920 --> 00:20:52.650 A transpose is as good a matrix as A. 00:20:52.650 --> 00:20:57.410 It follows the same rule that the this plus the dimension -- 00:20:57.410 --> 00:21:00.830 this dimension plus this dimension adds up to the number 00:21:00.830 --> 00:21:03.130 of columns. 00:21:03.130 --> 00:21:06.770 And over here, A transpose has m columns. 00:21:06.770 --> 00:21:09.290 OK. 00:21:09.290 --> 00:21:09.790 OK. 00:21:09.790 --> 00:21:13.580 So I gave you the easy answer, the dimensions. 00:21:13.580 --> 00:21:21.160 Now can I go back to check on a basis? 00:21:21.160 --> 00:21:25.540 We would like to think that -- say the row space, 00:21:25.540 --> 00:21:29.280 because we've got a basis for the column space. 00:21:29.280 --> 00:21:33.570 The pivot columns give a basis for the column space. 00:21:33.570 --> 00:21:36.840 Now I'm asking you to look at the row space. 00:21:36.840 --> 00:21:40.790 And I -- you could say, OK, I can produce a basis for the row 00:21:40.790 --> 00:21:45.540 space by transposing my matrix, making those columns, 00:21:45.540 --> 00:21:48.670 then doing elimination, row reduction, 00:21:48.670 --> 00:21:54.690 and checking out the pivot columns in this transposed 00:21:54.690 --> 00:21:55.350 matrix. 00:21:55.350 --> 00:21:57.760 But that means you had to do all that row 00:21:57.760 --> 00:22:00.750 reduction on A transpose. 00:22:00.750 --> 00:22:05.590 It ought to be possible, if we take a matrix A -- 00:22:05.590 --> 00:22:08.620 let me take the matrix -- maybe we had this matrix in the last 00:22:08.620 --> 00:22:09.120 lecture. 00:22:09.120 --> 00:22:15.530 1 1 1, 2 1 2, 3 2 3, 1 1 1. 00:22:21.950 --> 00:22:22.670 OK. 00:22:22.670 --> 00:22:24.770 That, that matrix was so easy. 00:22:24.770 --> 00:22:29.100 We spotted its pivot columns, one and two, without actually 00:22:29.100 --> 00:22:30.930 doing row reduction. 00:22:30.930 --> 00:22:35.070 But now let's do the job properly. 00:22:35.070 --> 00:22:38.980 So I subtract this away from this to produce a zero. 00:22:38.980 --> 00:22:42.810 So one 2 3 1 is fine. 00:22:42.810 --> 00:22:47.419 Subtracting that away leaves me minus 1 -1 0, right? 00:22:47.419 --> 00:22:49.960 And subtracting that from the last row, oh, well that's easy. 00:22:53.140 --> 00:22:53.690 OK? 00:22:53.690 --> 00:22:56.680 I'm doing row reduction. 00:22:56.680 --> 00:23:00.700 Now I've -- the first column is all set. 00:23:00.700 --> 00:23:04.230 The second column I now see the pivot. 00:23:04.230 --> 00:23:07.800 And I can clean up, if I -- 00:23:07.800 --> 00:23:08.480 actually, 00:23:08.480 --> 00:23:09.860 OK. 00:23:09.860 --> 00:23:13.070 Why don't I make the pivot into a 1. 00:23:13.070 --> 00:23:18.805 I'll multiply that row through by by -1, and then I have 1 1. 00:23:22.200 --> 00:23:24.370 That was an elementary operation I'm allowed, 00:23:24.370 --> 00:23:26.790 multiply a row by a number. 00:23:26.790 --> 00:23:28.150 And now I'll do elimination. 00:23:28.150 --> 00:23:31.300 Two of those away from that will knock this guy out 00:23:31.300 --> 00:23:33.120 and make this into a 1. 00:23:33.120 --> 00:23:36.560 So that's now a 0 and that's a 00:23:36.560 --> 00:23:37.620 OK. 00:23:37.620 --> 00:23:39.230 Done. 00:23:39.230 --> 00:23:42.470 That's R. 00:23:42.470 --> 00:23:45.930 I'm seeing the identity matrix here. 00:23:45.930 --> 00:23:48.530 I'm seeing zeros below. 00:23:48.530 --> 00:23:49.990 And I'm seeing F there. 00:23:53.151 --> 00:23:53.650 OK. 00:23:56.340 --> 00:24:00.110 What about its row space? 00:24:00.110 --> 00:24:02.650 What happened to its row space -- well, what happened -- 00:24:02.650 --> 00:24:04.610 let me first ask, just because this is, is -- 00:24:04.610 --> 00:24:06.780 sometimes something does happen. 00:24:06.780 --> 00:24:09.010 Its column space changed. 00:24:09.010 --> 00:24:18.820 The column space of R is not the column space of A, right? 00:24:18.820 --> 00:24:22.000 Because 1 1 1 is certainly in the column space of A 00:24:22.000 --> 00:24:26.460 and certainly not in the column space of R. 00:24:26.460 --> 00:24:29.770 I did row operations. 00:24:29.770 --> 00:24:33.750 Those row operations preserve the row space. 00:24:33.750 --> 00:24:36.640 So the row, so the column spaces are different. 00:24:36.640 --> 00:24:39.550 Different column spaces, different column spaces. 00:24:45.950 --> 00:24:50.480 But I believe that they have the same row space. 00:24:55.080 --> 00:24:55.880 Same row space. 00:25:00.030 --> 00:25:04.040 I believe that the row space of that matrix and the row space 00:25:04.040 --> 00:25:06.210 of this matrix are identical. 00:25:06.210 --> 00:25:09.640 They have exactly the same vectors in them. 00:25:09.640 --> 00:25:14.400 Those vectors are vectors with four components, right? 00:25:14.400 --> 00:25:17.940 They're all combinations of those rows. 00:25:17.940 --> 00:25:19.800 Or I believe you get the same thing 00:25:19.800 --> 00:25:22.030 by taking all combinations of these rows. 00:25:24.690 --> 00:25:29.960 And if true, what's a basis? 00:25:29.960 --> 00:25:32.430 What's a basis for the row space of R, 00:25:32.430 --> 00:25:42.450 and it'll be a basis for the row space of the original A, 00:25:42.450 --> 00:25:45.120 but it's obviously a basis for the row space of R. 00:25:45.120 --> 00:25:47.840 What's a basis for the row space of that matrix? 00:25:47.840 --> 00:25:48.820 The first two rows. 00:25:51.930 --> 00:25:57.550 So a basis for the row -- so a basis is, 00:25:57.550 --> 00:26:15.500 for the row space of A or of R, is, is the first R rows of R. 00:26:15.500 --> 00:26:18.050 Not of A. 00:26:18.050 --> 00:26:21.960 Sometimes it's true for A, but not necessarily. 00:26:21.960 --> 00:26:29.280 But R, we definitely have a matrix here whose row space we 00:26:29.280 --> 00:26:32.060 can, we can identify. 00:26:32.060 --> 00:26:36.790 The row space is spanned by the three rows, 00:26:36.790 --> 00:26:40.580 but if we want a basis we want independence. 00:26:40.580 --> 00:26:43.380 So out goes row three. 00:26:43.380 --> 00:26:46.820 The row space is also spanned by the first two rows. 00:26:46.820 --> 00:26:48.920 This guy didn't contribute anything. 00:26:48.920 --> 00:26:52.706 And of course over here this 1 2 3 1 in the bottom 00:26:52.706 --> 00:26:53.830 didn't contribute anything. 00:26:53.830 --> 00:26:56.510 We had it already. 00:26:56.510 --> 00:26:58.260 So this, here is a basis. 00:26:58.260 --> 00:27:01.260 1 0 1 1 and 0 1 1 0. 00:27:04.710 --> 00:27:06.860 I believe those are in the row space. 00:27:06.860 --> 00:27:08.090 I know they're independent. 00:27:08.090 --> 00:27:10.680 Why are they in the row space? 00:27:10.680 --> 00:27:13.720 Why are those two vectors in the row space? 00:27:13.720 --> 00:27:17.440 Because all those operations we did, 00:27:17.440 --> 00:27:22.680 which started with these rows and took combinations of them 00:27:22.680 --> 00:27:23.900 -- 00:27:23.900 --> 00:27:28.910 I took this row minus this row, that gave me something 00:27:28.910 --> 00:27:30.780 that's still in the row space. 00:27:30.780 --> 00:27:32.400 That's the point. 00:27:32.400 --> 00:27:36.760 When I took a row minus a multiple of another row, 00:27:36.760 --> 00:27:38.380 I'm staying in the row space. 00:27:38.380 --> 00:27:41.330 The row space is not changing. 00:27:41.330 --> 00:27:43.240 My little basis for it is changing, 00:27:43.240 --> 00:27:46.680 and I've ended up with, sort of the best basis. 00:27:49.380 --> 00:27:53.240 If the columns of the identity matrix are the best basis 00:27:53.240 --> 00:28:00.490 for R^3 or R^n, the rows of this matrix are the best basis 00:28:00.490 --> 00:28:02.510 for the row space. 00:28:02.510 --> 00:28:06.250 Best in the sense of being as clean as I can make it. 00:28:06.250 --> 00:28:09.310 Starting off with the identity and then finishing up 00:28:09.310 --> 00:28:11.420 with whatever has to be in there. 00:28:11.420 --> 00:28:12.730 OK. 00:28:12.730 --> 00:28:16.720 Do you see then that the dimension is r? 00:28:16.720 --> 00:28:23.330 For sure, because we've got r pivots, r non-zero rows. 00:28:23.330 --> 00:28:26.420 We've got the right number of vectors, r. 00:28:26.420 --> 00:28:30.080 They're in the row space, they're independent. 00:28:30.080 --> 00:28:31.600 That's it. 00:28:31.600 --> 00:28:34.160 They are a basis for the row space. 00:28:34.160 --> 00:28:36.220 And we can even pin that down further. 00:28:36.220 --> 00:28:41.340 How do I know that every row of A is a combination? 00:28:41.340 --> 00:28:45.110 How do I know they span the row space? 00:28:45.110 --> 00:28:48.110 Well, somebody says, I've got the right number of them, 00:28:48.110 --> 00:28:48.740 so they must. 00:28:48.740 --> 00:28:49.870 But -- and that's true. 00:28:49.870 --> 00:28:54.400 But let me just say, how do I know that this row is 00:28:54.400 --> 00:28:57.300 a combination of these? 00:28:57.300 --> 00:29:01.870 By just reversing the steps of row reduction. 00:29:01.870 --> 00:29:07.760 If I just reverse the steps and go from A -- from R back to A, 00:29:07.760 --> 00:29:10.010 then what do I, what I doing? 00:29:10.010 --> 00:29:12.030 I'm starting with these rows, I'm 00:29:12.030 --> 00:29:15.730 taking combinations of them. 00:29:15.730 --> 00:29:19.670 After a couple of steps, undoing the subtractions 00:29:19.670 --> 00:29:22.710 that I did before, I'm back to these rows. 00:29:22.710 --> 00:29:25.500 So these rows are combinations of those rows. 00:29:25.500 --> 00:29:28.020 Those rows are combinations of those rows. 00:29:28.020 --> 00:29:31.740 The two row spaces are the same. 00:29:31.740 --> 00:29:34.830 The bases are the same. 00:29:34.830 --> 00:29:38.540 And the natural basis is this guy. 00:29:38.540 --> 00:29:41.530 Is that all right for the row space? 00:29:41.530 --> 00:29:45.030 The row space is sitting there in R 00:29:45.030 --> 00:29:47.750 in its cleanest possible form. 00:29:47.750 --> 00:29:48.800 OK. 00:29:48.800 --> 00:29:56.875 Now what about the fourth guy, the null space of A transpose? 00:29:59.630 --> 00:30:03.440 First of all, why do I call that the left null space? 00:30:03.440 --> 00:30:11.610 So let me save that and bring that down. 00:30:11.610 --> 00:30:14.050 OK. 00:30:14.050 --> 00:30:20.940 So the fourth space is the null space of A transpose. 00:30:23.700 --> 00:30:27.200 So it has in it vectors, let me call them y, 00:30:27.200 --> 00:30:30.470 so that A transpose y equals 0. 00:30:30.470 --> 00:30:35.360 If A transpose y equals 0, then y 00:30:35.360 --> 00:30:39.890 is in the null space of A transpose, of course. 00:30:39.890 --> 00:30:47.770 So this is a matrix times a column equaling zero. 00:30:50.480 --> 00:30:56.550 And now, because I want y to sit on the left 00:30:56.550 --> 00:31:00.200 and I want A instead of A transpose, 00:31:00.200 --> 00:31:03.380 I'll just transpose that equation. 00:31:03.380 --> 00:31:06.180 Can I just transpose that? 00:31:06.180 --> 00:31:10.705 On the right, it makes the zero vector lie down. 00:31:14.590 --> 00:31:21.510 And on the left, it's a product, A, A transpose times y. 00:31:21.510 --> 00:31:24.510 If I take the transpose, then they come in opposite order, 00:31:24.510 --> 00:31:25.540 right? 00:31:25.540 --> 00:31:30.156 So that's y transpose times A transpose transpose. 00:31:33.210 --> 00:31:35.540 But nobody's going to leave it like that. 00:31:35.540 --> 00:31:39.620 That's -- A transpose transpose is just A, of course. 00:31:39.620 --> 00:31:43.360 When I transposed A transpose I got back to A. 00:31:43.360 --> 00:31:45.870 Now do you see what I have now? 00:31:45.870 --> 00:31:51.820 I have a row vector, y transpose, 00:31:51.820 --> 00:31:58.540 multiplying A, and multiplying from the left. 00:31:58.540 --> 00:32:02.180 That's why I call it the left null space. 00:32:02.180 --> 00:32:05.590 But by making it -- putting it on the left, 00:32:05.590 --> 00:32:09.720 I had to make it into a row instead of a column vector, 00:32:09.720 --> 00:32:15.250 and so my convention is I usually don't do that. 00:32:15.250 --> 00:32:18.960 I usually stay with A transpose y equals 0. 00:32:18.960 --> 00:32:20.290 OK. 00:32:20.290 --> 00:32:27.990 And you might ask, how do we get a basis -- or I might ask, 00:32:27.990 --> 00:32:31.470 how do we get a basis for this fourth space, 00:32:31.470 --> 00:32:32.610 this left null space? 00:32:36.150 --> 00:32:36.720 OK. 00:32:36.720 --> 00:32:40.050 I'll do it in the example. 00:32:40.050 --> 00:32:43.350 As always -- not that one. 00:32:49.880 --> 00:32:53.310 The left null space is not jumping out at me here. 00:32:57.060 --> 00:33:00.220 I know which are the free variables -- 00:33:00.220 --> 00:33:03.890 the special solutions, but those are special solutions to A x 00:33:03.890 --> 00:33:06.230 equals zero, and now I'm looking at A transpose, 00:33:06.230 --> 00:33:08.420 and I'm not seeing it here. 00:33:08.420 --> 00:33:12.760 So -- but somehow you feel that the work that you did which 00:33:12.760 --> 00:33:19.310 simplified A to R should have revealed the left null space 00:33:19.310 --> 00:33:20.760 too. 00:33:20.760 --> 00:33:25.970 And it's slightly less immediate, but it's there. 00:33:25.970 --> 00:33:31.010 So from A to R, I took some steps, 00:33:31.010 --> 00:33:34.880 and I guess I'm interested in what were those steps, 00:33:34.880 --> 00:33:36.600 or what were all of them together. 00:33:36.600 --> 00:33:43.160 I don't -- I'm not interested in what particular ones they were. 00:33:43.160 --> 00:33:45.910 I'm interested in what was the whole matrix that 00:33:45.910 --> 00:33:51.620 took me from A to R. 00:33:51.620 --> 00:33:52.880 How would you find that? 00:33:55.640 --> 00:33:58.970 Do you remember Gauss-Jordan, where you 00:33:58.970 --> 00:34:02.260 tack on the identity matrix? 00:34:02.260 --> 00:34:03.810 Let's do that again. 00:34:03.810 --> 00:34:06.640 So I, I'll do it above, here. 00:34:06.640 --> 00:34:13.620 So this is now, this is now the idea of -- 00:34:13.620 --> 00:34:17.570 I take the matrix A, which is m by n. 00:34:20.389 --> 00:34:22.639 In Gauss-Jordan, when we saw him before -- 00:34:22.639 --> 00:34:25.150 A was a square invertible matrix and we 00:34:25.150 --> 00:34:27.900 were finding its inverse. 00:34:27.900 --> 00:34:29.409 Now the matrix isn't square. 00:34:29.409 --> 00:34:32.360 It's probably rectangular. 00:34:32.360 --> 00:34:36.989 But I'll still tack on the identity matrix, and of course 00:34:36.989 --> 00:34:42.179 since these have length m it better be m by m. 00:34:42.179 --> 00:34:49.374 And now I'll do the reduced row echelon form of this matrix. 00:34:52.480 --> 00:34:56.120 And what do I get? 00:35:01.640 --> 00:35:04.950 The reduced row echelon form starts with these columns, 00:35:04.950 --> 00:35:11.680 starts with the first columns, works like mad, and produces R. 00:35:11.680 --> 00:35:13.920 Of course, still that same size, m by n. 00:35:13.920 --> 00:35:15.700 And we did it before. 00:35:15.700 --> 00:35:19.140 And then whatever it did to get R, 00:35:19.140 --> 00:35:22.520 something else is going to show up here. 00:35:22.520 --> 00:35:26.690 Let me call it E, m by m. 00:35:26.690 --> 00:35:30.170 It's whatever -- do you see that E is just going to contain 00:35:30.170 --> 00:35:32.640 a record of what we did? 00:35:32.640 --> 00:35:38.780 We did whatever it took to get A to become R. 00:35:38.780 --> 00:35:40.930 And at the same time, we were doing it 00:35:40.930 --> 00:35:44.410 to the identity matrix. 00:35:44.410 --> 00:35:48.590 So we started with the identity matrix, we buzzed along. 00:35:48.590 --> 00:35:51.190 So we took some -- 00:35:51.190 --> 00:35:55.860 all this row reduction amounted to multiplying on the left 00:35:55.860 --> 00:36:00.040 by some matrix, some series of elementary matrices 00:36:00.040 --> 00:36:05.780 that altogether gave us one matrix, and that matrix is E. 00:36:05.780 --> 00:36:11.450 So all this row reduction stuff amounted to multiplying by E. 00:36:11.450 --> 00:36:13.120 How do I know that? 00:36:13.120 --> 00:36:16.680 It certainly amounted to multiply it by something. 00:36:16.680 --> 00:36:21.660 And that something took I to E, so that something was E. 00:36:21.660 --> 00:36:29.790 So now look at the first part, E A is R. 00:36:29.790 --> 00:36:31.520 No big deal. 00:36:31.520 --> 00:36:38.710 All I've said is that the row reduction steps that we all 00:36:38.710 --> 00:36:45.680 know -- well, taking A to R, are in a, in some matrix, 00:36:45.680 --> 00:36:49.340 and I can find out what that matrix is by just tacking I 00:36:49.340 --> 00:36:51.810 on and seeing what comes out. 00:36:51.810 --> 00:36:54.570 What comes out is E. 00:36:54.570 --> 00:36:58.400 Let's just review the invertible square case. 00:36:58.400 --> 00:37:00.860 What happened then? 00:37:00.860 --> 00:37:04.350 Because I was interested in it in chapter two also. 00:37:04.350 --> 00:37:08.260 When A was square and invertible, I took A I, 00:37:08.260 --> 00:37:10.480 I did row, row elimination. 00:37:10.480 --> 00:37:12.210 And what was the R that came out? 00:37:12.210 --> 00:37:14.770 It was I. 00:37:14.770 --> 00:37:24.530 So in chapter two, in chapter two, R was I. 00:37:24.530 --> 00:37:27.310 The row the, the reduced row echelon 00:37:27.310 --> 00:37:31.730 form of a nice invertible square matrix is the identity. 00:37:31.730 --> 00:37:41.510 So if R was I in that case, then E was -- then E was A inverse, 00:37:41.510 --> 00:37:44.430 because E A is I. 00:37:44.430 --> 00:37:45.200 Good. 00:37:45.200 --> 00:37:48.540 That's, that was good and easy. 00:37:48.540 --> 00:37:52.990 Now what I'm saying is that there still is an E. 00:37:52.990 --> 00:37:55.740 It's not A inverse any more, because A is rectangular. 00:37:55.740 --> 00:37:57.730 It hasn't got an inverse. 00:37:57.730 --> 00:38:05.010 But there is still some matrix E that connected this to this -- 00:38:05.010 --> 00:38:09.260 oh, I should have figured out in advanced what it was. 00:38:09.260 --> 00:38:11.810 Shoot. 00:38:11.810 --> 00:38:12.800 I didn't -- 00:38:12.800 --> 00:38:16.620 I did those steps and sort of erased as I went -- 00:38:16.620 --> 00:38:20.370 and, I should have done them to the identity too. 00:38:20.370 --> 00:38:22.500 Can I do that? 00:38:22.500 --> 00:38:23.460 Can I do that? 00:38:23.460 --> 00:38:26.140 I'll keep the identity matrix, like I'm supposed to do, 00:38:26.140 --> 00:38:29.750 and I'll do the same operations on it, and see what I end up 00:38:29.750 --> 00:38:30.340 with. 00:38:30.340 --> 00:38:31.300 OK. 00:38:31.300 --> 00:38:32.810 So I'm starting with the identity -- 00:38:32.810 --> 00:38:40.511 which I'll write in light, light enough, but -- 00:38:40.511 --> 00:38:41.010 OK. 00:38:41.010 --> 00:38:42.520 What did I do? 00:38:42.520 --> 00:38:45.950 I subtracted that row from that one and that row from that one. 00:38:45.950 --> 00:38:47.950 OK, I'll do that to the identity. 00:38:47.950 --> 00:38:52.990 So I subtract that first row from row two and row three. 00:38:52.990 --> 00:38:55.310 Good. 00:38:55.310 --> 00:38:56.950 Then I think I multiplied -- 00:38:56.950 --> 00:38:57.620 Do you remember? 00:38:57.620 --> 00:39:01.890 I multiplied row two by minus one. 00:39:01.890 --> 00:39:05.270 Let me just do that. 00:39:05.270 --> 00:39:06.610 Then what did I do? 00:39:06.610 --> 00:39:14.880 I subtracted two of row two away from row one. 00:39:14.880 --> 00:39:15.770 I better do that. 00:39:15.770 --> 00:39:17.720 Subtract two of this away from this. 00:39:17.720 --> 00:39:24.180 That's minus one, two of these away leaves a plus 2 and 0. 00:39:24.180 --> 00:39:28.440 I believe that's E. 00:39:28.440 --> 00:39:35.870 The way to check is to see, multiply that E by this A, 00:39:35.870 --> 00:39:37.420 just to see did I do it right. 00:39:40.570 --> 00:39:49.285 So I believe E was -1 2 0, 1 -1 0, and -1 0 1. 00:39:53.020 --> 00:39:53.520 OK. 00:39:53.520 --> 00:39:58.030 That's my E, that's my A, and that's R. 00:39:58.030 --> 00:40:00.110 All right. 00:40:00.110 --> 00:40:02.950 All I'm struggling to do is right, 00:40:02.950 --> 00:40:09.660 the reason I wanted this blasted E was so that I could figure 00:40:09.660 --> 00:40:14.570 out the left null space, not only its dimension, 00:40:14.570 --> 00:40:17.020 which I know -- 00:40:17.020 --> 00:40:19.427 actually, what is the dimension of the left null space? 00:40:19.427 --> 00:40:20.260 So here's my matrix. 00:40:23.180 --> 00:40:24.465 What's the rank of the matrix? 00:40:27.560 --> 00:40:30.640 And the dimension of the null -- of the left null space is 00:40:30.640 --> 00:40:33.470 supposed to be m-r. 00:40:33.470 --> 00:40:34.930 It's 3 -2, 1. 00:40:34.930 --> 00:40:39.090 I believe that the left null space is one dimensional. 00:40:39.090 --> 00:40:42.990 There is one combination of those three rows 00:40:42.990 --> 00:40:46.840 that produces the zero row. 00:40:46.840 --> 00:40:52.270 There's a basis -- a basis for the left null space has only 00:40:52.270 --> 00:40:54.200 got one vector in it. 00:40:54.200 --> 00:40:55.790 And what is that vector? 00:40:55.790 --> 00:40:58.710 It's here in the last row of E. 00:40:58.710 --> 00:41:01.400 But we could have seen it earlier. 00:41:01.400 --> 00:41:05.220 What combination of those rows gives the zero row? 00:41:05.220 --> 00:41:09.110 -1 of that plus one of that. 00:41:09.110 --> 00:41:14.460 So a basis for the left null space of this matrix -- 00:41:14.460 --> 00:41:18.350 I'm looking for combinations of rows that give the zero row 00:41:18.350 --> 00:41:22.080 if I'm looking at the left null space. 00:41:22.080 --> 00:41:24.980 For the null space, I'm looking at combinations of columns 00:41:24.980 --> 00:41:26.780 to get the zero column. 00:41:26.780 --> 00:41:29.760 Now I'm looking at combinations of these three rows 00:41:29.760 --> 00:41:34.370 to get the zero row, and of course there is my zero row, 00:41:34.370 --> 00:41:37.160 and here is my vector that produced it. 00:41:37.160 --> 00:41:40.010 -1 of that row and one of that 00:41:40.010 --> 00:41:40.510 row. 00:41:40.510 --> 00:41:41.650 Obvious. 00:41:41.650 --> 00:41:42.210 OK. 00:41:42.210 --> 00:41:45.940 So in that example -- and actually in all examples, 00:41:45.940 --> 00:41:51.310 we have seen how to produce a basis for the left null space. 00:41:51.310 --> 00:41:54.800 I won't ask you that all the time, because -- 00:41:54.800 --> 00:41:58.790 it didn't come out immediately from R. 00:41:58.790 --> 00:42:03.850 We had to keep track of E for that left null space. 00:42:03.850 --> 00:42:07.520 But at least it didn't require us to transpose the matrix 00:42:07.520 --> 00:42:10.220 and start all over again. 00:42:10.220 --> 00:42:12.390 OK, those are the four subspaces. 00:42:12.390 --> 00:42:15.520 Can I review them? 00:42:15.520 --> 00:42:18.950 The row space and the null space are in R^n. 00:42:18.950 --> 00:42:22.220 Their dimensions add to n. 00:42:22.220 --> 00:42:27.470 The column space and the left null space are in R^m, 00:42:27.470 --> 00:42:30.900 and their dimensions add to m. 00:42:30.900 --> 00:42:33.700 OK. 00:42:33.700 --> 00:42:39.080 So let me close these last minutes 00:42:39.080 --> 00:42:49.620 by pushing you a little bit more to a new type of vector space. 00:42:49.620 --> 00:42:53.230 All our vector spaces, all the ones that we took seriously, 00:42:53.230 --> 00:43:01.570 have been subspaces of some real three or n dimensional space. 00:43:01.570 --> 00:43:04.880 Now I'm going to write down another vector 00:43:04.880 --> 00:43:06.720 space, a new vector space. 00:43:14.120 --> 00:43:18.610 Say all three by three matrices. 00:43:26.420 --> 00:43:27.990 My matrices are the vectors. 00:43:31.830 --> 00:43:33.240 Is that all right? 00:43:33.240 --> 00:43:34.130 I'm just naming them. 00:43:34.130 --> 00:43:36.570 You can put quotes around vectors. 00:43:36.570 --> 00:43:40.050 Every three by three matrix is one of my vectors. 00:43:40.050 --> 00:43:43.060 Now how I entitled to call those things vectors? 00:43:43.060 --> 00:43:46.380 I mean, they look very much like matrices. 00:43:46.380 --> 00:43:49.980 But they are vectors in my vector space because they obey 00:43:49.980 --> 00:43:50.640 the rules. 00:43:50.640 --> 00:43:55.850 All I'm supposed to be able to do with vectors is add them -- 00:43:55.850 --> 00:43:58.190 I can add matrices -- 00:43:58.190 --> 00:44:01.580 I'm supposed to be able to multiply them by scalar numbers 00:44:01.580 --> 00:44:09.290 like seven -- well, I can multiply a matrix by And that 00:44:09.290 --> 00:44:11.960 -- and I can take combinations of matrices, 00:44:11.960 --> 00:44:15.060 I can take three of one matrix minus five of another 00:44:15.060 --> 00:44:15.990 matrix. 00:44:15.990 --> 00:44:21.260 And those combinations, there's a zero matrix, the matrix 00:44:21.260 --> 00:44:23.570 that has all zeros in it. 00:44:23.570 --> 00:44:26.050 If I add that to another matrix, it doesn't change it. 00:44:26.050 --> 00:44:26.960 All the good stuff. 00:44:26.960 --> 00:44:30.270 If I multiply a matrix by one it doesn't change it. 00:44:30.270 --> 00:44:32.800 All those eight rules for a vector space 00:44:32.800 --> 00:44:37.460 that we never wrote down, all easily satisfied. 00:44:37.460 --> 00:44:41.150 So now we have a different -- 00:44:41.150 --> 00:44:46.450 now of course you can say you can multiply those matrices. 00:44:46.450 --> 00:44:47.240 I don't care. 00:44:47.240 --> 00:44:50.070 For the moment, I'm only thinking of these matrices 00:44:50.070 --> 00:44:57.960 as forming a vector space -- so I only doing A plus B and c 00:44:57.960 --> 00:44:59.350 times A. 00:44:59.350 --> 00:45:03.050 I'm not interested in A B for now. 00:45:06.580 --> 00:45:09.340 The fact that I can multiply is not 00:45:09.340 --> 00:45:13.721 relevant to th- to a vector space. 00:45:13.721 --> 00:45:14.220 OK. 00:45:14.220 --> 00:45:15.636 So I have three by three matrices. 00:45:18.180 --> 00:45:21.600 And how about subspaces? 00:45:21.600 --> 00:45:26.420 What's -- tell me a subspace of this matrix space. 00:45:26.420 --> 00:45:30.270 Let me call this matrix space M. 00:45:30.270 --> 00:45:34.270 That's my matrix space, my space of all three by three matrices. 00:45:34.270 --> 00:45:37.730 Tell me a subspace of it. 00:45:37.730 --> 00:45:40.670 What about the upper triangular matrices? 00:45:40.670 --> 00:45:41.260 OK. 00:45:41.260 --> 00:45:43.330 So subspaces. 00:45:43.330 --> 00:45:50.660 Subspaces of M. 00:45:50.660 --> 00:45:53.675 All, all upper triangular matrices. 00:46:00.370 --> 00:46:01.750 Another subspace. 00:46:01.750 --> 00:46:03.085 All symmetric matrices. 00:46:11.610 --> 00:46:13.730 The intersection of two subspaces 00:46:13.730 --> 00:46:15.090 is supposed to be a subspace. 00:46:15.090 --> 00:46:20.310 We gave a little effort to the proof of that fact. 00:46:20.310 --> 00:46:23.150 If I look at the matrices that are in this subspace -- 00:46:23.150 --> 00:46:26.420 they're symmetric, and they're also in this subspace, 00:46:26.420 --> 00:46:31.280 they're upper triangular, what do they look like? 00:46:31.280 --> 00:46:33.550 Well, if they're symmetric but they 00:46:33.550 --> 00:46:35.660 have zeros below the diagonal, they better 00:46:35.660 --> 00:46:38.650 have zeros above the diagonal, so the intersection 00:46:38.650 --> 00:46:40.285 would be diagonal matrices. 00:46:44.820 --> 00:46:48.430 That's another subspace, smaller than those. 00:46:50.970 --> 00:46:53.740 How can I use the word smaller? 00:46:53.740 --> 00:46:56.390 Well, I'm now entitled to use the word smaller. 00:46:56.390 --> 00:47:00.410 I mean, well, one way to say is, OK, these 00:47:00.410 --> 00:47:02.570 are contained in those. 00:47:02.570 --> 00:47:05.210 These are contained in those. 00:47:05.210 --> 00:47:09.205 But more precisely, I could give the dimension of these spaces. 00:47:11.710 --> 00:47:14.740 So I could -- we can compute -- let's compute it next time, 00:47:14.740 --> 00:47:17.870 the dimension of all upper -- of the subspace of upper 00:47:17.870 --> 00:47:20.490 triangular three by three matrices. 00:47:20.490 --> 00:47:23.790 The dimension of symmetric three by three matrices. 00:47:23.790 --> 00:47:27.630 The dimension of diagonal three by three matrices. 00:47:27.630 --> 00:47:29.690 Well, to produce dimension, that means 00:47:29.690 --> 00:47:33.370 I'm supposed to produce a basis, and then 00:47:33.370 --> 00:47:37.230 I just count how many vecto- how many I needed in the basis. 00:47:37.230 --> 00:47:39.880 Let me give you the answer for this one. 00:47:39.880 --> 00:47:41.430 What's the dimension? 00:47:41.430 --> 00:47:44.610 The dimension of this -- say, this subspace, 00:47:44.610 --> 00:47:47.520 let me call it D, all diagonal matrices. 00:47:47.520 --> 00:47:54.610 The dimension of this subspace is -- 00:47:54.610 --> 00:47:57.610 as I write you're working it out -- 00:47:57.610 --> 00:47:58.970 three. 00:47:58.970 --> 00:48:09.150 Because here's a matrix in this -- it's a diagonal matrix. 00:48:09.150 --> 00:48:10.040 Here's another one. 00:48:15.560 --> 00:48:16.610 Here's another one. 00:48:20.350 --> 00:48:22.755 Better make it diagonal, let me put a seven there. 00:48:25.970 --> 00:48:28.130 That was not a very great choice, 00:48:28.130 --> 00:48:31.020 but it's three diagonal matrices, 00:48:31.020 --> 00:48:33.950 and I believe that they're a basis. 00:48:33.950 --> 00:48:37.190 I believe that those three matrices are independent 00:48:37.190 --> 00:48:40.680 and I believe that any diagonal matrix is 00:48:40.680 --> 00:48:42.220 a combination of those three. 00:48:42.220 --> 00:48:47.310 So they span the subspace of diagonal matrices. 00:48:47.310 --> 00:48:49.040 Do you see that idea? 00:48:49.040 --> 00:48:55.170 It's like stretching the idea from R^n to R^(n by n), 00:48:55.170 --> 00:48:57.360 three by three. 00:48:57.360 --> 00:49:02.310 But the -- we can still add, we can still multiply by numbers, 00:49:02.310 --> 00:49:06.370 and we just ignore the fact that we can multiply two matrices 00:49:06.370 --> 00:49:07.550 together. 00:49:07.550 --> 00:49:09.380 OK, thank you. 00:49:09.380 --> 00:49:11.930 That's lecture ten.