WEBVTT 00:00:10.740 --> 00:00:11.700 OK. 00:00:11.700 --> 00:00:17.050 This is linear algebra lecture eleven. 00:00:17.050 --> 00:00:22.760 And at the end of lecture ten, I was talking about some vector 00:00:22.760 --> 00:00:26.880 spaces, but they're -- 00:00:26.880 --> 00:00:29.070 the things in those vector spaces 00:00:29.070 --> 00:00:33.180 were not what we usually call vectors. 00:00:33.180 --> 00:00:36.010 Nevertheless, you could add them and you 00:00:36.010 --> 00:00:40.230 could multiply by numbers, so we can call them vectors. 00:00:40.230 --> 00:00:41.970 I think the example I was working 00:00:41.970 --> 00:00:44.100 with they were matrices. 00:00:44.100 --> 00:00:46.990 So the -- so we had like a matrix space, 00:00:46.990 --> 00:00:50.230 the space of all three by three matrices. 00:00:50.230 --> 00:00:55.530 And I'd like to just pick up on that, because -- 00:00:55.530 --> 00:01:00.600 we've been so specific about n dimensional space here, 00:01:00.600 --> 00:01:05.640 and you really want to see that the same ideas work as long 00:01:05.640 --> 00:01:09.270 as you can add and multiply by scalars. 00:01:09.270 --> 00:01:16.640 So these new, new vector spaces, the example I took 00:01:16.640 --> 00:01:24.085 was the space M of all three by three matrices. 00:01:26.900 --> 00:01:29.230 OK. 00:01:29.230 --> 00:01:32.630 I can add them, I can multiply by scalars. 00:01:32.630 --> 00:01:36.380 I can multiply two of them together, but I don't do that. 00:01:36.380 --> 00:01:38.710 That's not part of the vector space picture. 00:01:38.710 --> 00:01:43.420 The vector space part is just adding the matrices 00:01:43.420 --> 00:01:45.690 and multiplying by numbers. 00:01:45.690 --> 00:01:50.540 And that's fine, we stay within this space of three 00:01:50.540 --> 00:01:52.220 by three matrices. 00:01:52.220 --> 00:01:56.050 And I had some subspaces that were interesting, 00:01:56.050 --> 00:02:02.850 like the symmetric, the subspace of symmetric matrices, 00:02:02.850 --> 00:02:05.600 symmetric three by threes. 00:02:05.600 --> 00:02:13.420 Or the subspace of upper triangular three by threes. 00:02:13.420 --> 00:02:21.540 Now I, I use the word subspace because it follows the rule. 00:02:21.540 --> 00:02:25.240 If I add two symmetric matrices, I'm still symmetric. 00:02:25.240 --> 00:02:27.680 If I multiply two symmetric matrices, 00:02:27.680 --> 00:02:30.900 is the product automatically symmetric? 00:02:30.900 --> 00:02:31.870 No. 00:02:31.870 --> 00:02:34.510 But I'm not multiplying matrices. 00:02:34.510 --> 00:02:35.180 I'm just adding. 00:02:35.180 --> 00:02:36.900 So I'm fine. 00:02:36.900 --> 00:02:38.430 This is a subspace. 00:02:38.430 --> 00:02:42.320 Similarly, if I add two upper triangular matrices, 00:02:42.320 --> 00:02:45.400 I'm still upper triangular. 00:02:45.400 --> 00:02:48.630 And, that's a subspace. 00:02:48.630 --> 00:02:52.900 Now I just want to take these as example and ask, well, 00:02:52.900 --> 00:02:55.240 what's a basis for that subspace? 00:02:55.240 --> 00:02:57.570 What's the dimension of that subspace? 00:02:57.570 --> 00:03:00.900 And what's bd- dimension of the whole space? 00:03:00.900 --> 00:03:07.840 So, there's a natural basis for all three by three matrices, 00:03:07.840 --> 00:03:11.120 and why don't we just write it down. 00:03:11.120 --> 00:03:21.370 So, so M, a basis for M. 00:03:21.370 --> 00:03:23.760 Again, all three by threes. 00:03:27.940 --> 00:03:30.380 OK. 00:03:30.380 --> 00:03:34.400 And then I'll just count how many members are in that basis 00:03:34.400 --> 00:03:36.490 and I'll know the dimension. 00:03:36.490 --> 00:03:40.790 And OK, it's going to take me a little time. 00:03:40.790 --> 00:03:43.720 In fact, what is the dimension? 00:03:43.720 --> 00:03:47.840 Any idea of what I'm coming up with next? 00:03:47.840 --> 00:03:52.570 How many numbers does it take to specify that three 00:03:52.570 --> 00:03:53.250 by three matrix? 00:03:53.250 --> 00:03:53.750 Nine. 00:03:56.310 --> 00:03:59.550 Nine is the, is the dimension I'm going to find. 00:03:59.550 --> 00:04:04.630 And the most obvious basis would be the matrix that's 00:04:04.630 --> 00:04:12.880 that matrix and then this matrix with a one there 00:04:12.880 --> 00:04:21.490 and that's two of them, shall I put in the third one, 00:04:21.490 --> 00:04:27.130 and then onwards, and the last one maybe 00:04:27.130 --> 00:04:29.900 would end with the one. 00:04:29.900 --> 00:04:30.400 OK. 00:04:34.340 --> 00:04:36.240 That's like the standard basis. 00:04:36.240 --> 00:04:40.820 In fact, our space is practically the same 00:04:40.820 --> 00:04:43.850 as nine dimensional space. 00:04:43.850 --> 00:04:48.310 It's just the nine numbers are written in a square 00:04:48.310 --> 00:04:50.150 instead of in a column. 00:04:50.150 --> 00:04:54.510 But somehow it's different and, and ought to be thought 00:04:54.510 --> 00:04:56.186 of as -- 00:04:59.310 --> 00:05:01.280 natural for itself. 00:05:01.280 --> 00:05:05.760 Because now what about the symmetric three by threes? 00:05:05.760 --> 00:05:09.100 So that's a subspace. 00:05:09.100 --> 00:05:12.300 Just let's just think, what's the dimension of that subspace 00:05:12.300 --> 00:05:14.050 and what's a basis for that subspace. 00:05:16.630 --> 00:05:17.820 OK. 00:05:17.820 --> 00:05:21.300 And I guess this question occurs to me. 00:05:21.300 --> 00:05:24.830 If I look at this subspace of symmetric three 00:05:24.830 --> 00:05:32.040 by threes, well, how many of these original basis 00:05:32.040 --> 00:05:36.060 members belong to the subspace? 00:05:36.060 --> 00:05:38.050 I think only three of them do. 00:05:38.050 --> 00:05:40.660 This one is symmetric. 00:05:40.660 --> 00:05:43.490 This last one is symmetric. 00:05:43.490 --> 00:05:48.250 And the one in the middle with a, with a one in that position 00:05:48.250 --> 00:05:52.210 -- in the two two position, would be symmetric. 00:05:52.210 --> 00:05:56.680 But so I've got three of these original nine are symmetric, 00:05:56.680 --> 00:06:03.800 but, so this is an example where -- 00:06:03.800 --> 00:06:05.600 but that's, that's not all, right? 00:06:05.600 --> 00:06:06.610 What's the dimension? 00:06:06.610 --> 00:06:08.450 Let's put the dimensions down. 00:06:08.450 --> 00:06:13.580 Dimension of the, of M, was nine. 00:06:13.580 --> 00:06:16.830 What's the dimension of -- shall we call this S -- is what? 00:06:19.630 --> 00:06:21.340 What's the dimension of this? 00:06:21.340 --> 00:06:25.660 I'm sort of taking simple examples where we can, we can, 00:06:25.660 --> 00:06:30.130 spot the answer to these questions. 00:06:30.130 --> 00:06:32.690 So how many -- if I have a symmetric -- 00:06:32.690 --> 00:06:36.750 think of all symmetric matrices as a subspace, 00:06:36.750 --> 00:06:40.760 how many parameters do I choose in three by three symmetric 00:06:40.760 --> 00:06:42.720 matrices? 00:06:42.720 --> 00:06:45.020 Six, right. 00:06:45.020 --> 00:06:49.220 If I choose the diagonal that's three, 00:06:49.220 --> 00:06:53.340 and the three entries above the diagonal, 00:06:53.340 --> 00:06:55.230 then I know what the three entries below. 00:06:55.230 --> 00:06:58.550 So the dimension is six. 00:06:58.550 --> 00:07:00.820 I guess what's the dimension of this here? 00:07:00.820 --> 00:07:05.180 Let's call this space U for upper triangular. 00:07:05.180 --> 00:07:09.440 So what's the dimension of that space of all upper triangular 00:07:09.440 --> 00:07:11.390 three by threes? 00:07:11.390 --> 00:07:13.870 Again six. 00:07:13.870 --> 00:07:14.585 Again six. 00:07:17.620 --> 00:07:24.490 And, but we haven't got a -- we haven't seen -- well, actually, 00:07:24.490 --> 00:07:29.620 maybe we have got a basis here for, the upper triangulars. 00:07:29.620 --> 00:07:35.070 I guess six of these guys, one, two, three, four, 00:07:35.070 --> 00:07:39.120 and a, and a couple more, would be upper triangular. 00:07:39.120 --> 00:07:44.390 So there's a accidental case where the big basis contains 00:07:44.390 --> 00:07:46.930 in it a basis for the subspace. 00:07:46.930 --> 00:07:50.720 But with the symmetric guy, it didn't have. 00:07:50.720 --> 00:07:53.570 The symmetric guy the, basis -- so you see -- 00:07:53.570 --> 00:07:56.690 a basis is the basis for the big space, 00:07:56.690 --> 00:08:00.800 we generally need to think it all over again to get a basis 00:08:00.800 --> 00:08:03.020 for the subspace. 00:08:03.020 --> 00:08:06.050 And then how do I get other subspaces? 00:08:06.050 --> 00:08:15.100 Well, we spoke before about, the subspace the symmetric matrices 00:08:15.100 --> 00:08:18.050 and the upper triangular. 00:08:18.050 --> 00:08:24.920 This is symmetric and upper triangular. 00:08:30.280 --> 00:08:36.409 What's the, what's the dimension of that space? 00:08:36.409 --> 00:08:36.909 OK. 00:08:36.909 --> 00:08:39.390 Well, what's in that space? 00:08:39.390 --> 00:08:42.480 So what's -- if a matrix is symmetric and also upper 00:08:42.480 --> 00:08:46.950 triangular, that makes it diagonal. 00:08:46.950 --> 00:08:51.640 So this is the same as the diagonal matrices, 00:08:51.640 --> 00:08:55.230 diagonal three by threes. 00:08:55.230 --> 00:09:03.260 And the dimension of this, of S intersect U, right -- 00:09:03.260 --> 00:09:06.090 you're OK with that symbol? 00:09:06.090 --> 00:09:10.040 That's, that's the vectors that are in both S and U, 00:09:10.040 --> 00:09:12.010 and that's D. 00:09:12.010 --> 00:09:15.320 So S intersect U is the diagonals. 00:09:15.320 --> 00:09:21.250 And the dimension of the diagonal matrices is three. 00:09:21.250 --> 00:09:24.880 And we've got a basis, no problem. 00:09:24.880 --> 00:09:33.430 OK, as I write that, I think, OK, what about putting -- 00:09:33.430 --> 00:09:36.460 so this is like, this intersection -- 00:09:36.460 --> 00:09:40.820 is taking all the vectors that are 00:09:40.820 --> 00:09:45.720 in both, that are symmetric and also upper triangular. 00:09:45.720 --> 00:09:50.620 Now we looked at the union. 00:09:50.620 --> 00:09:55.070 Suppose I take the matrices that are symmetric 00:09:55.070 --> 00:09:59.070 or upper triangular. 00:09:59.070 --> 00:10:01.620 What -- why was that no good? 00:10:01.620 --> 00:10:08.900 So why is it no -- why I not interested in the union, 00:10:08.900 --> 00:10:13.060 putting together those two subspaces? 00:10:13.060 --> 00:10:18.740 So this, these are matrices that are in S or in U, or possibly 00:10:18.740 --> 00:10:21.760 both, so they, the diagonals included. 00:10:21.760 --> 00:10:22.970 But what's bad about this? 00:10:22.970 --> 00:10:27.670 It's not a subspace. 00:10:27.670 --> 00:10:30.420 It's like having, taking, you know, 00:10:30.420 --> 00:10:34.145 a couple of lines in the plane and stopping there. 00:10:36.680 --> 00:10:40.550 A line -- this is -- so there's a three dimensional subspace 00:10:40.550 --> 00:10:44.030 of a nine dimensional space, there's -- ooh, sorry, six. 00:10:44.030 --> 00:10:46.150 There's a six dimensional subspace 00:10:46.150 --> 00:10:47.910 of a nine dimensional space. 00:10:47.910 --> 00:10:49.410 There's another one. 00:10:49.410 --> 00:10:52.670 But they, they're headed in different directions, 00:10:52.670 --> 00:10:55.360 so we, we can't just put them together. 00:10:55.360 --> 00:10:56.990 We have to fill in. 00:10:56.990 --> 00:10:58.870 So that's what we do. 00:10:58.870 --> 00:11:03.000 To get this bigger space that I'll write with a plus sign, 00:11:03.000 --> 00:11:09.590 this is combinations of things in S and things in U. 00:11:09.590 --> 00:11:10.410 OK. 00:11:10.410 --> 00:11:15.300 So that's the final space I'm going to introduce. 00:11:15.300 --> 00:11:17.420 I have a couple of subspaces. 00:11:17.420 --> 00:11:19.830 I can take their intersection. 00:11:19.830 --> 00:11:24.840 And now I'm interested in not their union but their sum. 00:11:24.840 --> 00:11:28.130 So this would be the, this is the intersection, 00:11:28.130 --> 00:11:31.580 and this will be their sum. 00:11:31.580 --> 00:11:35.195 So what do I need for a subspace here? 00:11:38.010 --> 00:11:43.260 I take anything in S plus anything in U. 00:11:43.260 --> 00:11:46.010 I don't just take things that are in S and pop 00:11:46.010 --> 00:11:48.600 in also, separately, things that are in U. 00:11:48.600 --> 00:11:56.190 This is the sum of any element of S, 00:11:56.190 --> 00:12:02.310 that is, any symmetric matrix, plus any 00:12:02.310 --> 00:12:08.450 in U, any element of U. 00:12:08.450 --> 00:12:09.050 OK. 00:12:09.050 --> 00:12:11.950 Now as long as we've got an example here, 00:12:11.950 --> 00:12:15.020 tell me what we get. 00:12:15.020 --> 00:12:18.270 If I take every symmetric matrix, 00:12:18.270 --> 00:12:20.310 take all symmetric matrices, and add them 00:12:20.310 --> 00:12:22.770 to all upper triangular matrices, 00:12:22.770 --> 00:12:27.520 then I've got a whole lot of matrices and it is a subspace. 00:12:27.520 --> 00:12:30.170 And what's -- it's a vector space, 00:12:30.170 --> 00:12:34.280 and what vector space would I then have? 00:12:34.280 --> 00:12:36.670 Any idea what, what matrices can I 00:12:36.670 --> 00:12:42.200 get out of a symmetric plus an upper triangular? 00:12:42.200 --> 00:12:44.200 I can get anything. 00:12:44.200 --> 00:12:45.600 I get all matrices. 00:12:45.600 --> 00:12:50.300 I get all three by threes. 00:12:50.300 --> 00:12:54.380 It's worth thinking about that. 00:12:54.380 --> 00:12:58.330 It's just like stretch your mind a little, just a little, 00:12:58.330 --> 00:13:05.700 to, to think of these subspaces and what their intersection is 00:13:05.700 --> 00:13:07.110 and what their sum is. 00:13:07.110 --> 00:13:08.950 And now can I give you a little -- oh, well, 00:13:08.950 --> 00:13:10.810 let's figure out the dimension. 00:13:10.810 --> 00:13:17.220 So what's the dimension of S plus U? 00:13:17.220 --> 00:13:25.470 In this example is nine, because we got all three by threes. 00:13:25.470 --> 00:13:31.690 So the original spaces had, the original symmetric space had 00:13:31.690 --> 00:13:36.010 dimension six and the original upper triangular space 00:13:36.010 --> 00:13:38.010 had dimension six. 00:13:38.010 --> 00:13:42.975 And actually I'm seeing here a nice formula. 00:13:48.360 --> 00:13:54.810 That the dimension of S plus the dimension of U -- 00:13:54.810 --> 00:13:58.020 if I have two subspaces, the dimension of one plus 00:13:58.020 --> 00:14:03.750 the dimension of the other -- equals the dimension 00:14:03.750 --> 00:14:07.850 of their intersection plus the dimension of their sum. 00:14:07.850 --> 00:14:12.530 Six plus six is three plus nine. 00:14:16.550 --> 00:14:20.850 That's kind of satisfying, that these natural operations -- 00:14:20.850 --> 00:14:23.360 and we've -- this is it, actually, 00:14:23.360 --> 00:14:27.650 this is the set of natural things to do with, 00:14:27.650 --> 00:14:29.390 with subspaces. 00:14:29.390 --> 00:14:36.730 That, the dimensions come out in a good way. 00:14:36.730 --> 00:14:37.640 OK. 00:14:37.640 --> 00:14:43.540 Maybe I'll take just one more example of a vector space 00:14:43.540 --> 00:14:48.270 that doesn't have vectors in it. 00:14:48.270 --> 00:14:51.296 It's come from differential equations. 00:14:54.200 --> 00:14:57.120 So this is a one more new vector space 00:14:57.120 --> 00:14:59.580 that we'll give just a few minutes to. 00:14:59.580 --> 00:15:06.380 Suppose I have a differential equation like d^2y/dx^2+ y=0. 00:15:09.590 --> 00:15:12.200 OK. 00:15:12.200 --> 00:15:14.175 I look at the solutions to that equation. 00:15:16.790 --> 00:15:21.230 So what are the solutions to that equation? 00:15:21.230 --> 00:15:24.815 y=cos(x) is a solution. 00:15:28.250 --> 00:15:31.960 y=sin(x) is a solution. 00:15:31.960 --> 00:15:37.730 y equals -- well, e to the (ix) is a solution, if you want, 00:15:37.730 --> 00:15:41.150 if you allow me to put that in. 00:15:41.150 --> 00:15:42.520 But why should I put that in? 00:15:42.520 --> 00:15:45.890 It's already there. 00:15:45.890 --> 00:15:48.650 You see, I'm really looking at a null space here. 00:15:48.650 --> 00:15:52.730 I'm looking at the null space of a differential equation. 00:15:52.730 --> 00:15:55.210 That's the solution space. 00:15:55.210 --> 00:16:01.000 And describe the solution space, all solutions 00:16:01.000 --> 00:16:03.290 to this differential equation. 00:16:03.290 --> 00:16:07.450 So the equation is y''+y=0. 00:16:07.450 --> 00:16:11.630 Cosine's, cosine's a solution, sine is a solution. 00:16:11.630 --> 00:16:14.430 Now tell me all the solutions. 00:16:14.430 --> 00:16:18.930 They're -- so I don't need e^(ix). 00:16:18.930 --> 00:16:21.220 Forget that. 00:16:21.220 --> 00:16:23.420 What are all the complete solutions? 00:16:30.510 --> 00:16:32.860 Is what? 00:16:32.860 --> 00:16:34.840 A combination of these. 00:16:34.840 --> 00:16:38.070 The complete solution is y equals 00:16:38.070 --> 00:16:48.350 some multiple of the cosine plus some multiple of the sine. 00:16:48.350 --> 00:16:51.520 That's a vector space. 00:16:51.520 --> 00:16:52.660 That's a vector space. 00:16:52.660 --> 00:16:54.530 What's the dimension of that space? 00:16:54.530 --> 00:16:57.500 What's a basis for that space? 00:16:57.500 --> 00:17:00.100 OK, let me ask you a basis first. 00:17:00.100 --> 00:17:03.850 If I take the set of solutions to that second order 00:17:03.850 --> 00:17:06.450 differential equation -- 00:17:06.450 --> 00:17:10.520 there it is, those are the solutions. 00:17:10.520 --> 00:17:12.010 What's a basis for that space? 00:17:14.680 --> 00:17:16.480 Now remember, what's the, what question I 00:17:16.480 --> 00:17:16.980 asking? 00:17:16.980 --> 00:17:18.890 Because if you know the question I'm asking, 00:17:18.890 --> 00:17:21.750 you'll see the answer. 00:17:21.750 --> 00:17:25.990 A basis means all the guys in the space 00:17:25.990 --> 00:17:29.170 are combinations of these basis vectors. 00:17:29.170 --> 00:17:31.320 Well, this is a basis. 00:17:31.320 --> 00:17:35.870 sin x, cos x there is a basis. 00:17:38.640 --> 00:17:44.090 Those two -- they're like the special solutions, right? 00:17:44.090 --> 00:17:47.320 We had special solutions to Ax=b. 00:17:47.320 --> 00:17:53.730 Now we've got special solutions to differential equations. 00:17:53.730 --> 00:17:59.270 Sorry, we had special solutions to Ax=0, I misspoke. 00:17:59.270 --> 00:18:01.760 The special solutions were for the null space 00:18:01.760 --> 00:18:04.040 just as here we're talking about the null space. 00:18:04.040 --> 00:18:07.360 Do you see that here is a -- those two -- 00:18:07.360 --> 00:18:13.170 and what's the dimension of the solution space? 00:18:13.170 --> 00:18:23.360 How many vectors in this basis? 00:18:23.360 --> 00:18:27.430 Two, the sine and cosine. 00:18:27.430 --> 00:18:30.960 Are those the only basis for this space? 00:18:30.960 --> 00:18:32.720 By no means. 00:18:32.720 --> 00:18:37.070 e^(ix) and e^(-ix) would be another basis. 00:18:37.070 --> 00:18:38.220 Lots of bases. 00:18:38.220 --> 00:18:43.960 But do you see that really what a course in differential -- 00:18:43.960 --> 00:18:50.720 in linear differential equations is about is finding a basis 00:18:50.720 --> 00:18:52.460 for the solution space. 00:18:52.460 --> 00:18:56.640 The dimension of the solution space will always be -- 00:18:56.640 --> 00:19:01.410 will be two, because we have a second order equation. 00:19:01.410 --> 00:19:04.470 So that's, like there's 18.03 in -- 00:19:04.470 --> 00:19:10.310 five minutes of 18.06 is enough to, to take care of 18.03. 00:19:10.310 --> 00:19:12.590 So there's a -- that's one more example. 00:19:12.590 --> 00:19:13.090 OK. 00:19:13.090 --> 00:19:16.200 And of course the point of the example 00:19:16.200 --> 00:19:24.020 is these things don't look like vectors. 00:19:24.020 --> 00:19:26.080 They look like functions. 00:19:26.080 --> 00:19:31.520 But we can call them vectors, because we can add them 00:19:31.520 --> 00:19:34.320 and we can multiply by constants, 00:19:34.320 --> 00:19:36.470 so we can take linear combinations. 00:19:36.470 --> 00:19:39.800 That's all we have to be allowed to do. 00:19:39.800 --> 00:19:43.920 So that's really why this idea of linear algebra and basis 00:19:43.920 --> 00:19:51.120 and dimension and so on plays a wider role than -- 00:19:51.120 --> 00:19:56.820 our constant discussions of m by n matrices. 00:19:56.820 --> 00:19:57.520 OK. 00:19:57.520 --> 00:20:00.360 That's what I wanted to say about that topic. 00:20:00.360 --> 00:20:09.740 Now of course the key, number associated with matrices, 00:20:09.740 --> 00:20:13.510 to go back to that number, is the rank. 00:20:13.510 --> 00:20:17.990 And the rank, what do we know about the rank? 00:20:20.970 --> 00:20:23.360 Well, we know it's not bigger than m 00:20:23.360 --> 00:20:25.310 and it's not bigger than n. 00:20:25.310 --> 00:20:29.060 So but I'd like to have a little discussion on the rank. 00:20:29.060 --> 00:20:30.780 Maybe I'll put that here. 00:20:30.780 --> 00:20:33.895 So I'm picking up this topic of rank one matrices. 00:20:38.360 --> 00:20:46.070 And the reason I'm interested in rank one matrices 00:20:46.070 --> 00:20:48.900 is that they ought to be simple. 00:20:48.900 --> 00:20:55.450 If the rank is only one, the matrix can't get away from 00:20:55.450 --> 00:20:55.950 us. 00:20:55.950 --> 00:20:59.840 So for example, let me take -- let me create a rank one 00:20:59.840 --> 00:21:00.960 matrix. 00:21:00.960 --> 00:21:01.460 OK. 00:21:01.460 --> 00:21:04.840 Suppose it's three -- suppose it's two by three. 00:21:07.820 --> 00:21:10.943 And let me give you the first row. 00:21:16.080 --> 00:21:19.750 What can the second row be? 00:21:19.750 --> 00:21:24.060 Tell me a possible second row here, for, for this matrix 00:21:24.060 --> 00:21:27.570 to have rank one. 00:21:27.570 --> 00:21:30.380 A possible second row is? 00:21:30.380 --> 00:21:33.150 Two eight ten. 00:21:33.150 --> 00:21:41.390 The second row is a multiple of the first row. 00:21:41.390 --> 00:21:43.380 It's not independent. 00:21:43.380 --> 00:21:45.580 So tell me a basis for the -- oh yeah, 00:21:45.580 --> 00:21:50.250 sorry to keep bringing up these same questions. 00:21:50.250 --> 00:21:53.750 After the quiz I'll stop, but for now, 00:21:53.750 --> 00:21:57.200 tell me a basis for the row space. 00:21:57.200 --> 00:22:03.470 A basis for the row space of that matrix is the first row, 00:22:03.470 --> 00:22:03.980 right? 00:22:03.980 --> 00:22:06.860 The first row, one four five. 00:22:06.860 --> 00:22:11.020 A basis for the column space of this matrix is? 00:22:11.020 --> 00:22:15.110 What's the dimension of the column space? 00:22:15.110 --> 00:22:19.140 The dimension of the column space is also one, 00:22:19.140 --> 00:22:19.640 right? 00:22:19.640 --> 00:22:21.130 Because it's also the rank. 00:22:21.130 --> 00:22:24.980 The dimension -- you remember the dimension of the column 00:22:24.980 --> 00:22:33.090 space equals the rank equals the dimension of the column space 00:22:33.090 --> 00:22:38.510 of the transpose, which is the row space of A. 00:22:38.510 --> 00:22:44.620 OK, and in this case it's one, r is one. 00:22:44.620 --> 00:22:48.080 And sure enough, all the columns are -- 00:22:48.080 --> 00:22:51.390 all the other columns are multiples of that column. 00:22:51.390 --> 00:22:57.000 Now there's -- there ought to be a nice way to see that, 00:22:57.000 --> 00:22:59.700 and here it is. 00:22:59.700 --> 00:23:06.530 I can write that matrix as its pivot column, one two, 00:23:06.530 --> 00:23:10.520 times its -- 00:23:10.520 --> 00:23:11.730 times one four five. 00:23:14.470 --> 00:23:18.570 A column times a row, one column times one row 00:23:18.570 --> 00:23:21.340 gives me a matrix, right? 00:23:21.340 --> 00:23:25.370 If I multiply a column by a row, that, 00:23:25.370 --> 00:23:30.500 g- that's a two by one matrix times a one by three matrix, 00:23:30.500 --> 00:23:34.780 and the result of the multiplication is two by three. 00:23:34.780 --> 00:23:36.630 And it comes out right. 00:23:36.630 --> 00:23:45.450 So what I want to -- my point is the rank one matrices that 00:23:45.450 --> 00:23:52.830 every rank one matrix has the form some column times some 00:23:52.830 --> 00:23:55.520 row. 00:23:55.520 --> 00:23:59.180 So U is a column vector, V is a column vector -- 00:23:59.180 --> 00:24:03.570 but I make it into a row by putting in V transpose. 00:24:03.570 --> 00:24:12.330 So that's the -- complete picture of rank one matrices. 00:24:12.330 --> 00:24:14.580 We'll be interested in rank one matrices. 00:24:14.580 --> 00:24:19.270 Later we'll find, oh, their determinant, that'll be easy, 00:24:19.270 --> 00:24:23.620 their eigenvalues, that'll be interesting. 00:24:23.620 --> 00:24:26.450 Rank one matrices are like the building blocks 00:24:26.450 --> 00:24:27.410 for all matrices. 00:24:27.410 --> 00:24:30.620 And actually maybe you can guess. 00:24:37.220 --> 00:24:47.890 If I took any matrix, a five by seventeen matrix of rank four, 00:24:47.890 --> 00:24:51.890 then it seems pretty likely -- and it's true, 00:24:51.890 --> 00:24:56.060 that I could break that five by seventeen matrix down 00:24:56.060 --> 00:24:59.840 as a combination of rank one matrices. 00:24:59.840 --> 00:25:03.310 And probably how many of those would I need? 00:25:03.310 --> 00:25:07.810 If I have a five by seventeen matrix of rank four, 00:25:07.810 --> 00:25:11.300 I'll need four of them, right. 00:25:11.300 --> 00:25:13.490 Four rank one matrices. 00:25:13.490 --> 00:25:17.710 So the rank one matrices are the, are the building blocks. 00:25:17.710 --> 00:25:23.680 And out -- I can produce every, I can produce every five by -- 00:25:23.680 --> 00:25:29.070 every rank four matrix out of four rank one matrices. 00:25:29.070 --> 00:25:33.170 That brings me to a question, of course. 00:25:33.170 --> 00:25:33.670 OK. 00:25:33.670 --> 00:25:36.406 Would the rank four matrices form a subspace? 00:25:38.990 --> 00:25:43.450 Let me take all five by seventeen matrices and think 00:25:43.450 --> 00:25:48.589 about rank four -- the subset of rank four matrices. 00:25:48.589 --> 00:25:49.880 Let me -- I'll write this down. 00:25:53.670 --> 00:25:57.010 You seem I'm reviewing for the quiz, 00:25:57.010 --> 00:26:00.800 because I'm asking the kind of questions that are short enough 00:26:00.800 --> 00:26:04.780 but -- that bring out do you know what these words mean. 00:26:04.780 --> 00:26:06.400 So I take -- 00:26:06.400 --> 00:26:13.080 my matrix space M now is all five by seventeen matrices. 00:26:13.080 --> 00:26:27.010 And now the question I ask is the subset of, of rank four 00:26:27.010 --> 00:26:34.340 matrices, is that a subspace? 00:26:34.340 --> 00:26:38.650 If I add a matrix of -- so if I multiply a matrix of rank four 00:26:38.650 --> 00:26:39.880 by -- 00:26:39.880 --> 00:26:43.520 of rank four or less, let's say, because I 00:26:43.520 --> 00:26:49.490 have to let the zero matrix in if it's going to be a subspace. 00:26:49.490 --> 00:26:52.330 But, but that doesn't just because the zero matrix 00:26:52.330 --> 00:26:56.610 got in there doesn't mean I have a subspace. 00:26:56.610 --> 00:26:59.840 So if I -- so the, the question really comes down to -- 00:26:59.840 --> 00:27:05.605 if I add two rank four matrices, is the sum rank four? 00:27:09.050 --> 00:27:09.800 What do you think? 00:27:12.730 --> 00:27:15.110 If -- no, not usually. 00:27:15.110 --> 00:27:16.540 Not usually. 00:27:16.540 --> 00:27:20.200 If I add two rank four matrices, the sum is probably -- 00:27:20.200 --> 00:27:22.750 what could I say about the sum? 00:27:22.750 --> 00:27:27.905 Well, actually, well, the rank could be five. 00:27:30.790 --> 00:27:35.610 It's a general fact, actually, that the rank of A plus B 00:27:35.610 --> 00:27:40.880 can't be more than rank of A plus the rank of B. 00:27:40.880 --> 00:27:42.880 So this would say if I added two of those, 00:27:42.880 --> 00:27:45.820 the rank couldn't be larger than eight, but I know actually 00:27:45.820 --> 00:27:48.420 the rank couldn't be as large as eight anyway. 00:27:48.420 --> 00:27:50.650 What -- how big could the rank be, 00:27:50.650 --> 00:27:52.480 for, for the rank of a matrix in M? 00:27:52.480 --> 00:27:57.450 Could be as large as five, right, right. 00:27:57.450 --> 00:28:00.460 So they're all sort of natural ideas. 00:28:00.460 --> 00:28:05.570 So it's rank four matrices or rank one matrices -- 00:28:05.570 --> 00:28:09.600 let me, let me change that to rank one. 00:28:09.600 --> 00:28:12.670 Let me take the subset of rank one matrices. 00:28:12.670 --> 00:28:15.660 Is that a vector space? 00:28:15.660 --> 00:28:19.860 If I add a rank one matrix to a rank one matrix? 00:28:19.860 --> 00:28:20.460 No. 00:28:20.460 --> 00:28:23.050 It's most likely going to have rank two. 00:28:23.050 --> 00:28:23.790 So this is -- 00:28:23.790 --> 00:28:25.090 So I'll just make that point. 00:28:25.090 --> 00:28:27.690 Not a subspace. 00:28:32.030 --> 00:28:34.640 OK. 00:28:34.640 --> 00:28:35.140 OK. 00:28:35.140 --> 00:28:38.210 Those are topics that I wanted to, just 00:28:38.210 --> 00:28:42.530 fill out the, the previous lectures. 00:28:42.530 --> 00:28:46.730 The I'll ask one more subspace question, a, 00:28:46.730 --> 00:28:50.570 a more, a more, likely example. 00:28:50.570 --> 00:28:53.751 Suppose I'm in -- let me put, put this example on a new 00:28:53.751 --> 00:28:54.250 board. 00:28:56.850 --> 00:28:58.823 Suppose I'm in R, in R^4. 00:29:03.980 --> 00:29:10.850 So my typical vector in R^4 has four components, v1, v2, v3, 00:29:10.850 --> 00:29:11.350 and v4. 00:29:15.560 --> 00:29:21.280 Suppose I take the subspace of vectors 00:29:21.280 --> 00:29:23.850 whose components add to zero. 00:29:23.850 --> 00:29:34.070 So I let S be all v, all vectors v in four dimensional space 00:29:34.070 --> 00:29:37.050 with v1+v2+v3+v4=0. 00:29:37.050 --> 00:29:39.320 So I just want to consider that bunch of vectors. 00:29:39.320 --> 00:29:41.560 Is it a subspace, first of all? 00:29:41.560 --> 00:29:42.430 It is a subspace. 00:29:42.430 --> 00:29:57.880 It is a subspace. 00:29:57.880 --> 00:30:01.780 What's -- how do we see that? 00:30:01.780 --> 00:30:04.380 It is a subspace. 00:30:04.380 --> 00:30:06.890 I -- formally I should check. 00:30:06.890 --> 00:30:11.430 If I have one vector that with whose components add to zero 00:30:11.430 --> 00:30:13.990 and I multiply that vector by six -- 00:30:13.990 --> 00:30:16.980 the components still add to zero, just six times as -- 00:30:16.980 --> 00:30:19.250 six times zero. 00:30:19.250 --> 00:30:22.840 If I have a couple of v and a w and I add them, 00:30:22.840 --> 00:30:25.150 the, the components still add to zero. 00:30:25.150 --> 00:30:27.170 OK, it's a subspace. 00:30:27.170 --> 00:30:29.500 What's the dimension of that space 00:30:29.500 --> 00:30:32.730 and what's a basis for that space? 00:30:32.730 --> 00:30:37.370 So you see how I can just describe a space and we -- 00:30:37.370 --> 00:30:41.530 we can ask for the dimension -- ask for the basis first 00:30:41.530 --> 00:30:42.800 and the dimension. 00:30:42.800 --> 00:30:44.620 Of course, the dimension's the one 00:30:44.620 --> 00:30:48.710 that's easy to tell me in a single word. 00:30:48.710 --> 00:30:51.790 What's the dimension of our subspace S here? 00:30:55.910 --> 00:30:57.940 And a basis tell me -- 00:30:57.940 --> 00:31:01.250 some vectors in it. 00:31:01.250 --> 00:31:07.210 Well, I'm going to make ask you again to guess the dimension. 00:31:07.210 --> 00:31:09.190 Again I think I heard it. 00:31:09.190 --> 00:31:11.270 The dimension is three. 00:31:11.270 --> 00:31:12.650 Three. 00:31:12.650 --> 00:31:18.380 Now how does this connect to our Ax=0? 00:31:18.380 --> 00:31:21.370 Is this the null space of something? 00:31:21.370 --> 00:31:24.500 Is that the null space of a matrix? 00:31:24.500 --> 00:31:26.210 And then we can look at the matrix 00:31:26.210 --> 00:31:30.240 and, and we know everything about those subspaces. 00:31:30.240 --> 00:31:39.665 This is the null space of what matrix? 00:31:47.010 --> 00:31:53.030 What's the matrix where the null space is then Ab=0. 00:31:53.030 --> 00:31:57.195 So I want this equation to be Ab=0. 00:31:59.920 --> 00:32:02.510 b is now the vector. 00:32:02.510 --> 00:32:07.850 And what's the matrix that, that we're seeing there? 00:32:07.850 --> 00:32:14.580 It's the matrix of four ones. 00:32:14.580 --> 00:32:20.120 Do you see that that's -- that if I look at Ab=0 for this 00:32:20.120 --> 00:32:27.180 matrix A, I multiply by b and I get this requirement, 00:32:27.180 --> 00:32:29.280 that the components add to zero. 00:32:29.280 --> 00:32:33.670 So I'm really when I speak about S -- 00:32:33.670 --> 00:32:37.280 I'm speaking about the null space of that matrix. 00:32:37.280 --> 00:32:38.050 OK. 00:32:38.050 --> 00:32:41.350 Let's just say we've got a matrix now, 00:32:41.350 --> 00:32:44.000 we want its null space. 00:32:44.000 --> 00:32:47.610 Well, we -- tell me its rank first. 00:32:47.610 --> 00:32:54.900 The rank of that matrix is one, thanks. 00:32:54.900 --> 00:32:57.210 So r is one. 00:32:57.210 --> 00:32:59.610 What's the general formula for the dimension 00:32:59.610 --> 00:33:01.530 of the null space? 00:33:01.530 --> 00:33:07.640 The dimension of the null space of a matrix is -- 00:33:07.640 --> 00:33:12.420 in general, an m by n matrix of rank r? 00:33:12.420 --> 00:33:16.895 How many independent guys in the null space? 00:33:21.140 --> 00:33:22.920 n-r, right? 00:33:22.920 --> 00:33:25.780 n-r. 00:33:25.780 --> 00:33:31.390 In this case, n is four, four columns. 00:33:31.390 --> 00:33:35.330 The rank is one, so the null space is three dimensions. 00:33:35.330 --> 00:33:39.440 So of course y- you could see it in this case, 00:33:39.440 --> 00:33:44.160 but you can also see it here in our systematic way 00:33:44.160 --> 00:33:49.450 of dealing with the four fundamental subspaces 00:33:49.450 --> 00:33:51.730 of a matrix. 00:33:51.730 --> 00:33:55.410 So what actually what, what are all four subspaces 00:33:55.410 --> 00:33:56.050 then? 00:33:56.050 --> 00:33:58.840 The row space is clear. 00:33:58.840 --> 00:34:01.420 The row space is in R^4. 00:34:01.420 --> 00:34:05.530 Yeah, can we take the four fundamental subspaces 00:34:05.530 --> 00:34:06.910 of this matrix? 00:34:06.910 --> 00:34:08.815 Let's just kill this example. 00:34:11.860 --> 00:34:15.400 The row space is one dimensional. 00:34:15.400 --> 00:34:19.830 It's all multiples of that, of that row. 00:34:19.830 --> 00:34:22.780 The null space is three dimensional. 00:34:22.780 --> 00:34:26.260 Oh, you better give me a basis for the null space. 00:34:26.260 --> 00:34:28.080 So what's a basis for the null space? 00:34:28.080 --> 00:34:30.489 The special solutions. 00:34:30.489 --> 00:34:34.830 To find the special solutions, I look for the free variables. 00:34:34.830 --> 00:34:39.650 The free variables here are -- there's the pivot. 00:34:39.650 --> 00:34:44.010 The free variables are two, three, and four. 00:34:44.010 --> 00:34:52.730 So the basis, basis for S, for S will be -- 00:34:52.730 --> 00:35:01.710 I'm expecting three vectors, three special solutions. 00:35:01.710 --> 00:35:06.470 I give the value one to that free variable, 00:35:06.470 --> 00:35:12.000 and what's the pivot variable if the -- 00:35:12.000 --> 00:35:15.130 this is going to be a vector in S? 00:35:15.130 --> 00:35:16.390 Minus one. 00:35:16.390 --> 00:35:20.250 Now they're always added to -- the entries add to zero. 00:35:20.250 --> 00:35:22.870 The second special solution has a one 00:35:22.870 --> 00:35:26.540 in the second free variable, and again a minus one 00:35:26.540 --> 00:35:27.730 makes it right. 00:35:27.730 --> 00:35:30.940 The third one has a one in the third free variable, 00:35:30.940 --> 00:35:34.910 and again a minus one makes it right. 00:35:34.910 --> 00:35:35.780 That's my answer. 00:35:35.780 --> 00:35:39.770 That's the answer I would be looking for. 00:35:39.770 --> 00:35:43.300 The -- a basis for this subspace S, 00:35:43.300 --> 00:35:45.530 you would just list three vectors, 00:35:45.530 --> 00:35:48.350 and those would be the natural three to list. 00:35:48.350 --> 00:35:55.240 Not the only possible three, but those are the special three. 00:35:55.240 --> 00:35:58.960 OK, tell me about the column space, 00:35:58.960 --> 00:36:02.560 What's the column space of this matrix A? 00:36:07.390 --> 00:36:13.500 So the column space is a subspace of R^1, 00:36:13.500 --> 00:36:15.470 because m is only one. 00:36:15.470 --> 00:36:17.920 The columns only have one component. 00:36:17.920 --> 00:36:23.360 So the column space of S, the column space of A is somewhere 00:36:23.360 --> 00:36:26.410 in the space R^1, because we only have -- 00:36:26.410 --> 00:36:30.240 these columns are short. 00:36:30.240 --> 00:36:32.940 And what is the column space actually? 00:36:36.330 --> 00:36:42.280 I just, it's just talking with these words is what I'm doing. 00:36:42.280 --> 00:36:48.360 The column space for that matrix is R^1. 00:36:48.360 --> 00:36:52.780 The column space for that matrix is 00:36:52.780 --> 00:36:55.463 all multiples of that column. 00:36:58.410 --> 00:37:00.760 And all multiples give you all of R^1. 00:37:03.630 --> 00:37:07.140 And what's the, the remaining fourth space, 00:37:07.140 --> 00:37:12.680 the null space of A transpose is what? 00:37:17.560 --> 00:37:21.880 So we transpose A. 00:37:21.880 --> 00:37:26.290 We look for combinations of the columns 00:37:26.290 --> 00:37:30.310 now that give zero for A transpose. 00:37:30.310 --> 00:37:32.190 And there aren't any. 00:37:32.190 --> 00:37:36.620 The only thing, the only combination of these rows 00:37:36.620 --> 00:37:41.500 to give the zero row is the zero combination. 00:37:41.500 --> 00:37:42.380 OK. 00:37:42.380 --> 00:37:44.415 So let's just check dimensions. 00:37:47.180 --> 00:37:51.320 The null space has dimension three. 00:37:51.320 --> 00:37:53.420 The row space has dimension one. 00:37:53.420 --> 00:37:54.630 Three plus one is four. 00:37:57.180 --> 00:37:59.680 The column space has dimension one, 00:37:59.680 --> 00:38:02.790 and what's the dimension of this, like, 00:38:02.790 --> 00:38:05.510 smallest possible space? 00:38:05.510 --> 00:38:08.790 What's the dimension of the zero space? 00:38:08.790 --> 00:38:09.675 It's a subspace. 00:38:13.520 --> 00:38:14.190 Zero. 00:38:14.190 --> 00:38:15.190 What else could it be? 00:38:15.190 --> 00:38:17.970 I mean, let's -- we have to take a reasonable answer -- 00:38:17.970 --> 00:38:20.190 and the only reasonable answer is zero. 00:38:20.190 --> 00:38:25.900 So one plus zero gives -- this was n, the number of columns, 00:38:25.900 --> 00:38:30.480 and this is m, the number of rows. 00:38:30.480 --> 00:38:32.910 And let's just, let me just say again 00:38:32.910 --> 00:38:37.030 then the, the, the subspace that has only that one 00:38:37.030 --> 00:38:42.830 point, that point is zero dimensional, of course. 00:38:42.830 --> 00:38:46.920 And the basis is empty, because if the dimension is zero, 00:38:46.920 --> 00:38:49.480 there shouldn't be anybody in the basis. 00:38:49.480 --> 00:38:56.160 So the basis of that smallest subspace is the empty set. 00:38:56.160 --> 00:38:59.590 And the number of members in the empty set is zero, 00:38:59.590 --> 00:39:01.400 so that's the dimension. 00:39:01.400 --> 00:39:02.610 OK. 00:39:02.610 --> 00:39:04.040 Good. 00:39:04.040 --> 00:39:10.740 Now I have just five minutes to tell you about -- 00:39:10.740 --> 00:39:16.010 well, actually, about some, some, some, this is now, 00:39:16.010 --> 00:39:21.430 this last topic of small world graphs, and leads into, 00:39:21.430 --> 00:39:27.140 a lecture about graphs and linear algebra. 00:39:27.140 --> 00:39:29.700 But let me tell you -- 00:39:29.700 --> 00:39:32.860 in these last minutes the graph that I interested in. 00:39:32.860 --> 00:39:40.600 It's the graph where -- so what is a graph? 00:39:40.600 --> 00:39:42.020 Better tell you that first. 00:39:42.020 --> 00:39:42.880 OK. 00:39:42.880 --> 00:39:43.516 What's a graph? 00:39:47.380 --> 00:39:48.180 OK. 00:39:48.180 --> 00:39:49.980 This isn't calculus. 00:39:49.980 --> 00:39:54.010 We're not, I'm not thinking of, like, some sine curve. 00:39:54.010 --> 00:39:57.940 The word graph is used in a completely different way. 00:39:57.940 --> 00:40:07.490 It's a set of, a bunch of nodes and edges, 00:40:07.490 --> 00:40:10.060 edges connecting the nodes. 00:40:10.060 --> 00:40:17.280 So I have nodes like five nodes and edges -- 00:40:17.280 --> 00:40:21.880 I'll put in some edges, I could put, include them all. 00:40:21.880 --> 00:40:23.970 There's -- well, let me put in a couple more. 00:40:27.350 --> 00:40:31.380 There's a graph with five nodes and one two three four 00:40:31.380 --> 00:40:34.690 five six edges. 00:40:34.690 --> 00:40:38.190 And some five by six matrix is going to tell us 00:40:38.190 --> 00:40:41.100 everything about that graph. 00:40:41.100 --> 00:40:43.350 Let me leave that matrix to next time 00:40:43.350 --> 00:40:46.350 and tell you about the question I'm interested in. 00:40:46.350 --> 00:40:52.340 Suppose, suppose the graph isn't just, 00:40:52.340 --> 00:40:56.230 just doesn't have just five nodes, but suppose every, 00:40:56.230 --> 00:40:59.780 suppose every person in this room is a node. 00:41:03.200 --> 00:41:07.310 And suppose there's an edge between two nodes 00:41:07.310 --> 00:41:11.300 if those two people are friends. 00:41:11.300 --> 00:41:14.070 So have I described a graph? 00:41:14.070 --> 00:41:18.980 It's a pretty big graph, hundred, hundred nodes. 00:41:18.980 --> 00:41:21.190 And I don't know how many edges are in there. 00:41:24.920 --> 00:41:27.730 There's an edge if you're friends. 00:41:27.730 --> 00:41:29.950 So that's the graph for this class. 00:41:29.950 --> 00:41:35.120 A, a similar graph you could take for the whole country, 00:41:35.120 --> 00:41:38.680 so two hundred and sixty million nodes. 00:41:38.680 --> 00:41:43.340 And edges between friends. 00:41:43.340 --> 00:41:50.110 And the question for that graph is how many steps does it take 00:41:50.110 --> 00:41:52.210 to get from anybody to anybody? 00:41:56.780 --> 00:42:01.950 What two people are furthest apart in this friendship graph, 00:42:01.950 --> 00:42:04.070 say for the US? 00:42:04.070 --> 00:42:08.820 By furthest apart, I mean the distance from -- 00:42:08.820 --> 00:42:12.640 well, I'll tell you my distance to Clinton. 00:42:12.640 --> 00:42:14.040 It's two. 00:42:14.040 --> 00:42:18.440 I happened to go to college with somebody who knows Clinton. 00:42:18.440 --> 00:42:19.190 I don't know him. 00:42:19.190 --> 00:42:24.950 So my distance to Clinton is not one, because I don't, happily 00:42:24.950 --> 00:42:26.680 or not, don't know him. 00:42:26.680 --> 00:42:29.050 But I know somebody who does. 00:42:29.050 --> 00:42:32.541 He's a Senator and so I presume he knows him. 00:42:32.541 --> 00:42:33.040 OK. 00:42:33.040 --> 00:42:35.311 I don't know what your -- well, what's your distance 00:42:35.311 --> 00:42:35.810 to Clinton? 00:42:39.100 --> 00:42:40.950 Well, not more than three, right. 00:42:40.950 --> 00:42:43.000 Actually, true. 00:42:43.000 --> 00:42:44.430 You know me. 00:42:44.430 --> 00:42:50.880 I take credit for reducing your Clinton distance to three -- 00:42:50.880 --> 00:42:52.195 what's your distance to Monica. 00:42:54.780 --> 00:43:04.090 Not, anybody below -- below four is in trouble here. 00:43:04.090 --> 00:43:07.010 Or maybe three, but, right. 00:43:07.010 --> 00:43:14.740 So -- and what's Hillary's distance to Monica? 00:43:14.740 --> 00:43:18.020 I don't think we'd better put that on tape here. 00:43:18.020 --> 00:43:22.360 That's one or two, I guess. 00:43:22.360 --> 00:43:24.490 Is that right? 00:43:24.490 --> 00:43:28.980 I don't -- well, we won't, think more about that. 00:43:28.980 --> 00:43:32.490 So actually, the, the real question 00:43:32.490 --> 00:43:38.030 is what are large distances? 00:43:38.030 --> 00:43:41.910 How, how far apart could people be separated? 00:43:41.910 --> 00:43:46.520 And roughly this number six degrees of separation 00:43:46.520 --> 00:43:49.940 has kind of appeared as the movie title, as the book title, 00:43:49.940 --> 00:43:52.050 and it's with this meaning. 00:43:52.050 --> 00:43:56.050 That roughly speaking -- 00:43:56.050 --> 00:43:59.230 six might be a fairly -- 00:43:59.230 --> 00:44:01.860 not too many people. 00:44:01.860 --> 00:44:04.860 If you sit next to somebody on an airplane, 00:44:04.860 --> 00:44:07.850 you get talking to them. 00:44:07.850 --> 00:44:12.310 You begin to discuss mutual friends to sort of find out, 00:44:12.310 --> 00:44:15.220 OK, what connections do you have, 00:44:15.220 --> 00:44:17.450 and very often you'll find you're 00:44:17.450 --> 00:44:21.880 connected in, like, two or three or four steps. 00:44:21.880 --> 00:44:23.970 And you remark, it's a small world, 00:44:23.970 --> 00:44:27.850 and that's how this expression small world came up. 00:44:27.850 --> 00:44:31.450 But six, I don't know if you could find -- if it took six, 00:44:31.450 --> 00:44:34.810 I don't know if you would successfully discover those six 00:44:34.810 --> 00:44:36.980 in a, in an airplane conversation. 00:44:36.980 --> 00:44:40.150 But here's the math question, and I'll 00:44:40.150 --> 00:44:42.650 leave it for next, for lecture twelve, 00:44:42.650 --> 00:44:46.010 and do a lot of linear algebra in lecture twelve. 00:44:46.010 --> 00:44:54.860 But the interesting point is that with a few shortcuts, 00:44:54.860 --> 00:44:58.290 the distances come down dramatically. 00:44:58.290 --> 00:45:03.900 That, I mean, all your distances to Clinton immediately drop 00:45:03.900 --> 00:45:06.460 to three by taking linear algebra. 00:45:06.460 --> 00:45:11.580 That's, like, an extra bonus for taking linear algebra. 00:45:11.580 --> 00:45:17.550 And to understand mathematically what it is about these graphs 00:45:17.550 --> 00:45:18.120 -- 00:45:18.120 --> 00:45:21.710 or like the graphs of the World Wide Web. 00:45:21.710 --> 00:45:23.170 There's a fantastic graph. 00:45:23.170 --> 00:45:27.790 So many people would like to understand and model the web. 00:45:27.790 --> 00:45:34.650 What the -- where the edges are links and the nodes are, sites, 00:45:34.650 --> 00:45:37.530 websites. 00:45:37.530 --> 00:45:39.960 I'll leave you with that graph, and I'll see you -- 00:45:39.960 --> 00:45:42.740 have a good weekend, and see you on Monday.