WEBVTT 00:00:09.860 --> 00:00:10.620 Okay. 00:00:10.620 --> 00:00:14.290 This is lecture five in linear algebra. 00:00:14.290 --> 00:00:19.015 And, it will complete this chapter of the book. 00:00:21.620 --> 00:00:24.030 So the last section of this chapter 00:00:24.030 --> 00:00:30.060 is two point seven that talks about permutations, which 00:00:30.060 --> 00:00:34.240 finished the previous lecture, and transposes, 00:00:34.240 --> 00:00:36.650 which also came in the previous lecture. 00:00:36.650 --> 00:00:40.750 There's a little more to do with those guys, permutations 00:00:40.750 --> 00:00:42.860 and transposes. 00:00:42.860 --> 00:00:48.180 But then the heart of the lecture will be the beginning 00:00:48.180 --> 00:00:51.910 of what you could say is the beginning of linear algebra, 00:00:51.910 --> 00:00:57.470 the beginning of real linear algebra which is seeing 00:00:57.470 --> 00:01:01.590 a bigger picture with vector spaces -- not just vectors, 00:01:01.590 --> 00:01:06.650 but spaces of vectors and sub-spaces of those spaces. 00:01:06.650 --> 00:01:11.110 So we're a little ahead of the syllabus, which 00:01:11.110 --> 00:01:13.150 is good, because we're coming to the place 00:01:13.150 --> 00:01:15.780 where, there's a lot to 00:01:15.780 --> 00:01:16.620 do. 00:01:16.620 --> 00:01:17.800 Okay. 00:01:17.800 --> 00:01:22.400 So, to begin with permutations. 00:01:22.400 --> 00:01:26.790 Can I just -- 00:01:26.790 --> 00:01:36.850 so these permutations, those are matrices P and they execute 00:01:36.850 --> 00:01:38.400 row exchanges. 00:01:42.160 --> 00:01:44.910 And we may need them. 00:01:44.910 --> 00:01:47.120 We may have a perfectly good matrix, 00:01:47.120 --> 00:01:52.310 a perfect matrix A that's invertible that we can solve A 00:01:52.310 --> 00:01:55.580 x=b, but to do it -- 00:01:55.580 --> 00:02:00.050 I've got to allow myself that extra freedom 00:02:00.050 --> 00:02:05.070 that if a zero shows up in the pivot position I move it away. 00:02:05.070 --> 00:02:06.630 I get a non-zero. 00:02:06.630 --> 00:02:12.460 I get a proper pivot there by exchanging from a row below. 00:02:12.460 --> 00:02:15.000 And you've seen that already, and I just 00:02:15.000 --> 00:02:18.200 want to collect the ideas together. 00:02:18.200 --> 00:02:21.310 And principle, I could even have to do 00:02:21.310 --> 00:02:24.750 that two times, or more times. 00:02:24.750 --> 00:02:27.290 So I have to allow -- 00:02:27.290 --> 00:02:29.330 to complete the -- 00:02:29.330 --> 00:02:34.370 the theory, the possibility that I take my matrix A, 00:02:34.370 --> 00:02:38.540 I start elimination, I find out that I need row exchanges 00:02:38.540 --> 00:02:41.620 and I do it and continue and I finish. 00:02:41.620 --> 00:02:42.750 Okay. 00:02:42.750 --> 00:02:49.150 Then all I want to do is say -- and I won't make a big project 00:02:49.150 --> 00:02:51.160 out of this -- 00:02:51.160 --> 00:02:54.300 what happens to A equal L U? 00:02:54.300 --> 00:02:59.570 So A equal L U -- 00:02:59.570 --> 00:03:05.130 this was a matrix L with ones on the diagonal and zeroes 00:03:05.130 --> 00:03:09.450 above and multipliers below, and this U 00:03:09.450 --> 00:03:13.425 we know, with zeroes down here. 00:03:16.940 --> 00:03:19.080 That's only possible. 00:03:19.080 --> 00:03:21.770 That description of elimination assumes 00:03:21.770 --> 00:03:26.480 that we don't have a P, that we don't have any row exchanges. 00:03:26.480 --> 00:03:29.570 And now I just want to say, okay, how 00:03:29.570 --> 00:03:31.220 do I account for row exchanges? 00:03:31.220 --> 00:03:33.990 Because that doesn't. 00:03:33.990 --> 00:03:40.030 The P in this factorization is the identity matrix. 00:03:40.030 --> 00:03:44.140 The rows were in a good order, we left them there. 00:03:44.140 --> 00:03:47.830 Maybe I'll just add a little moment of reality, 00:03:47.830 --> 00:03:55.190 too, about how Matlab actually does elimination. 00:03:55.190 --> 00:03:59.520 Matlab not only checks whether that pivot is not zero, 00:03:59.520 --> 00:04:02.770 as every human would do. 00:04:02.770 --> 00:04:05.480 It checks for is that pivot big enough, 00:04:05.480 --> 00:04:09.380 because it doesn't like very, very small pivots. 00:04:09.380 --> 00:04:13.550 Pivots close to zero are numerically bad. 00:04:13.550 --> 00:04:16.660 So actually if we ask Matlab to solve a system, 00:04:16.660 --> 00:04:21.420 it will do some elimination some row exchanges, which 00:04:21.420 --> 00:04:23.180 we don't think are necessary. 00:04:23.180 --> 00:04:27.930 Algebra doesn't say they're necessary, but accuracy -- 00:04:27.930 --> 00:04:31.680 numerical accuracy says they are. 00:04:31.680 --> 00:04:35.930 Well, we're doing algebra, so here we 00:04:35.930 --> 00:04:39.380 will say, well, what do row exchanges do, 00:04:39.380 --> 00:04:42.230 but we won't do them unless we have to. 00:04:42.230 --> 00:04:45.340 But we may have to. 00:04:45.340 --> 00:04:50.380 And then, the result is -- 00:04:50.380 --> 00:04:51.850 it's hiding here. 00:04:51.850 --> 00:04:56.040 It's the main fact. 00:04:56.040 --> 00:05:00.100 This is the description of elimination with row exchanges. 00:05:00.100 --> 00:05:12.280 So A equal L U becomes P A equal L U. 00:05:12.280 --> 00:05:15.210 So this P is the matrix that does the row exchanges, 00:05:15.210 --> 00:05:18.080 and actually it does them -- 00:05:18.080 --> 00:05:20.780 it gets the rows into the right order, 00:05:20.780 --> 00:05:23.810 into the good order where pivots will not -- 00:05:23.810 --> 00:05:27.120 where zeroes won't appear in the pivot position, 00:05:27.120 --> 00:05:32.700 where L and U will come out right as up here. 00:05:32.700 --> 00:05:36.430 So, that's the point. 00:05:36.430 --> 00:05:39.650 Actually, I don't want to labor that point, 00:05:39.650 --> 00:05:43.110 that a permutation matrix -- 00:05:43.110 --> 00:05:47.260 and you remember what those were. 00:05:47.260 --> 00:05:51.280 I'll remind you from last time of what the main points about 00:05:51.280 --> 00:05:57.820 permutation matrices were -- 00:05:57.820 --> 00:06:00.340 and then just leave this factorization 00:06:00.340 --> 00:06:03.820 as the general case. 00:06:03.820 --> 00:06:17.220 This is -- any invertible A we get this. 00:06:17.220 --> 00:06:20.090 For almost every one, we don't need a P. 00:06:20.090 --> 00:06:24.220 But there's that handful that do need row exchanges, 00:06:24.220 --> 00:06:26.480 and if we do need them, there they are. 00:06:26.480 --> 00:06:30.410 Okay, finally, just to remember what P was. 00:06:30.410 --> 00:06:46.440 So permutations, P is the identity matrix 00:06:46.440 --> 00:06:51.480 with reordered rows. 00:06:56.710 --> 00:06:59.707 I include in reordering the possibility that you just 00:06:59.707 --> 00:07:00.540 leave them the same. 00:07:00.540 --> 00:07:02.880 So the identity matrix is -- okay. 00:07:02.880 --> 00:07:08.490 That's, like, your basic permutation matrix -- 00:07:08.490 --> 00:07:12.840 your do-nothing permutation matrix is the identity. 00:07:12.840 --> 00:07:15.730 And then there are the ones that exchange two rows and then 00:07:15.730 --> 00:07:18.940 the ones that exchange three rows and then then ones that 00:07:18.940 --> 00:07:21.040 exchange four -- 00:07:21.040 --> 00:07:24.640 well, it gets a little -- 00:07:24.640 --> 00:07:27.310 it gets more interesting algebraically 00:07:27.310 --> 00:07:30.630 if you've got four rows, you might exchange them 00:07:30.630 --> 00:07:32.190 all in one big cycle. 00:07:32.190 --> 00:07:35.550 One to two, two to three, three to four, four to one. 00:07:35.550 --> 00:07:41.070 Or you might have -- exchange one and two and three and four. 00:07:41.070 --> 00:07:42.467 Lots of possibilities there. 00:07:42.467 --> 00:07:43.800 In fact, how many possibilities? 00:07:46.570 --> 00:07:49.310 The answer was (n)factorial. 00:07:49.310 --> 00:07:50.984 This is n(n-1)(n-2)... 00:07:56.920 --> 00:07:58.230 (3)(2)(1). 00:07:58.230 --> 00:08:05.100 That's the number of -- this counts the reorderings, 00:08:05.100 --> 00:08:07.990 the possible reorderings. 00:08:07.990 --> 00:08:15.730 So it counts all the n by n permutations. 00:08:23.170 --> 00:08:26.200 And all those matrices have these -- 00:08:26.200 --> 00:08:32.380 have this nice property that they're all invertible, 00:08:32.380 --> 00:08:39.140 because we can bring those rows back into the normal order. 00:08:39.140 --> 00:08:42.679 And the matrix that does that is just P -- 00:08:42.679 --> 00:08:46.590 is just the same as the transpose. 00:08:46.590 --> 00:08:49.180 You might take a permutation matrix, 00:08:49.180 --> 00:08:51.820 multiply by its transpose and you will see how -- 00:08:51.820 --> 00:08:57.220 that the ones hit the ones and give the ones in the identity 00:08:57.220 --> 00:08:58.010 matrix. 00:08:58.010 --> 00:08:59.980 So this is a -- 00:08:59.980 --> 00:09:02.590 we'll be highly interested in matrices 00:09:02.590 --> 00:09:04.980 that have nice properties. 00:09:04.980 --> 00:09:09.580 And one property that -- maybe I could rewrite that as P 00:09:09.580 --> 00:09:13.880 transpose P is the identity. 00:09:13.880 --> 00:09:16.950 That tells me in other words that this 00:09:16.950 --> 00:09:19.300 is the inverse of that. 00:09:19.300 --> 00:09:20.210 Okay. 00:09:20.210 --> 00:09:25.940 We'll be interested in matrices that have P transpose 00:09:25.940 --> 00:09:27.940 P equal the identity. 00:09:27.940 --> 00:09:30.470 There are more of them than just permutations, 00:09:30.470 --> 00:09:35.000 but my point right now is that permutations are like a little 00:09:35.000 --> 00:09:36.870 group in the middle -- 00:09:36.870 --> 00:09:42.180 in the center of these special matrices. 00:09:42.180 --> 00:09:43.270 Okay. 00:09:43.270 --> 00:09:47.590 So now we know how many there are. 00:09:47.590 --> 00:09:50.750 Twenty four in the case of -- there are twenty four four 00:09:50.750 --> 00:09:55.960 by four permutations, there are five factorial which is 00:09:55.960 --> 00:10:00.340 a hundred and twenty, five times twenty four would bump us up 00:10:00.340 --> 00:10:03.710 to a hundred and twenty -- so listing all the five by five 00:10:03.710 --> 00:10:09.990 permutations would be not so much fun. 00:10:09.990 --> 00:10:12.380 Okay. 00:10:12.380 --> 00:10:15.320 So that's permutations. 00:10:15.320 --> 00:10:20.960 Now also in section two seven is some discussion of transposes. 00:10:20.960 --> 00:10:22.895 And can I just complete that discussion. 00:10:25.610 --> 00:10:30.910 First of all, I haven't even transposed a matrix 00:10:30.910 --> 00:10:32.110 on the board here, have I? 00:10:32.110 --> 00:10:33.710 So I'd better do it. 00:10:33.710 --> 00:10:39.975 So suppose I take a matrix like (1 2 4; 3 3 1). 00:10:43.660 --> 00:10:47.640 It's a rectangular matrix, three by two. 00:10:47.640 --> 00:10:51.030 And I want to transpose it. 00:10:51.030 --> 00:10:53.380 So what's -- 00:10:53.380 --> 00:10:59.040 I'll use a T, also Matlab would use a prime. 00:10:59.040 --> 00:11:02.050 And the result will be -- 00:11:02.050 --> 00:11:07.770 I'll right it here, because this was three rows and two columns, 00:11:07.770 --> 00:11:10.060 this was a three by two matrix. 00:11:10.060 --> 00:11:13.080 The transpose will be two rows and three columns, 00:11:13.080 --> 00:11:15.380 two by three. 00:11:15.380 --> 00:11:20.180 So it's short and wider. 00:11:20.180 --> 00:11:25.730 And, of course, that row -- that column becomes a row -- 00:11:25.730 --> 00:11:27.510 that column becomes the other row. 00:11:30.060 --> 00:11:36.200 And at the same time, that row became a column. 00:11:36.200 --> 00:11:37.970 This row became a column. 00:11:37.970 --> 00:11:41.610 Oh, what's the general formula for the transpose? 00:11:41.610 --> 00:11:48.920 So the transpose -- 00:11:48.920 --> 00:11:50.740 you see it in numbers. 00:11:50.740 --> 00:11:54.570 What I'm going to write is the same thing in symbols. 00:11:54.570 --> 00:11:57.920 The numbers are the clearest, of course. 00:11:57.920 --> 00:12:04.170 But in symbols, if I take A transpose 00:12:04.170 --> 00:12:10.135 and I ask what number is in row I and column J of A transpose? 00:12:14.600 --> 00:12:16.320 Well, it came out of A. 00:12:16.320 --> 00:12:22.080 It came out A by this flip across the main diagonal. 00:12:22.080 --> 00:12:28.030 And, actually, it was the number in A 00:12:28.030 --> 00:12:34.540 which was in row J, column I. 00:12:34.540 --> 00:12:36.920 So the row and column -- 00:12:36.920 --> 00:12:39.880 the row and column numbers just get reversed. 00:12:39.880 --> 00:12:42.030 The row number becomes the column number, 00:12:42.030 --> 00:12:44.450 the column number becomes the row number. 00:12:44.450 --> 00:12:45.610 No problem. 00:12:45.610 --> 00:12:46.260 Okay. 00:12:46.260 --> 00:12:49.280 Now, a special -- 00:12:49.280 --> 00:12:52.880 the best matrices, we could say. 00:12:52.880 --> 00:12:58.590 In a lot of applications, symmetric matrices show up. 00:12:58.590 --> 00:13:02.145 So can I just call attention to symmetric matrices? 00:13:07.960 --> 00:13:08.840 What does that mean? 00:13:08.840 --> 00:13:11.110 What does that word symmetric mean? 00:13:11.110 --> 00:13:16.000 It means that this transposing doesn't change the matrix. 00:13:16.000 --> 00:13:17.690 A transpose equals A. 00:13:17.690 --> 00:13:21.255 And an example. 00:13:25.740 --> 00:13:29.900 So, let's take a matrix that's symmetric, 00:13:29.900 --> 00:13:33.750 so whatever is sitting on the diagonal -- 00:13:33.750 --> 00:13:36.800 but now what's above the diagonal, like a one, 00:13:36.800 --> 00:13:41.630 had better be there, a seven had better be here, 00:13:41.630 --> 00:13:43.410 a nine had better be there. 00:13:43.410 --> 00:13:47.460 There's a symmetric matrix. 00:13:47.460 --> 00:13:52.420 I happened to use all positive numbers as its entries. 00:13:52.420 --> 00:13:54.030 That's not the point. 00:13:54.030 --> 00:13:57.160 The point is that if I transpose that matrix, 00:13:57.160 --> 00:13:58.880 I get it back again. 00:13:58.880 --> 00:14:03.370 So symmetric matrices have this property A transpose equals A. 00:14:03.370 --> 00:14:07.150 I guess at this point -- 00:14:07.150 --> 00:14:14.130 I'm just asking you to notice this family of matrices that 00:14:14.130 --> 00:14:17.830 are unchanged by transposing. 00:14:17.830 --> 00:14:22.950 And they're easy to identify, of course. 00:14:22.950 --> 00:14:28.060 You know, it's not maybe so easy before we had a case where 00:14:28.060 --> 00:14:31.830 the transpose gave the inverse. 00:14:31.830 --> 00:14:36.230 That's highly important, but not so simple to see. 00:14:36.230 --> 00:14:39.200 This is the case where the transpose gives the same matrix 00:14:39.200 --> 00:14:39.720 back again. 00:14:39.720 --> 00:14:42.510 That's totally simple to see. 00:14:42.510 --> 00:14:43.380 Okay. 00:14:43.380 --> 00:14:49.410 Could actually -- maybe I could even say when would we get such 00:14:49.410 --> 00:14:51.150 a matrix? 00:14:51.150 --> 00:14:55.740 For example, this -- that matrix is absolutely far from 00:14:55.740 --> 00:14:57.850 symmetric, right? 00:14:57.850 --> 00:15:01.150 The transpose isn't even the same shape -- 00:15:01.150 --> 00:15:03.700 because it's rectangular, it turns the -- 00:15:03.700 --> 00:15:06.580 lies down on its side. 00:15:06.580 --> 00:15:10.970 But let me tell you a way to get a symmetric matrix out of 00:15:10.970 --> 00:15:11.880 this. 00:15:11.880 --> 00:15:14.640 Multiply those together. 00:15:14.640 --> 00:15:17.210 If I multiply this rectangular, shall I 00:15:17.210 --> 00:15:18.910 call it R for rectangular? 00:15:18.910 --> 00:15:22.890 So let that be R for rectangular matrix 00:15:22.890 --> 00:15:27.580 and let that be R transpose, which it is. 00:15:27.580 --> 00:15:32.000 Then I think that if I multiply those together, 00:15:32.000 --> 00:15:34.390 I get a symmetric matrix. 00:15:34.390 --> 00:15:36.780 Can I just do it with the numbers 00:15:36.780 --> 00:15:44.700 and then ask you why, how did I know it would be symmetric? 00:15:44.700 --> 00:15:52.415 So my point is that R transpose R is always symmetric. 00:15:52.415 --> 00:15:52.914 Okay? 00:15:57.560 --> 00:16:01.470 And I'm going to do it for that particular R transpose R which 00:16:01.470 --> 00:16:02.750 was -- 00:16:02.750 --> 00:16:05.815 let's see, the column was one two four three three one. 00:16:09.700 --> 00:16:11.700 I called that one R transpose, didn't I, 00:16:11.700 --> 00:16:16.030 and I called this guy one two four three three one. 00:16:16.030 --> 00:16:17.550 I called that R. 00:16:17.550 --> 00:16:19.970 Shall we just do that multiplication? 00:16:19.970 --> 00:16:22.960 Okay, so up here I'm getting a ten. 00:16:22.960 --> 00:16:27.310 Next to it I'm getting two, a nine, I'm getting an eleven. 00:16:27.310 --> 00:16:30.390 Next to that I'm getting four and three, a seven. 00:16:30.390 --> 00:16:32.650 Now what do I get there? 00:16:32.650 --> 00:16:37.650 This eleven came from one three times two three, right? 00:16:37.650 --> 00:16:39.720 Row one, column two. 00:16:39.720 --> 00:16:41.530 What goes here? 00:16:41.530 --> 00:16:43.360 Row two, column one. 00:16:43.360 --> 00:16:46.050 But no difference. 00:16:46.050 --> 00:16:50.220 One three two three or two three one three, same thing. 00:16:50.220 --> 00:16:51.750 It's going to be an eleven. 00:16:51.750 --> 00:16:54.310 That's the symmetry. 00:16:54.310 --> 00:16:56.230 I can continue to fill it out. 00:16:56.230 --> 00:16:58.100 What -- oh, let's get that seven. 00:16:58.100 --> 00:17:00.520 That seven will show up down here, too, 00:17:00.520 --> 00:17:02.650 and then four more numbers. 00:17:02.650 --> 00:17:06.619 That seven will show up here because one three times four 00:17:06.619 --> 00:17:10.630 one gave the seven, but also four one times one three 00:17:10.630 --> 00:17:11.599 will give that seven. 00:17:11.599 --> 00:17:16.030 Do you see that it works? 00:17:16.030 --> 00:17:24.630 Actually, do you want to see it work also in matrix language? 00:17:24.630 --> 00:17:27.089 I mean, that's quite convincing, right? 00:17:27.089 --> 00:17:29.965 That seven is no accident. 00:17:32.630 --> 00:17:35.550 The eleven is no accident. 00:17:35.550 --> 00:17:40.680 But just tell me how do I know if I transpose this guy -- 00:17:40.680 --> 00:17:42.820 How do I know it's symmetric? 00:17:42.820 --> 00:17:45.620 Well, I'm going to transpose it. 00:17:45.620 --> 00:17:49.610 And when I transpose it, I'm hoping 00:17:49.610 --> 00:17:52.210 I get the matrix back again. 00:17:52.210 --> 00:17:54.630 So can I transpose R transpose R? 00:17:54.630 --> 00:17:56.157 So just -- so, why? 00:17:59.880 --> 00:18:02.945 Well, my suggestion is take the transpose. 00:18:09.800 --> 00:18:11.670 That's the only way to show it's symmetric. 00:18:11.670 --> 00:18:14.340 Take the transpose and see that it didn't change. 00:18:14.340 --> 00:18:19.970 Okay, so I take the transpose of R transpose R. 00:18:19.970 --> 00:18:20.470 Okay. 00:18:20.470 --> 00:18:23.190 How do I do that? 00:18:23.190 --> 00:18:27.680 This is our little practice on the rules for transposes. 00:18:27.680 --> 00:18:33.780 So the rule for transposes is the order gets reversed. 00:18:33.780 --> 00:18:38.300 Just like inverses, which we did prove, 00:18:38.300 --> 00:18:44.390 same rule for transposes and -- which we'll now use. 00:18:44.390 --> 00:18:46.110 So the order gets reversed. 00:18:46.110 --> 00:18:51.600 It's the transpose of that that comes first, 00:18:51.600 --> 00:18:56.880 and the transpose of this that comes -- no. 00:18:56.880 --> 00:18:58.010 Is that -- yeah. 00:18:58.010 --> 00:19:01.130 That's what I have to write, right? 00:19:01.130 --> 00:19:03.500 This is a product of two matrices and I want its 00:19:03.500 --> 00:19:04.350 transpose. 00:19:04.350 --> 00:19:06.880 So I put the matrices in the opposite order 00:19:06.880 --> 00:19:08.480 and I transpose them. 00:19:08.480 --> 00:19:09.780 But what have I got here? 00:19:09.780 --> 00:19:12.320 What is R transpose transpose? 00:19:15.300 --> 00:19:18.670 Well, don't all speak at once. 00:19:18.670 --> 00:19:22.080 R transpose transpose, I flipped over the diagonal, 00:19:22.080 --> 00:19:28.540 I flipped over the diagonal again, so I've got R. 00:19:28.540 --> 00:19:32.770 And that's just my point, that if I started with this matrix, 00:19:32.770 --> 00:19:34.740 I transposed it, I got it back again. 00:19:37.850 --> 00:19:46.810 So that's the check, without using numbers, but with -- 00:19:46.810 --> 00:19:52.510 it checked in two lines that I always get symmetric matrices 00:19:52.510 --> 00:19:53.480 this way. 00:19:53.480 --> 00:19:55.680 And actually, that's where they come 00:19:55.680 --> 00:20:00.220 from in so many practical applications. 00:20:00.220 --> 00:20:02.730 Okay. 00:20:02.730 --> 00:20:07.580 So now I've said something today about permutations and about 00:20:07.580 --> 00:20:14.540 transposes and about symmetry and I'm ready 00:20:14.540 --> 00:20:17.290 for chapter three. 00:20:17.290 --> 00:20:21.550 Can we take a breath -- 00:20:21.550 --> 00:20:25.250 the tape won't take a breath, but the lecturer will, 00:20:25.250 --> 00:20:31.980 because to tell you about vector spaces is -- 00:20:31.980 --> 00:20:38.220 we really have to start now and think, okay, listen up. 00:20:38.220 --> 00:20:39.500 What are vector spaces? 00:20:47.400 --> 00:20:48.400 And what are sub-spaces? 00:20:51.160 --> 00:20:51.890 Okay. 00:20:51.890 --> 00:21:01.430 So, the point is, The main operations that we do -- 00:21:01.430 --> 00:21:04.450 what do we do with vectors? 00:21:04.450 --> 00:21:05.820 We add them. 00:21:05.820 --> 00:21:07.270 We know how to add two vectors. 00:21:09.880 --> 00:21:13.930 We multiply them by numbers, usually called scalers. 00:21:13.930 --> 00:21:17.350 If we have a vector, we know what three V is. 00:21:17.350 --> 00:21:24.030 If we have a vector V and W, we know what V plus W is. 00:21:24.030 --> 00:21:26.670 Those are the two operations that we've 00:21:26.670 --> 00:21:29.090 got to be able to do. 00:21:29.090 --> 00:21:33.220 To legitimately talk about a space of vectors, 00:21:33.220 --> 00:21:35.870 the requirement is that we should 00:21:35.870 --> 00:21:39.990 be able to add the things and multiply by numbers 00:21:39.990 --> 00:21:44.380 and that there should be some decent rules satisfied. 00:21:44.380 --> 00:21:45.370 Okay. 00:21:45.370 --> 00:21:48.590 So let me start with examples. 00:21:48.590 --> 00:21:50.755 So I'm talking now about vector spaces. 00:21:56.810 --> 00:21:58.685 And I'm going to start with examples. 00:22:06.690 --> 00:22:09.480 Let me say again what this word space is meaning. 00:22:09.480 --> 00:22:14.010 When I say that word space, that means to me 00:22:14.010 --> 00:22:19.440 that I've got a bunch of vectors, a space of vectors. 00:22:19.440 --> 00:22:21.010 But not just any bunch of vectors. 00:22:24.010 --> 00:22:28.720 It has to be a space of vectors -- 00:22:28.720 --> 00:22:31.300 has to allow me to do the operations that vectors 00:22:31.300 --> 00:22:32.740 are for. 00:22:32.740 --> 00:22:37.110 I have to be able to add vectors and multiply by numbers. 00:22:37.110 --> 00:22:39.560 I have to be able to take linear combinations. 00:22:39.560 --> 00:22:43.120 Well, where did we meet linear combinations? 00:22:43.120 --> 00:22:48.960 We met them back in, say in R^2. 00:22:48.960 --> 00:22:51.370 So there's a vector space. 00:22:51.370 --> 00:22:54.010 What's that vector space? 00:22:54.010 --> 00:22:59.270 So R two is telling me I'm talking about real numbers 00:22:59.270 --> 00:23:01.370 and I'm talking about two real numbers. 00:23:01.370 --> 00:23:11.470 So this is all two dimensional vectors -- 00:23:11.470 --> 00:23:16.580 real, such as -- 00:23:16.580 --> 00:23:18.750 well, I'm not going to be able to list them all. 00:23:18.750 --> 00:23:20.210 But let me put a few down. 00:23:20.210 --> 00:23:30.420 |3; 2|, |0;0|, |pi; e|. 00:23:30.420 --> 00:23:30.920 So on. 00:23:35.450 --> 00:23:39.890 And it's natural -- okay. 00:23:39.890 --> 00:23:44.270 Let's see, I guess I should do algebra first. 00:23:44.270 --> 00:23:46.980 Algebra means what can I do to these vectors? 00:23:46.980 --> 00:23:48.130 I can add them. 00:23:48.130 --> 00:23:50.520 I can add that to that. 00:23:50.520 --> 00:23:51.660 And how do I do it? 00:23:51.660 --> 00:23:54.460 A component at a time, of course. 00:23:54.460 --> 00:23:58.240 Three two added to zero zero gives me, three two. 00:23:58.240 --> 00:24:00.190 Sorry about that. 00:24:00.190 --> 00:24:05.780 Three two added to pi e gives me three plus pi, two plus e. 00:24:05.780 --> 00:24:07.260 Oh, you know what it does. 00:24:07.260 --> 00:24:11.240 And you know the picture that goes with it. 00:24:11.240 --> 00:24:14.830 There's the vector three two. 00:24:14.830 --> 00:24:19.520 And often, the picture has an arrow. 00:24:19.520 --> 00:24:22.550 The vector zero zero, which is a highly important vector -- 00:24:22.550 --> 00:24:24.610 it's got, like, the most important here 00:24:24.610 --> 00:24:25.930 -- is there. 00:24:25.930 --> 00:24:29.840 And of course there's not much of an arrow. 00:24:29.840 --> 00:24:35.110 Pi -- I'll have to remember -- pi is about three and a little 00:24:35.110 --> 00:24:37.570 more, e is about two and a little more. 00:24:37.570 --> 00:24:41.090 So maybe there's pi e. 00:24:41.090 --> 00:24:44.690 I never drew pi e before. 00:24:44.690 --> 00:24:47.030 It's just natural to -- 00:24:47.030 --> 00:24:55.560 this is the first component on the horizontal 00:24:55.560 --> 00:24:59.470 and this is the second component, 00:24:59.470 --> 00:25:02.010 going up the vertical. 00:25:02.010 --> 00:25:02.910 Okay. 00:25:02.910 --> 00:25:07.570 And the whole plane is R two. 00:25:07.570 --> 00:25:14.980 So R two is, we could say, the plane. 00:25:14.980 --> 00:25:17.710 The xy plane. 00:25:17.710 --> 00:25:18.920 That's what everybody thinks. 00:25:24.770 --> 00:25:32.800 But the point is it's a vector space because all those vectors 00:25:32.800 --> 00:25:34.140 are in there. 00:25:34.140 --> 00:25:37.380 If I removed one of them -- 00:25:37.380 --> 00:25:39.420 Suppose I removed zero zero. 00:25:39.420 --> 00:25:43.480 Suppose I tried to take the -- considered the X Y plane with 00:25:43.480 --> 00:25:46.360 a puncture, with a point removed. 00:25:46.360 --> 00:25:47.200 Like the origin. 00:25:47.200 --> 00:25:50.470 That would be, like, awful to take the origin away. 00:25:50.470 --> 00:25:52.570 Why is that? 00:25:52.570 --> 00:25:54.550 Why do I need the origin there? 00:25:54.550 --> 00:25:59.570 Because I have to be allowed -- if I had these other vectors, 00:25:59.570 --> 00:26:03.330 I have to be allowed to multiply three two -- 00:26:03.330 --> 00:26:05.610 this was three two -- 00:26:05.610 --> 00:26:09.820 by anything, by any scaler, including zero. 00:26:09.820 --> 00:26:12.020 I've got to be allowed to multiply by zero 00:26:12.020 --> 00:26:15.010 and the result's got to be there. 00:26:15.010 --> 00:26:18.110 I can't do without that point. 00:26:18.110 --> 00:26:23.670 And I have to be able to add three two to the opposite guy, 00:26:23.670 --> 00:26:26.800 minus three minus two. 00:26:26.800 --> 00:26:29.230 And if I add those I'm back to the origin again. 00:26:29.230 --> 00:26:31.360 No way I can do without the origin. 00:26:31.360 --> 00:26:36.280 Every vector space has got that zero vector in it. 00:26:36.280 --> 00:26:38.650 Okay, that's an easy vector space, 00:26:38.650 --> 00:26:42.520 because we have a natural picture of it. 00:26:42.520 --> 00:26:43.840 Okay. 00:26:43.840 --> 00:26:46.260 Similarly easy is R^3. 00:26:50.340 --> 00:26:54.630 This would be all -- let me go up a little here. 00:26:54.630 --> 00:26:57.820 This would be -- 00:26:57.820 --> 00:27:02.670 R three would be all three dimensional vectors -- 00:27:02.670 --> 00:27:09.645 or shall I say vectors with three real components. 00:27:14.320 --> 00:27:15.000 Okay. 00:27:15.000 --> 00:27:21.030 Let me just to be sure we're together, 00:27:21.030 --> 00:27:23.661 let me take the vector three two zero. 00:27:29.410 --> 00:27:33.390 Is that a vector in R^2 or R^3? 00:27:33.390 --> 00:27:38.490 Definitely it's in R^3. 00:27:38.490 --> 00:27:40.150 It's got three components. 00:27:40.150 --> 00:27:43.040 One of them happens to be zero, but that's a perfectly okay 00:27:43.040 --> 00:27:43.850 number. 00:27:43.850 --> 00:27:48.290 So that's a vector in R^3. 00:27:48.290 --> 00:27:51.590 We don't want to mix up the -- 00:27:51.590 --> 00:27:55.090 I mean, keep these vectors straight and keep R^n straight. 00:27:55.090 --> 00:27:57.630 So what's R^n? 00:27:57.630 --> 00:27:59.150 R^n. 00:27:59.150 --> 00:28:05.990 So this is our big example, is all vectors with n components. 00:28:05.990 --> 00:28:11.170 And I'm making these darn things column vectors. 00:28:11.170 --> 00:28:14.050 Can I try to follow that convention, 00:28:14.050 --> 00:28:17.530 that they'll be column vectors, and their components should 00:28:17.530 --> 00:28:20.690 be real numbers. 00:28:20.690 --> 00:28:24.830 Later we'll need complex numbers and complex vectors, 00:28:24.830 --> 00:28:26.721 but much later. 00:28:26.721 --> 00:28:27.220 Okay. 00:28:27.220 --> 00:28:28.780 So that's a vector space. 00:28:31.420 --> 00:28:33.670 Now, let's see. 00:28:33.670 --> 00:28:35.910 What do I have to tell you about vector spaces? 00:28:35.910 --> 00:28:44.090 I said the most important thing, which is that we can add any 00:28:44.090 --> 00:28:46.760 two of these and we -- still in R^2. 00:28:46.760 --> 00:28:50.220 We can multiply by any number and we're still in R^2. 00:28:50.220 --> 00:28:53.380 We can take any combination and we're still in R^2. 00:28:53.380 --> 00:28:55.290 And same goes for R^n. 00:28:55.290 --> 00:29:02.240 It's -- honesty requires me to mention that these operations 00:29:02.240 --> 00:29:08.300 of adding and multiplying have to obey a few rules. 00:29:08.300 --> 00:29:12.790 Like, we can't just arbitrarily say, okay, the sum of three two 00:29:12.790 --> 00:29:15.610 and pi e is zero zero. 00:29:15.610 --> 00:29:18.410 It's not. 00:29:18.410 --> 00:29:22.650 The sum of three two and minus three two is zero zero. 00:29:22.650 --> 00:29:27.030 So -- oh, I'm not going to -- the book, actually, 00:29:27.030 --> 00:29:32.420 lists the eight rules that the addition and multiplication 00:29:32.420 --> 00:29:34.680 have to satisfy, but they do. 00:29:34.680 --> 00:29:38.810 They certainly satisfy it in R^n and usually it's not those 00:29:38.810 --> 00:29:42.170 eight rules that are in doubt. 00:29:42.170 --> 00:29:50.070 What's -- the question is, can we do those additions and do we 00:29:50.070 --> 00:29:51.250 stay in the space? 00:29:51.250 --> 00:29:55.580 Let me show you a case where you can't. 00:29:55.580 --> 00:29:59.810 So suppose this is going to be not a vector space. 00:30:05.490 --> 00:30:08.780 Suppose I take the xy plane -- so there's R^2. 00:30:08.780 --> 00:30:11.240 That is a vector space. 00:30:11.240 --> 00:30:15.940 Now suppose I just take part of it. 00:30:15.940 --> 00:30:17.670 Just this. 00:30:17.670 --> 00:30:22.270 Just this one -- this is one quarter of the vector space. 00:30:24.910 --> 00:30:29.965 All the vectors with positive or at least not negative 00:30:29.965 --> 00:30:30.465 components. 00:30:33.070 --> 00:30:37.540 Can I add those safely? 00:30:37.540 --> 00:30:38.410 Yes. 00:30:38.410 --> 00:30:41.690 If I add a vector with, like, two -- 00:30:41.690 --> 00:30:45.030 three two to another vector like five six, 00:30:45.030 --> 00:30:48.950 I'm still up in this quarter, no problem with adding. 00:30:48.950 --> 00:30:54.860 But there's a heck of a problem with multiplying by scalers, 00:30:54.860 --> 00:30:58.690 because there's a lot of scalers that will take me out 00:30:58.690 --> 00:31:02.280 of this quarter plane, like negative ones. 00:31:02.280 --> 00:31:05.820 If I took three two and I multiplied by minus five, 00:31:05.820 --> 00:31:08.240 I'm way down here. 00:31:08.240 --> 00:31:12.220 So that's not a vector space, because it's not -- 00:31:12.220 --> 00:31:14.250 closed is the right word. 00:31:14.250 --> 00:31:17.870 It's not closed under multiplication 00:31:17.870 --> 00:31:19.850 by all real numbers. 00:31:22.500 --> 00:31:27.150 So a vector space has to be closed under multiplication 00:31:27.150 --> 00:31:29.010 and addition of vectors. 00:31:29.010 --> 00:31:31.680 In other words, linear combinations. 00:31:31.680 --> 00:31:37.560 It -- so, it means that if I give you a few vectors -- 00:31:37.560 --> 00:31:39.980 yeah look, here's an important -- here -- 00:31:39.980 --> 00:31:42.420 now we're getting to some really important vector spaces. 00:31:42.420 --> 00:31:47.460 Well, R^n -- like, they are the most important. 00:31:47.460 --> 00:31:52.520 But we will be interested in so- in vector spaces that are 00:31:52.520 --> 00:31:55.700 inside R^n. 00:31:55.700 --> 00:32:01.790 Vector spaces that follow the rules, but they -- 00:32:01.790 --> 00:32:10.140 we don't need all of -- see, there we started with R^2 here, 00:32:10.140 --> 00:32:15.060 and took part of it and messed it up. 00:32:15.060 --> 00:32:17.420 What we got was not a vector space. 00:32:17.420 --> 00:32:25.670 Now tell me a vector space that is part of R^2 and is still 00:32:25.670 --> 00:32:31.480 safely -- we can multiply, we can add and we stay in this 00:32:31.480 --> 00:32:32.880 smaller vector space. 00:32:32.880 --> 00:32:35.680 So it's going to be called a subspace. 00:32:35.680 --> 00:32:40.990 So I'm going to change this bad example to a good one. 00:32:40.990 --> 00:32:42.440 Okay. 00:32:42.440 --> 00:32:45.620 So I'm going to start again with R^2, 00:32:45.620 --> 00:32:50.120 but I'm going to take an example -- it is a vector space, 00:32:50.120 --> 00:32:53.805 so it'll be a vector space inside R^2. 00:32:56.970 --> 00:33:03.560 And we'll call that a subspace of R^2. 00:33:06.450 --> 00:33:07.040 Okay. 00:33:07.040 --> 00:33:09.010 What can I do? 00:33:09.010 --> 00:33:11.960 It's got something in it. 00:33:11.960 --> 00:33:14.730 Suppose it's got this vector in it. 00:33:14.730 --> 00:33:17.070 Okay. 00:33:17.070 --> 00:33:19.740 If that vector's in my little subspace 00:33:19.740 --> 00:33:23.500 and it's a true subspace, then there's 00:33:23.500 --> 00:33:24.990 got to be some more in it, 00:33:24.990 --> 00:33:25.640 right? 00:33:25.640 --> 00:33:28.900 I have to be able to multiply that by two, 00:33:28.900 --> 00:33:33.660 and that double vector has to be included. 00:33:33.660 --> 00:33:36.610 Have to be able to multiply by zero, that vector, 00:33:36.610 --> 00:33:39.420 or by half, or by three quarters. 00:33:39.420 --> 00:33:40.310 All these vectors. 00:33:40.310 --> 00:33:44.470 Or by minus a half, or by minus one. 00:33:44.470 --> 00:33:48.730 I have to be able to multiply by any number. 00:33:48.730 --> 00:33:52.250 So that is going to say that I have to have that whole line. 00:33:52.250 --> 00:33:56.410 Do you see that? 00:33:56.410 --> 00:33:58.440 Once I get a vector in there -- 00:33:58.440 --> 00:34:03.070 I've got the whole line of all multiples of that vector. 00:34:03.070 --> 00:34:09.320 I can't have a vector space without extending to get 00:34:09.320 --> 00:34:10.770 those multiples in there. 00:34:10.770 --> 00:34:12.399 Now I still have to check addition. 00:34:15.000 --> 00:34:16.179 But that comes out okay. 00:34:16.179 --> 00:34:20.560 This line is going to work, because I could add something 00:34:20.560 --> 00:34:23.219 on the line to something else on the line 00:34:23.219 --> 00:34:26.540 and I'm still on the line. 00:34:26.540 --> 00:34:28.469 So, example. 00:34:28.469 --> 00:34:33.340 So this is all examples of a subspace -- 00:34:33.340 --> 00:34:45.199 our example is a line in R^2 actually -- not just any line. 00:34:45.199 --> 00:34:50.239 If I took this line, would that -- 00:34:50.239 --> 00:34:51.960 so all the vectors on that line. 00:34:51.960 --> 00:34:56.929 So that vector and that vector and this vector and this vector 00:34:56.929 --> 00:34:58.380 -- 00:34:58.380 --> 00:35:05.450 in lighter type, I'm drawing something that doesn't work. 00:35:05.450 --> 00:35:07.510 It's not a subspace. 00:35:07.510 --> 00:35:09.890 The line in R^2 -- to be a subspace, 00:35:09.890 --> 00:35:15.220 the line in R^2 must go through the zero vector. 00:35:19.400 --> 00:35:21.700 Because -- why is this line no good? 00:35:21.700 --> 00:35:23.140 Let me do a dashed line. 00:35:27.500 --> 00:35:31.290 Because if I multiplied that vector on the dashed line 00:35:31.290 --> 00:35:34.490 by zero, then I'm down here, I'm not on the dashed line. 00:35:34.490 --> 00:35:36.620 Z- zero's got to be. 00:35:36.620 --> 00:35:39.920 Every subspace has got to contain zero -- 00:35:39.920 --> 00:35:43.230 because I must be allowed to multiply by zero and that will 00:35:43.230 --> 00:35:46.300 always give me the zero vector. 00:35:46.300 --> 00:35:48.020 Okay. 00:35:48.020 --> 00:35:51.410 Now, I was going to make -- 00:35:51.410 --> 00:35:54.460 create some subspaces. 00:35:54.460 --> 00:35:59.610 Oh, while I'm in R^2, why don't we think of all 00:35:59.610 --> 00:36:01.110 the possibilities. 00:36:01.110 --> 00:36:03.760 R two, there can't be that many. 00:36:03.760 --> 00:36:07.510 So what are the possible subspaces of R^2? 00:36:07.510 --> 00:36:08.720 Let me list them. 00:36:11.480 --> 00:36:16.220 So I'm listing now the subspaces of R^2. 00:36:19.660 --> 00:36:23.750 And one possibility that we always allow 00:36:23.750 --> 00:36:29.760 is all of R two, the whole thing, the whole space. 00:36:29.760 --> 00:36:34.010 That counts as a subspace of itself. 00:36:34.010 --> 00:36:35.830 You always want to allow that. 00:36:35.830 --> 00:36:39.750 Then the others are lines -- 00:36:39.750 --> 00:36:45.690 any line, meaning infinitely far in both directions 00:36:45.690 --> 00:36:49.810 through the zero. 00:36:55.110 --> 00:36:57.790 So that's like the whole space -- 00:36:57.790 --> 00:37:00.550 that's like whole two D space. 00:37:00.550 --> 00:37:02.860 This is like one dimension. 00:37:02.860 --> 00:37:05.320 Is this line the same as R^1 ? 00:37:05.320 --> 00:37:07.470 No. 00:37:07.470 --> 00:37:11.200 You could say it looks a lot like R^1. 00:37:11.200 --> 00:37:14.380 R^1 was just a line and this is a line. 00:37:14.380 --> 00:37:17.460 But this is a line inside R^2. 00:37:17.460 --> 00:37:20.440 The vectors here have two components. 00:37:20.440 --> 00:37:23.600 So that's not the same as R^1, because there the vectors only 00:37:23.600 --> 00:37:25.570 have one component. 00:37:25.570 --> 00:37:29.590 Very close, you could say, but not the same. 00:37:29.590 --> 00:37:30.320 Okay. 00:37:30.320 --> 00:37:32.250 And now there's a third possibility. 00:37:36.550 --> 00:37:40.940 There's a third subspace that's -- 00:37:40.940 --> 00:37:47.970 of R^2 that's not the whole thing, and it's not a line. 00:37:47.970 --> 00:37:50.170 It's even less. 00:37:50.170 --> 00:37:52.840 It's just the zero vector alone. 00:37:52.840 --> 00:37:55.170 The zero vector alone, only. 00:38:01.250 --> 00:38:05.550 I'll often call this subspace Z, just for zero. 00:38:05.550 --> 00:38:07.700 Here's a line, L. 00:38:07.700 --> 00:38:10.010 Here's a plane, all of R^2. 00:38:10.010 --> 00:38:14.680 So, do you see that the zero vector's okay? 00:38:14.680 --> 00:38:16.970 You would just -- to understand subspaces, 00:38:16.970 --> 00:38:20.820 we have to know the rules -- and knowing the rules means that we 00:38:20.820 --> 00:38:25.040 have to see that yes, the zero vector by itself, 00:38:25.040 --> 00:38:27.990 just this guy alone satisfies the rules. 00:38:27.990 --> 00:38:28.690 Why's that? 00:38:28.690 --> 00:38:31.320 Oh, it's too dumb to tell you. 00:38:31.320 --> 00:38:36.430 If I took that and added it to itself, I'm still there. 00:38:36.430 --> 00:38:40.320 If I took that and multiplied by seventeen, I'm still there. 00:38:40.320 --> 00:38:44.070 So I've done the operations, adding and multiplying 00:38:44.070 --> 00:38:47.010 by numbers, that are required, and I didn't go 00:38:47.010 --> 00:38:50.300 outside this one point space. 00:38:53.570 --> 00:38:57.170 So that's always -- that's the littlest subspace. 00:38:57.170 --> 00:39:00.930 And the largest subspace is the whole thing and in-between come 00:39:00.930 --> 00:39:02.370 all -- 00:39:02.370 --> 00:39:04.080 whatever's in between. 00:39:04.080 --> 00:39:04.580 Okay. 00:39:04.580 --> 00:39:07.610 So for example, what's in between for R^3? 00:39:07.610 --> 00:39:12.100 So if I'm in ordinary three dimensions, the subspace is R, 00:39:12.100 --> 00:39:18.250 all of R^3 at one extreme, the zero vector at the bottom. 00:39:18.250 --> 00:39:23.430 And then a plane, a plane through the origin. 00:39:23.430 --> 00:39:26.510 Or a line, a line through the origin. 00:39:26.510 --> 00:39:32.970 So with R^3, the subspaces were R^3, plane through the origin, 00:39:32.970 --> 00:39:37.560 line through the origin and a zero vector by itself, 00:39:37.560 --> 00:39:43.030 zero zero zero, just that single vector. 00:39:43.030 --> 00:39:44.360 Okay, you've got the idea. 00:39:47.470 --> 00:39:51.080 But, now comes -- 00:39:51.080 --> 00:39:53.350 the reality is -- 00:39:53.350 --> 00:39:57.530 what are these -- where do these subspaces come -- 00:39:57.530 --> 00:40:00.950 how do they come out of matrices? 00:40:00.950 --> 00:40:06.080 And I want to take this matrix -- 00:40:06.080 --> 00:40:08.350 oh, let me take that matrix. 00:40:08.350 --> 00:40:17.430 So I want to create some subspaces out of that matrix. 00:40:17.430 --> 00:40:26.980 Well, one subspace is from the columns. 00:40:26.980 --> 00:40:29.760 Okay. 00:40:29.760 --> 00:40:34.050 So this is the important subspace, 00:40:34.050 --> 00:40:38.190 the first important subspace that comes from that matrix -- 00:40:38.190 --> 00:40:40.750 I'm going to -- let me call it A again. 00:40:40.750 --> 00:40:44.370 Back to -- okay. 00:40:44.370 --> 00:40:48.150 I'm looking at the columns of A. 00:40:48.150 --> 00:40:50.530 Those are vectors in R^3. 00:40:50.530 --> 00:40:52.380 So the columns are in R^3. 00:40:52.380 --> 00:40:58.100 The columns are in R^3. 00:41:02.280 --> 00:41:04.585 So I want those columns to be in my subspace. 00:41:08.970 --> 00:41:11.960 Now I can't just put two columns in my subspace 00:41:11.960 --> 00:41:14.512 and call it a subspace. 00:41:14.512 --> 00:41:16.970 What do I have to throw in -- if I'm going to put those two 00:41:16.970 --> 00:41:21.460 columns in, what else has got to be there to have a subspace? 00:41:21.460 --> 00:41:25.050 I must be able to add those things. 00:41:25.050 --> 00:41:28.460 So the sum of those columns -- 00:41:28.460 --> 00:41:34.970 so these columns are in R^3, and I have to be able -- 00:41:34.970 --> 00:41:37.330 I'm, you know, I want that to be in my subspace, 00:41:37.330 --> 00:41:39.080 I want that to be in my subspace, 00:41:39.080 --> 00:41:42.880 but therefore I have to be able to multiply them by anything. 00:41:42.880 --> 00:41:45.910 Zero zero zero has got to be in my subspace. 00:41:45.910 --> 00:41:48.630 I have to be able to add them so that four five five 00:41:48.630 --> 00:41:50.150 is in the subspace. 00:41:50.150 --> 00:41:53.054 I've got to be able to add one of these plus three of these. 00:41:53.054 --> 00:41:54.470 That'll give me some other vector. 00:41:57.100 --> 00:42:02.180 I have to be able to take all the linear combinations. 00:42:02.180 --> 00:42:14.200 So these are columns in R^3 and all there linear combinations 00:42:14.200 --> 00:42:16.920 form a subspace. 00:42:21.260 --> 00:42:23.400 What do I mean by linear combinations? 00:42:23.400 --> 00:42:26.060 I mean multiply that by something, 00:42:26.060 --> 00:42:28.290 multiply that by something and add. 00:42:28.290 --> 00:42:33.350 The two operations of linear algebra, multiplying by numbers 00:42:33.350 --> 00:42:36.060 and adding vectors. 00:42:36.060 --> 00:42:38.930 And, if I include all the results, 00:42:38.930 --> 00:42:40.875 then I'm guaranteed to have a subspace. 00:42:43.570 --> 00:42:46.860 I've done the job. 00:42:46.860 --> 00:42:49.210 And we'll give it a name -- 00:42:49.210 --> 00:42:49.960 the column space. 00:42:53.740 --> 00:42:54.380 Column space. 00:43:01.220 --> 00:43:05.920 And maybe I'll call it C of A. 00:43:05.920 --> 00:43:07.120 C for column space. 00:43:11.580 --> 00:43:15.020 There's an idea there that -- 00:43:15.020 --> 00:43:22.750 Like, the central idea for today's lecture is -- 00:43:22.750 --> 00:43:25.220 got a few vectors. 00:43:25.220 --> 00:43:27.130 Not satisfied with a few vectors, 00:43:27.130 --> 00:43:29.800 we want a space of vectors. 00:43:29.800 --> 00:43:33.160 The vectors, they're in -- these vectors in -- are in R^3 , 00:43:33.160 --> 00:43:37.050 so our space of vectors will be vectors in R^3. 00:43:37.050 --> 00:43:40.940 The key idea's -- we have to be able to take 00:43:40.940 --> 00:43:42.400 their combinations. 00:43:42.400 --> 00:43:47.300 So tell me, geometrically, if I drew all these things -- 00:43:47.300 --> 00:43:50.060 like if I drew one two four, that would be somewhere maybe 00:43:50.060 --> 00:43:50.930 there. 00:43:50.930 --> 00:43:54.740 If I drew three three one, who knows, might be -- 00:43:54.740 --> 00:43:57.140 I don't know, I'll say there. 00:43:57.140 --> 00:44:01.690 There's column one, there's column two. 00:44:01.690 --> 00:44:06.700 What else -- what's in the whole column space? 00:44:06.700 --> 00:44:11.160 How do I draw the whole column space now? 00:44:11.160 --> 00:44:13.430 I take all combinations of those two vectors. 00:44:15.970 --> 00:44:18.220 Do I get -- well, I guess I actually listed 00:44:18.220 --> 00:44:19.160 the possibilities. 00:44:19.160 --> 00:44:21.940 Do I get the whole space? 00:44:21.940 --> 00:44:24.190 Do I get a plane? 00:44:24.190 --> 00:44:26.984 I get more than a line, that's for sure. 00:44:26.984 --> 00:44:28.900 And I certainly get more than the zero vector, 00:44:28.900 --> 00:44:31.610 but I do get the zero vector included. 00:44:31.610 --> 00:44:34.160 What do I get if I combine -- 00:44:34.160 --> 00:44:39.115 take all the combinations of two vectors in R^3 ? 00:44:44.040 --> 00:44:46.450 So I've got all this stuff on -- 00:44:46.450 --> 00:44:49.040 that whole line gets filled out, that whole line gets filled 00:44:49.040 --> 00:44:51.190 out, but all in-between gets filled out -- 00:44:51.190 --> 00:44:52.900 between the two lines because I -- 00:44:52.900 --> 00:44:56.610 I allowed to add something from one line, something 00:44:56.610 --> 00:44:57.850 from the other. 00:44:57.850 --> 00:44:58.810 You see what's coming? 00:44:58.810 --> 00:44:59.643 I'm getting a plane. 00:45:05.060 --> 00:45:06.790 That's my -- and it's through the origin. 00:45:10.210 --> 00:45:17.950 Those two vectors, namely one two four and three three one, 00:45:17.950 --> 00:45:20.590 when I take all their combinations, 00:45:20.590 --> 00:45:21.770 I fill out a whole plane. 00:45:21.770 --> 00:45:25.240 Please think about that. 00:45:25.240 --> 00:45:28.280 That's the picture you have to see. 00:45:28.280 --> 00:45:31.940 You sure have to see it in R^3 , because we're going to do it 00:45:31.940 --> 00:45:36.880 in R^10, and we may take a combination of five vectors 00:45:36.880 --> 00:45:40.740 in R^10, and what will we have? 00:45:40.740 --> 00:45:41.630 God knows. 00:45:41.630 --> 00:45:44.910 It's some subspace. 00:45:44.910 --> 00:45:46.880 We'll have five vectors. 00:45:46.880 --> 00:45:49.010 They'll all have ten components. 00:45:49.010 --> 00:45:52.320 We take their combinations. 00:45:52.320 --> 00:45:58.240 We don't have R^5 , because our vectors have ten components. 00:45:58.240 --> 00:46:05.020 And we possibly have, like, some five dimensional flat thing 00:46:05.020 --> 00:46:06.680 going through the origin for sure. 00:46:09.220 --> 00:46:12.110 Well, of course, if those five vectors were all on the line, 00:46:12.110 --> 00:46:13.710 then we would only get that line. 00:46:13.710 --> 00:46:16.840 So, you see, there are, like, other possibilities here. 00:46:16.840 --> 00:46:21.690 It depends what -- it depends on those five vectors. 00:46:21.690 --> 00:46:25.440 Just like if our two columns had been on the same line, 00:46:25.440 --> 00:46:28.640 then the column space would have been only a line. 00:46:28.640 --> 00:46:31.440 Here it was a plane. 00:46:31.440 --> 00:46:31.940 Okay. 00:46:35.700 --> 00:46:37.610 I'm going to stop at that point. 00:46:37.610 --> 00:46:44.220 That's the central idea of -- the great example of how 00:46:44.220 --> 00:46:48.960 to create a subspace from a matrix. 00:46:48.960 --> 00:46:51.990 Take its columns, take their combinations, 00:46:51.990 --> 00:46:57.360 all their linear combinations and you get the column space. 00:46:57.360 --> 00:47:01.060 And that's the central sort of -- 00:47:01.060 --> 00:47:04.320 we're looking at linear algebra at a higher level. 00:47:04.320 --> 00:47:07.600 When I look at A -- now, I want to look at Ax=b. 00:47:07.600 --> 00:47:10.090 That'll be the first thing in the next lecture. 00:47:13.650 --> 00:47:17.300 How do I understand Ax=b in this language -- 00:47:17.300 --> 00:47:22.580 in this new language of vector spaces and column spaces. 00:47:22.580 --> 00:47:24.830 And what are other subspaces? 00:47:24.830 --> 00:47:30.230 So the column space is a big one, there are others to come. 00:47:30.230 --> 00:47:32.270 Okay, thanks.