WEBVTT 00:00:09.840 --> 00:00:14.610 OK, this is linear algebra lecture nine. 00:00:14.610 --> 00:00:21.130 And this is a key lecture, this is where we get these ideas 00:00:21.130 --> 00:00:27.480 of linear independence, when a bunch of vectors are 00:00:27.480 --> 00:00:29.600 independent -- 00:00:29.600 --> 00:00:33.890 or dependent, that's the opposite. 00:00:33.890 --> 00:00:36.680 The space they span. 00:00:36.680 --> 00:00:40.550 A basis for a subspace or a basis for a vector 00:00:40.550 --> 00:00:42.730 space, that's a central idea. 00:00:42.730 --> 00:00:47.060 And then the dimension of that subspace. 00:00:47.060 --> 00:00:50.240 So this is the day that those words 00:00:50.240 --> 00:00:52.960 get assigned clear meanings. 00:00:52.960 --> 00:00:58.380 And emphasize that we talk about a bunch of vectors 00:00:58.380 --> 00:00:59.550 being independent. 00:00:59.550 --> 00:01:02.780 Wouldn't talk about a matrix being independent. 00:01:02.780 --> 00:01:05.150 A bunch of vectors being independent. 00:01:05.150 --> 00:01:07.840 A bunch of vectors spanning a space. 00:01:07.840 --> 00:01:11.480 A bunch of vectors being a basis. 00:01:11.480 --> 00:01:15.070 And the dimension is some number. 00:01:15.070 --> 00:01:17.480 OK, so what are the definitions? 00:01:17.480 --> 00:01:21.840 Can I begin with a fact, a highly important fact, 00:01:21.840 --> 00:01:28.240 that, I didn't call directly attention to earlier. 00:01:28.240 --> 00:01:35.900 Suppose I have a matrix and I look at Ax equals zero. 00:01:35.900 --> 00:01:39.530 Suppose the matrix has a lot of columns, 00:01:39.530 --> 00:01:43.340 so that n is bigger than m. 00:01:43.340 --> 00:01:46.320 So I'm looking at n equations -- 00:01:46.320 --> 00:01:48.920 I mean, sorry, m equations, a small number 00:01:48.920 --> 00:01:52.650 of equations m, and more unknowns. 00:01:52.650 --> 00:01:55.960 I have more unknowns than equations. 00:01:55.960 --> 00:01:57.040 Let me write that down. 00:01:57.040 --> 00:01:59.850 More unknowns than equations. 00:01:59.850 --> 00:02:03.170 More unknown x-s than equations. 00:02:03.170 --> 00:02:15.440 Then the conclusion is that there's 00:02:15.440 --> 00:02:19.330 something in the null space of A, other 00:02:19.330 --> 00:02:21.350 than just the zero vector. 00:02:21.350 --> 00:02:26.320 The conclusion is there are some non-zero x-s 00:02:26.320 --> 00:02:28.220 such that Ax is zero. 00:02:28.220 --> 00:02:30.030 There are some special solutions. 00:02:30.030 --> 00:02:31.060 And why? 00:02:31.060 --> 00:02:33.590 We know why. 00:02:33.590 --> 00:02:38.120 I mean, it sort of like seems like a reasonable thing, more 00:02:38.120 --> 00:02:42.150 unknowns than equations, then it seems reasonable 00:02:42.150 --> 00:02:43.540 that we can solve them. 00:02:43.540 --> 00:02:49.210 But we have a, a clear algorithm which starts with a system 00:02:49.210 --> 00:02:55.440 and does elimination, gets the thing into an echelon 00:02:55.440 --> 00:03:01.630 form with some pivots and pivot columns, 00:03:01.630 --> 00:03:06.050 and possibly some free columns that don't have pivots. 00:03:06.050 --> 00:03:09.290 And the point is here there will be some free columns. 00:03:09.290 --> 00:03:14.940 The reason, so the reason is there must -- 00:03:14.940 --> 00:03:20.080 there will be free variables, at least one. 00:03:25.010 --> 00:03:27.580 That's the reason. 00:03:27.580 --> 00:03:30.130 That we now have this -- 00:03:30.130 --> 00:03:36.200 a complete, algorithm, a complete systematic way to say, 00:03:36.200 --> 00:03:38.860 OK, we take the system Ax equals zero, 00:03:38.860 --> 00:03:42.370 we row reduce, we identify the free variables, 00:03:42.370 --> 00:03:48.690 and, since there are n variables and at most m pivots, 00:03:48.690 --> 00:03:51.840 there will be some free variables, at least one, 00:03:51.840 --> 00:03:55.090 at least n-m in fact, left over. 00:03:55.090 --> 00:04:00.480 And those variables I can assign non-zero values to. 00:04:00.480 --> 00:04:02.110 I don't have to set those to zero. 00:04:02.110 --> 00:04:04.580 I can take them to be one or whatever I like, 00:04:04.580 --> 00:04:07.970 and then I can solve for the pivot variables. 00:04:07.970 --> 00:04:11.870 So then it gives me a solution to Ax equals zero. 00:04:11.870 --> 00:04:15.820 And it's a solution that isn't all zeros. 00:04:15.820 --> 00:04:21.769 So, that's an important point that we'll 00:04:21.769 --> 00:04:26.350 use now in this lecture. 00:04:26.350 --> 00:04:30.110 So now I want to say what does it mean for a bunch of vectors 00:04:30.110 --> 00:04:31.470 to be independent. 00:04:31.470 --> 00:04:32.370 OK. 00:04:32.370 --> 00:04:36.276 So this is like the background that we know. 00:04:36.276 --> 00:04:37.900 Now I want to speak about independence. 00:04:41.500 --> 00:04:42.000 OK. 00:04:46.110 --> 00:04:48.740 Let's see. 00:04:48.740 --> 00:04:55.430 I can give you the abstract definition, and I will, 00:04:55.430 --> 00:05:00.325 but I would also like to give you the direct meaning. 00:05:03.700 --> 00:05:12.700 So the question is, when vectors x1, x2 up to -- 00:05:12.700 --> 00:05:19.860 Suppose I have n vectors are independent if. 00:05:19.860 --> 00:05:22.350 Now I have to give you -- 00:05:22.350 --> 00:05:25.560 or linearly independent -- 00:05:25.560 --> 00:05:30.470 I'll often just say and write independent for short. 00:05:30.470 --> 00:05:31.300 OK. 00:05:31.300 --> 00:05:33.580 I'll give you the full definition. 00:05:33.580 --> 00:05:36.920 These are just vectors in some vector space. 00:05:36.920 --> 00:05:39.360 I can take combinations of them. 00:05:39.360 --> 00:05:45.070 The question is, do any combinations give zero? 00:05:45.070 --> 00:05:47.210 If some combination of those vectors 00:05:47.210 --> 00:05:51.030 gives the zero vector, other than the combination 00:05:51.030 --> 00:05:55.520 of all zeros, then they're dependent. 00:05:55.520 --> 00:06:06.100 They're independent if no combination gives the zero 00:06:06.100 --> 00:06:08.550 vector -- 00:06:08.550 --> 00:06:17.070 and then I have, I'll have to put in an except the zero 00:06:17.070 --> 00:06:18.840 combination. 00:06:18.840 --> 00:06:21.160 So what do I mean by that? 00:06:21.160 --> 00:06:23.720 No combination gives the zero vector. 00:06:23.720 --> 00:06:29.950 Any combination c1 x1+c2 x2 plus, 00:06:29.950 --> 00:06:39.260 plus cn xn is not zero except for the zero combination. 00:06:39.260 --> 00:06:45.610 This is when all the c-s, all the c-s are zero. 00:06:45.610 --> 00:06:46.315 Then of course. 00:06:49.290 --> 00:06:52.100 That combination -- I know I'll get zero. 00:06:52.100 --> 00:06:55.950 But the question is, does any other combination give zero? 00:06:55.950 --> 00:06:59.970 If not, then the vectors are independent. 00:06:59.970 --> 00:07:02.630 If some other combination does give zero, 00:07:02.630 --> 00:07:04.880 the vectors are dependent. 00:07:04.880 --> 00:07:05.510 OK. 00:07:05.510 --> 00:07:08.420 Let's just take examples. 00:07:08.420 --> 00:07:14.510 Suppose I'm in, say, in two dimensional space. 00:07:14.510 --> 00:07:15.590 OK. 00:07:15.590 --> 00:07:16.940 I give you -- 00:07:16.940 --> 00:07:19.900 I'd like to first take an example -- 00:07:19.900 --> 00:07:24.480 let me take an example where I have a vector and twice that 00:07:24.480 --> 00:07:26.290 vector. 00:07:26.290 --> 00:07:29.720 So that's two vectors, V and 2V. 00:07:29.720 --> 00:07:33.830 Are those dependent or independent? 00:07:33.830 --> 00:07:35.820 Those are dependent for sure, right, 00:07:35.820 --> 00:07:40.380 because there's one vector is twice the other. 00:07:40.380 --> 00:07:42.250 One vector is twice as long as the other, 00:07:42.250 --> 00:07:44.780 so if the word dependent means anything, 00:07:44.780 --> 00:07:46.670 these should be dependent. 00:07:46.670 --> 00:07:47.910 And they are. 00:07:47.910 --> 00:07:51.860 And in fact, I would take two of the first -- 00:07:51.860 --> 00:07:56.750 so here's, here is a vector V and the other guy is a vector 00:07:56.750 --> 00:07:59.350 2V, that's my -- 00:07:59.350 --> 00:08:04.730 so there's a vector V1 and my next vector V2 is 2V1. 00:08:04.730 --> 00:08:08.630 Of course those are dependent, because two 00:08:08.630 --> 00:08:14.120 of these first vectors minus the second vector is zero. 00:08:14.120 --> 00:08:17.620 That's a combination of these two vectors that gives the zero 00:08:17.620 --> 00:08:18.120 vector. 00:08:18.120 --> 00:08:20.060 OK, that was clear. 00:08:20.060 --> 00:08:24.430 Suppose, suppose I have a vector -- 00:08:24.430 --> 00:08:25.060 here's another 00:08:25.060 --> 00:08:25.800 example. 00:08:25.800 --> 00:08:26.950 It's easy example. 00:08:26.950 --> 00:08:31.010 Suppose I have a vector and the other guy is the zero vector. 00:08:31.010 --> 00:08:36.820 Suppose I have a vector V1 and V2 is the zero vector. 00:08:36.820 --> 00:08:43.309 Then are those vectors dependent or independent? 00:08:43.309 --> 00:08:46.230 They're dependent again. 00:08:46.230 --> 00:08:50.480 You could say, well, this guy is zero times that one. 00:08:50.480 --> 00:08:52.970 This one is some combination of those. 00:08:52.970 --> 00:08:54.900 But let me write it the other way. 00:08:54.900 --> 00:09:01.110 Let me say -- what combination, how many V1s and how many V2s 00:09:01.110 --> 00:09:02.671 shall I take to get the zero 00:09:02.671 --> 00:09:03.170 vector? 00:09:05.810 --> 00:09:11.490 If, if V1 is like the vector two one and V2 is the zero vector, 00:09:11.490 --> 00:09:15.740 zero zero, then I would like to show 00:09:15.740 --> 00:09:18.640 that some combination of those gives the zero vector. 00:09:18.640 --> 00:09:19.820 What shall I take? 00:09:19.820 --> 00:09:22.660 How many V1s shall I take? 00:09:22.660 --> 00:09:23.540 Zero of them. 00:09:23.540 --> 00:09:25.310 Yeah, no, take no V1s. 00:09:25.310 --> 00:09:28.290 But how many V2s? 00:09:28.290 --> 00:09:28.970 Six. 00:09:28.970 --> 00:09:31.700 OK. 00:09:31.700 --> 00:09:34.790 Or five. 00:09:34.790 --> 00:09:38.590 Then -- in other words, the point is if the zero 00:09:38.590 --> 00:09:41.340 vector's in there, if the zero -- 00:09:41.340 --> 00:09:44.760 if one of these vectors is the zero vector, 00:09:44.760 --> 00:09:47.490 independence is dead, right? 00:09:47.490 --> 00:09:50.440 If one of those vectors is the zero vector then I could always 00:09:50.440 --> 00:09:51.970 take -- 00:09:51.970 --> 00:09:54.120 include that one and none of the others, 00:09:54.120 --> 00:09:59.370 and I would get the zero answer, and I would show dependence. 00:09:59.370 --> 00:10:00.230 OK. 00:10:00.230 --> 00:10:04.890 Now, let me, let me finally draw an example 00:10:04.890 --> 00:10:06.860 where they will be independent. 00:10:06.860 --> 00:10:10.900 Suppose that's V1 and that's V2. 00:10:10.900 --> 00:10:14.820 Those are surely independent, right? 00:10:14.820 --> 00:10:19.400 Any combination of V1 and V2, will not 00:10:19.400 --> 00:10:22.780 be zero except, the zero combination. 00:10:22.780 --> 00:10:24.390 So those would be independent. 00:10:24.390 --> 00:10:28.040 But now let me, let me stick in a third vector, V3. 00:10:31.100 --> 00:10:34.200 Independent or dependent now, those three vectors? 00:10:34.200 --> 00:10:37.390 So now n is three here. 00:10:37.390 --> 00:10:42.150 I'm in two dimensional space, whatever, I'm in the plane. 00:10:42.150 --> 00:10:47.390 I have three vectors that I didn't draw so carefully. 00:10:47.390 --> 00:10:50.520 I didn't even tell you what exactly they were. 00:10:50.520 --> 00:10:55.820 But what's this answer on dependent or independent? 00:10:55.820 --> 00:10:57.200 Dependent. 00:10:57.200 --> 00:11:00.570 How do I know those are dependent? 00:11:00.570 --> 00:11:05.570 How do I know that some combination of V1, V2, and V3 00:11:05.570 --> 00:11:09.010 gives me the zero vector? 00:11:09.010 --> 00:11:12.130 I know because of that. 00:11:12.130 --> 00:11:20.160 That's the key fact that tells me 00:11:20.160 --> 00:11:23.990 that three vectors in the plane have to be dependent. 00:11:23.990 --> 00:11:25.120 Why's that? 00:11:25.120 --> 00:11:28.650 What's the connection between the dependence of these three 00:11:28.650 --> 00:11:30.620 vectors and that fact? 00:11:30.620 --> 00:11:31.140 OK. 00:11:31.140 --> 00:11:33.350 So here's the connection. 00:11:33.350 --> 00:11:42.260 I take the matrix A that has V1 in its first column, V2 00:11:42.260 --> 00:11:46.560 in its second column, V3 in its third column. 00:11:46.560 --> 00:11:48.560 So it's got three columns. 00:11:48.560 --> 00:11:49.600 And V1 -- 00:11:49.600 --> 00:11:51.950 I don't know, that looks like about two one 00:11:51.950 --> 00:11:52.880 to me. 00:11:52.880 --> 00:11:56.430 V2 looks like it might be one two. 00:11:56.430 --> 00:12:01.990 V3 looks like it might be maybe two, maybe two and a half, 00:12:01.990 --> 00:12:03.570 minus one. 00:12:07.520 --> 00:12:08.230 OK. 00:12:08.230 --> 00:12:13.900 Those are my three vectors, and I put them in the columns of A. 00:12:13.900 --> 00:12:18.530 Now that matrix A is two by three. 00:12:18.530 --> 00:12:22.960 It fits this pattern, that where we know we've got extra 00:12:22.960 --> 00:12:25.900 variables, we know we have some free variables, 00:12:25.900 --> 00:12:28.640 we know that there's some combination -- 00:12:28.640 --> 00:12:34.650 and let me instead of x-s, let me call them c1, c2, and c3 -- 00:12:34.650 --> 00:12:39.380 that gives the zero vector. 00:12:39.380 --> 00:12:41.830 Sorry that my little bit of art got in the way. 00:12:41.830 --> 00:12:44.470 Do you see the point? 00:12:44.470 --> 00:12:48.600 When I have a matrix, I'm interested 00:12:48.600 --> 00:12:53.710 in whether its columns are dependent or independent. 00:12:53.710 --> 00:12:56.470 The columns are dependent if there 00:12:56.470 --> 00:12:58.480 is something in the null space. 00:12:58.480 --> 00:13:00.860 The columns are dependent because this, 00:13:00.860 --> 00:13:03.090 this thing in the null space says 00:13:03.090 --> 00:13:09.190 that c1 of that plus c2 of that plus c3 of this is zero. 00:13:09.190 --> 00:13:13.670 So in other words, I can go out some V1, out some more V2, 00:13:13.670 --> 00:13:16.260 back on V3, and end up zero. 00:13:19.010 --> 00:13:19.850 OK. 00:13:19.850 --> 00:13:25.290 So let -- here I've give the general, abstract definition, 00:13:25.290 --> 00:13:28.360 but let me repeat that definition -- 00:13:28.360 --> 00:13:31.840 this is like repeat -- 00:13:31.840 --> 00:13:39.270 let me call them Vs now. 00:13:39.270 --> 00:13:47.450 V1 up to Vn are the columns of a matrix A. 00:13:47.450 --> 00:13:50.120 In other words, this is telling me 00:13:50.120 --> 00:13:57.510 that if I'm in m dimensional space, 00:13:57.510 --> 00:14:00.630 like two dimensional space in the example, 00:14:00.630 --> 00:14:03.940 I can answer the dependence-independence 00:14:03.940 --> 00:14:10.180 question directly by putting those vectors 00:14:10.180 --> 00:14:11.580 in the columns of a matrix. 00:14:14.330 --> 00:14:30.380 They are independent if the null space of A, of A, is what? 00:14:33.230 --> 00:14:36.370 If I have a bunch of columns in a matrix, 00:14:36.370 --> 00:14:38.870 I'm looking at their combinations, 00:14:38.870 --> 00:14:44.850 but that's just A times the vector of c-s. 00:14:44.850 --> 00:14:48.020 And these columns will be independent 00:14:48.020 --> 00:14:56.390 if the null space of A is the zero vector. 00:15:01.270 --> 00:15:11.960 They are dependent if there's something else in there. 00:15:11.960 --> 00:15:17.380 If there's something else in the null space, if A times c 00:15:17.380 --> 00:15:25.240 gives the zero vector for some non-zero vector 00:15:25.240 --> 00:15:26.575 c in the null space. 00:15:29.160 --> 00:15:30.770 Then they're dependent, because that's 00:15:30.770 --> 00:15:34.380 telling me a combination of the columns gives the zero column. 00:15:34.380 --> 00:15:38.040 I think you're with be, because we've seen, 00:15:38.040 --> 00:15:40.340 like, lecture after lecture, we're 00:15:40.340 --> 00:15:43.500 looking at the combinations of the columns and asking, 00:15:43.500 --> 00:15:45.550 do we get zero or don't we? 00:15:45.550 --> 00:15:49.030 And now we're giving the official name, 00:15:49.030 --> 00:15:54.010 dependent if we do, independent if we don't. 00:15:54.010 --> 00:15:57.890 So I could express this in other words now. 00:15:57.890 --> 00:16:01.420 I could say the rank -- what's the rank in this independent 00:16:01.420 --> 00:16:02.450 case? 00:16:02.450 --> 00:16:05.020 The rank r of the, of the matrix, 00:16:05.020 --> 00:16:08.235 in the case of independent columns, is? 00:16:11.570 --> 00:16:14.740 So the columns are independent. 00:16:14.740 --> 00:16:18.560 So how many pivot columns have I got. 00:16:18.560 --> 00:16:19.850 All n. 00:16:19.850 --> 00:16:23.310 All the columns would be pivot columns, 00:16:23.310 --> 00:16:25.980 because free columns are telling me 00:16:25.980 --> 00:16:29.820 that they're a combination of earlier columns. 00:16:29.820 --> 00:16:33.080 So this would be the case where the rank is n. 00:16:33.080 --> 00:16:36.910 This would be the case where the rank is smaller than n. 00:16:39.650 --> 00:16:44.770 So in this case the rank is n and the null space of A 00:16:44.770 --> 00:16:48.750 is only the zero vector. 00:16:48.750 --> 00:16:50.800 And no free variables. 00:16:50.800 --> 00:16:52.400 No free variables. 00:16:56.520 --> 00:16:58.975 And this is the case yes free variables. 00:17:04.300 --> 00:17:09.589 If you'll allow me to stretch the English language that far. 00:17:09.589 --> 00:17:16.290 That's the case where we have, a combination 00:17:16.290 --> 00:17:19.040 that gives the zero column. 00:17:19.040 --> 00:17:23.089 I'm often interested in the case when my vectors are 00:17:23.089 --> 00:17:25.560 popped into a matrix. 00:17:25.560 --> 00:17:28.349 So the, the definition over there of independence 00:17:28.349 --> 00:17:31.270 didn't talk about any matrix. 00:17:31.270 --> 00:17:36.130 The vectors didn't have to be vectors in N dimensional space. 00:17:36.130 --> 00:17:38.530 And I want to give you some examples 00:17:38.530 --> 00:17:41.710 of vectors that aren't what you think 00:17:41.710 --> 00:17:43.970 of immediately as vectors. 00:17:43.970 --> 00:17:49.420 But most of the time, this is -- the vectors we think of are 00:17:49.420 --> 00:17:51.200 columns. 00:17:51.200 --> 00:17:54.570 And we can put them in a matrix. 00:17:54.570 --> 00:17:57.330 And then independence or dependence 00:17:57.330 --> 00:18:01.650 comes back to the null space. 00:18:01.650 --> 00:18:02.162 OK. 00:18:02.162 --> 00:18:03.620 So that's the idea of independence. 00:18:06.410 --> 00:18:13.770 Can I just, yeah, let me go on to spanning a 00:18:13.770 --> 00:18:19.480 What does it mean for a bunch of vectors to span a space? 00:18:19.480 --> 00:18:20.580 space. 00:18:20.580 --> 00:18:23.800 Well, actually, we've seen it already. 00:18:23.800 --> 00:18:28.040 You remember, if we had a columns in a matrix, 00:18:28.040 --> 00:18:33.130 we took all their combinations and that gave us 00:18:33.130 --> 00:18:36.640 the column space. 00:18:36.640 --> 00:18:41.250 Those vectors that we started with span that column space. 00:18:41.250 --> 00:18:44.490 So spanning a space means -- 00:18:44.490 --> 00:18:48.440 so let me move that important stuff right up. 00:18:53.130 --> 00:18:54.050 OK. 00:18:54.050 --> 00:19:02.400 So vectors -- let me call them, say, V1 up to -- 00:19:02.400 --> 00:19:06.360 call you some different letter, say Vl -- 00:19:06.360 --> 00:19:15.680 span a space, a subspace, or just a vector space 00:19:15.680 --> 00:19:24.790 I could say, span a space means, means 00:19:24.790 --> 00:19:40.270 the space consists of all combinations of those vectors. 00:19:46.464 --> 00:19:48.505 That's exactly what we did with the column space. 00:19:51.040 --> 00:19:54.740 So now I could say in shorthand the columns of a matrix 00:19:54.740 --> 00:19:57.090 span the column space. 00:19:57.090 --> 00:20:00.750 So you remember it's a bunch of vectors that have this property 00:20:00.750 --> 00:20:05.450 that they span a space, and actually if I give you a bunch 00:20:05.450 --> 00:20:06.480 of vectors and say -- 00:20:06.480 --> 00:20:10.720 OK, let S be the space that they span, 00:20:10.720 --> 00:20:14.870 in other words let S contain all their combinations, 00:20:14.870 --> 00:20:17.750 that space S will be the smallest 00:20:17.750 --> 00:20:21.830 space with those vectors in it, right? 00:20:21.830 --> 00:20:24.300 Because any space with those vectors in it 00:20:24.300 --> 00:20:28.730 must have all the combinations of those vectors in it. 00:20:28.730 --> 00:20:34.540 And if I stop there, then I've got the smallest space, 00:20:34.540 --> 00:20:37.300 and that's the space that they span. 00:20:37.300 --> 00:20:37.800 OK. 00:20:37.800 --> 00:20:39.150 So I'm just -- 00:20:39.150 --> 00:20:44.150 rather than, needing to say, take all linear combinations 00:20:44.150 --> 00:20:48.950 and put them in a space, I'm compressing that 00:20:48.950 --> 00:20:50.740 into the word span. 00:20:53.169 --> 00:20:53.835 Straightforward. 00:20:57.520 --> 00:21:00.015 So if I think of a, of the column space of a 00:21:00.015 --> 00:21:00.515 OK. matrix. 00:21:03.890 --> 00:21:07.380 I've got their -- so I start with the columns. 00:21:07.380 --> 00:21:08.930 I take all their combinations. 00:21:08.930 --> 00:21:10.760 That gives me the columns space. 00:21:10.760 --> 00:21:13.440 They span the column space. 00:21:13.440 --> 00:21:16.230 Now are they independent? 00:21:16.230 --> 00:21:19.620 Maybe yes, maybe no. 00:21:19.620 --> 00:21:23.060 It depends on the particular columns that went into that 00:21:23.060 --> 00:21:24.470 matrix. 00:21:24.470 --> 00:21:29.700 But obviously I'm highly interested in a set 00:21:29.700 --> 00:21:36.870 of vectors that spans a space and is independent. 00:21:36.870 --> 00:21:41.630 That's, that means like I've got the right number of vectors. 00:21:41.630 --> 00:21:47.580 If I didn't have all of them, I wouldn't have my whole space. 00:21:47.580 --> 00:21:50.290 If I had more than that, they probably wouldn't -- 00:21:50.290 --> 00:21:52.450 they wouldn't be independent. 00:21:52.450 --> 00:21:56.860 So, like, basis -- and that's the word that's coming -- 00:21:56.860 --> 00:21:58.600 is just right. 00:21:58.600 --> 00:22:01.310 So here let me put what that word means. 00:22:01.310 --> 00:22:14.270 A basis for a vector space is, is a, is a sequence of vectors 00:22:14.270 --> 00:22:14.770 -- 00:22:20.050 --> 00:22:27.040 shall I call them V1, V2, up to let me say Vd now, 00:22:27.040 --> 00:22:33.600 I'll stop with that letters -- that has two properties. 00:22:33.600 --> 00:22:36.230 I've got enough vectors and not too many. 00:22:36.230 --> 00:22:39.340 It's a natural idea of a basis. 00:22:39.340 --> 00:22:42.570 So a basis is a bunch of vectors in the space 00:22:42.570 --> 00:22:47.200 and it's a so it's a sequence of vectors with two properties, 00:22:47.200 --> 00:22:50.050 with two properties. 00:22:54.760 --> 00:22:57.675 One, they are independent. 00:23:05.020 --> 00:23:07.990 And two -- you know what's coming? 00:23:07.990 --> 00:23:10.040 -- they span the space. 00:23:20.560 --> 00:23:21.880 OK. 00:23:21.880 --> 00:23:25.050 Let me take -- 00:23:25.050 --> 00:23:28.510 so time for examples, of course. 00:23:28.510 --> 00:23:32.320 So I'm asking you now to put definition one, 00:23:32.320 --> 00:23:38.030 the definition of independence, together with definition two, 00:23:38.030 --> 00:23:41.800 and let's look at examples, because this is -- 00:23:41.800 --> 00:23:44.840 this combination means the set I've -- 00:23:44.840 --> 00:23:47.690 of vectors I have is just right, and the -- 00:23:47.690 --> 00:23:51.210 so that this idea of a basis will be central. 00:23:51.210 --> 00:23:54.480 I'll always be asking you now for a basis. 00:23:54.480 --> 00:23:58.760 Whenever I look at a subspace, if I ask you for -- 00:23:58.760 --> 00:24:00.760 if you give me a basis for that subspace, 00:24:00.760 --> 00:24:02.920 you've told me what it is. 00:24:02.920 --> 00:24:07.430 You've told me everything I need to know about that subspace. 00:24:07.430 --> 00:24:10.520 Those -- I take their combinations and I know that I 00:24:10.520 --> 00:24:12.460 need all the combinations. 00:24:12.460 --> 00:24:13.210 OK. 00:24:13.210 --> 00:24:13.790 Examples. 00:24:13.790 --> 00:24:16.640 OK, so examples of a basis. 00:24:16.640 --> 00:24:20.060 Let me start with two dimensional space. 00:24:20.060 --> 00:24:22.350 Suppose the space -- say example. 00:24:26.520 --> 00:24:31.150 The space is, oh, let's make it R^3. 00:24:31.150 --> 00:24:35.980 Real three dimensional space. 00:24:35.980 --> 00:24:37.330 Give me one basis. 00:24:37.330 --> 00:24:38.705 One basis is? 00:24:43.970 --> 00:24:49.010 So I want some vectors, because if I ask you for a basis, 00:24:49.010 --> 00:24:53.390 I'm asking you for vectors, a little list of vectors. 00:24:53.390 --> 00:24:57.610 And it should be just right. 00:24:57.610 --> 00:25:02.880 So what would be a basis for three dimensional space? 00:25:02.880 --> 00:25:05.540 Well, the first basis that comes to mind, why don't we 00:25:05.540 --> 00:25:07.240 write that down. 00:25:07.240 --> 00:25:09.220 The first basis that comes to mind 00:25:09.220 --> 00:25:18.850 is this vector, this vector, and this vector. 00:25:18.850 --> 00:25:19.790 OK. 00:25:19.790 --> 00:25:23.540 That's one basis. 00:25:23.540 --> 00:25:27.190 Not the only basis, that's going to be my point. 00:25:27.190 --> 00:25:30.930 But let's just see -- yes, that's a basis. 00:25:30.930 --> 00:25:33.500 Are, are those vectors independent? 00:25:36.110 --> 00:25:40.120 So that's the like the x, y, z axes, so if those are not 00:25:40.120 --> 00:25:41.780 independent, we're in trouble. 00:25:41.780 --> 00:25:43.580 Certainly, they are. 00:25:43.580 --> 00:25:48.630 Take a combination c1 of this vector plus c2 of this vector 00:25:48.630 --> 00:25:51.340 plus c3 of that vector and try to make 00:25:51.340 --> 00:25:55.290 it give the zero vector. 00:25:55.290 --> 00:25:57.110 What are the c-s? 00:25:57.110 --> 00:26:03.080 If c1 of that plus c2 of that plus c3 of that gives me 0 0 0, 00:26:03.080 --> 00:26:05.600 then the c-s are all -- 00:26:05.600 --> 00:26:06.920 0, right. 00:26:06.920 --> 00:26:09.980 So that's the test for independence. 00:26:09.980 --> 00:26:16.210 In the language of matrices, which was under that board, 00:26:16.210 --> 00:26:19.700 I could make those the columns of a matrix. 00:26:19.700 --> 00:26:22.880 Well, it would be the identity matrix. 00:26:22.880 --> 00:26:25.590 Then I would ask, what's the null space of the identity 00:26:25.590 --> 00:26:26.760 matrix? 00:26:26.760 --> 00:26:30.560 And you would say it's only the zero vector. 00:26:30.560 --> 00:26:34.740 And I would say, fine, then the columns are independent. 00:26:34.740 --> 00:26:38.370 The only thing -- the identity times a vector giving zero, 00:26:38.370 --> 00:26:40.810 the only vector that does that is zero. 00:26:40.810 --> 00:26:41.980 OK. 00:26:41.980 --> 00:26:45.710 Now that's not the only basis. 00:26:45.710 --> 00:26:46.660 Far from it. 00:26:46.660 --> 00:26:50.680 Tell me another basis, a second basis, another basis. 00:26:57.560 --> 00:27:03.030 So, give me -- well, I'll just start it out. 00:27:03.030 --> 00:27:04.090 One one two. 00:27:06.950 --> 00:27:10.780 Two two five. 00:27:10.780 --> 00:27:14.420 Suppose I stopped there. 00:27:14.420 --> 00:27:20.980 Has that little bunch of vectors got the properties that 00:27:20.980 --> 00:27:24.350 I'm asking for in a basis for R^3? 00:27:24.350 --> 00:27:26.880 We're looking for a basis for R^3. 00:27:26.880 --> 00:27:30.430 Are they independent, those two column vectors? 00:27:30.430 --> 00:27:31.330 Yes. 00:27:31.330 --> 00:27:33.610 Do they span R^3? 00:27:33.610 --> 00:27:34.340 No. 00:27:34.340 --> 00:27:36.440 Our feeling is no. 00:27:36.440 --> 00:27:37.550 Our feeling is no. 00:27:37.550 --> 00:27:41.640 Our feeling is that there're some vectors in R3 that 00:27:41.640 --> 00:27:44.120 are not combinations of those. 00:27:44.120 --> 00:27:44.930 OK. 00:27:44.930 --> 00:27:47.830 So suppose I add in -- 00:27:47.830 --> 00:27:50.390 I need another vector then, because these two 00:27:50.390 --> 00:27:52.021 don't span the space. 00:27:52.021 --> 00:27:52.520 OK. 00:27:52.520 --> 00:27:56.360 Now it would be foolish for me to put in three three seven, 00:27:56.360 --> 00:27:58.810 right, as the third vector. 00:27:58.810 --> 00:27:59.950 That would be a goof. 00:27:59.950 --> 00:28:03.380 Because that, if I put in three three seven, 00:28:03.380 --> 00:28:07.680 those vectors would be dependent, right? 00:28:07.680 --> 00:28:09.440 If I put in three three seven, it 00:28:09.440 --> 00:28:12.170 would be the sum of those two, it 00:28:12.170 --> 00:28:15.020 would lie in the same plane as those. 00:28:15.020 --> 00:28:18.590 It wouldn't be independent. 00:28:18.590 --> 00:28:21.930 My attempt to create a basis would be dead. 00:28:21.930 --> 00:28:26.420 But if I take -- so what vector can I take? 00:28:26.420 --> 00:28:30.340 I can take any vector that's not in that plane. 00:28:30.340 --> 00:28:31.430 Let me try -- 00:28:31.430 --> 00:28:33.590 I hope that 3 3 8 would do it. 00:28:37.340 --> 00:28:40.470 At least it's not the sum of those two vectors. 00:28:40.470 --> 00:28:44.160 But I believe that's a basis. 00:28:44.160 --> 00:28:49.310 And what's the test then, for that to be a basis? 00:28:49.310 --> 00:28:53.160 Because I just picked those numbers, and if I had picked, 00:28:53.160 --> 00:29:02.300 5 7 -14 how would we know do we have a basis or don't we? 00:29:02.300 --> 00:29:05.430 You would put them in the columns of a matrix, 00:29:05.430 --> 00:29:09.280 and you would do elimination, row reduction -- 00:29:09.280 --> 00:29:15.340 and you would see do you get any free variables 00:29:15.340 --> 00:29:18.500 or are all the columns pivot columns. 00:29:18.500 --> 00:29:20.590 Well now actually we have a square -- 00:29:20.590 --> 00:29:22.760 the matrix would be three by three. 00:29:22.760 --> 00:29:26.260 So, what's the test on the matrix then? 00:29:29.280 --> 00:29:34.790 The matrix -- so in this case, when my space is R^3 and I have 00:29:34.790 --> 00:29:44.740 three vectors, my matrix is square and what I asking about 00:29:44.740 --> 00:29:49.780 that matrix in order for those columns to be a basis? 00:29:49.780 --> 00:29:51.190 So in this -- 00:29:51.190 --> 00:30:06.380 for R^n, if I have -- n vectors give a basis if the n by n 00:30:06.380 --> 00:30:19.850 matrix with those columns, with those columns, is what? 00:30:19.850 --> 00:30:24.070 What's the requirement on that matrix? 00:30:24.070 --> 00:30:27.010 Invertible, right, right. 00:30:27.010 --> 00:30:28.590 The matrix should be invertible. 00:30:28.590 --> 00:30:33.187 For a square matrix, that's the, that's the perfect answer. 00:30:33.187 --> 00:30:33.770 Is invertible. 00:30:38.260 --> 00:30:43.380 So that's when, that's when the space is the whole space R^n. 00:30:46.140 --> 00:30:50.360 Let me, let me be sure you're with me here. 00:30:50.360 --> 00:30:53.540 Let me remove that. 00:30:53.540 --> 00:31:01.580 Are those two vectors a basis for any space at all? 00:31:01.580 --> 00:31:04.020 Is there a vector space that those really 00:31:04.020 --> 00:31:08.540 are a basis for, those, that pair of vectors, this guy 00:31:08.540 --> 00:31:11.580 and this 1, 1 1 2 and 2 2 5? 00:31:11.580 --> 00:31:15.360 Is there a space for which that's a basis? 00:31:15.360 --> 00:31:16.220 Sure. 00:31:16.220 --> 00:31:21.970 They're independent, so they satisfy the first requirement, 00:31:21.970 --> 00:31:24.740 so what space shall I take for them to be a basis 00:31:24.740 --> 00:31:25.240 of? 00:31:25.240 --> 00:31:29.080 What spaces will they be a basis for? 00:31:29.080 --> 00:31:31.180 The one they span. 00:31:31.180 --> 00:31:32.350 Their combinations. 00:31:32.350 --> 00:31:33.920 It's a plane, right? 00:31:33.920 --> 00:31:36.340 It'll be a plane inside R^3. 00:31:36.340 --> 00:31:40.930 So if I take this vector 1 1 2, say it goes there, 00:31:40.930 --> 00:31:44.240 and this vector 2 2 5, say it goes there, 00:31:44.240 --> 00:31:50.797 those are a basis for -- because they span a plane. 00:31:50.797 --> 00:31:52.880 And they're a basis for the plane, because they're 00:31:52.880 --> 00:31:53.550 independent. 00:31:53.550 --> 00:31:56.800 If I stick in some third guy, like 3 3 7, 00:31:56.800 --> 00:32:01.050 which is in the plane -- suppose I put in, try to put in 3 3 7, 00:32:01.050 --> 00:32:05.740 then the three vectors would still span the plane, 00:32:05.740 --> 00:32:09.180 but they wouldn't be a basis anymore because they're not 00:32:09.180 --> 00:32:12.260 independent anymore. 00:32:12.260 --> 00:32:19.420 So, we're looking at the question of -- 00:32:19.420 --> 00:32:21.360 again, 00:32:21.360 --> 00:32:25.390 OK. the case with independent columns 00:32:25.390 --> 00:32:32.090 is the case where the column vectors span the column space. 00:32:32.090 --> 00:32:35.830 They're independent, so they're a basis for the column space. 00:32:35.830 --> 00:32:36.520 OK. 00:32:36.520 --> 00:32:42.490 So now there's one bit of intuition. 00:32:42.490 --> 00:32:46.420 Let me go back to all of R^n. 00:32:46.420 --> 00:32:48.150 So I -- where I put 3 3 8. 00:32:51.110 --> 00:32:51.740 OK. 00:32:51.740 --> 00:32:56.510 The first message is that the basis is not unique, right. 00:32:56.510 --> 00:32:58.090 There's zillions of bases. 00:32:58.090 --> 00:33:02.790 I take any invertible three by three matrix, 00:33:02.790 --> 00:33:07.800 its columns are a basis for R^3. 00:33:07.800 --> 00:33:11.200 The column space is R^3, and if those, 00:33:11.200 --> 00:33:15.180 if that matrix is invertible, those columns are independent, 00:33:15.180 --> 00:33:17.100 I've got a basis for R^3. 00:33:17.100 --> 00:33:19.860 So there're many, many bases. 00:33:19.860 --> 00:33:27.730 But there is something in common for all those bases. 00:33:27.730 --> 00:33:32.810 There's something that this basis shares with that basis 00:33:32.810 --> 00:33:35.880 and every other basis for R^3. 00:33:35.880 --> 00:33:37.515 And what's that? 00:33:40.040 --> 00:33:47.410 Well, you saw it coming, because when I stopped here and asked 00:33:47.410 --> 00:33:51.180 if that was a basis for R^3, you said no. 00:33:51.180 --> 00:33:54.730 And I know that you said no because you knew there 00:33:54.730 --> 00:33:57.410 weren't enough vectors there. 00:33:57.410 --> 00:34:03.670 And the great fact is that there're many, many bases, 00:34:03.670 --> 00:34:12.139 but -- let me put in somebody else, just for variety. 00:34:12.139 --> 00:34:14.850 There are many, many bases, but they all 00:34:14.850 --> 00:34:18.639 have the same number of vectors. 00:34:18.639 --> 00:34:21.350 If we're talking about the space R^3, 00:34:21.350 --> 00:34:25.040 then that number of vectors is three. 00:34:25.040 --> 00:34:27.650 If we're talking about the space R^n, 00:34:27.650 --> 00:34:31.320 then that number of vectors is n. 00:34:31.320 --> 00:34:36.389 If we're talking about some other space, 00:34:36.389 --> 00:34:41.570 the column space of some matrix, or the null space of some 00:34:41.570 --> 00:34:45.420 matrix, or some other space that we haven't even thought 00:34:45.420 --> 00:34:52.820 of, then that still is true that every basis -- 00:34:52.820 --> 00:34:57.270 that there're lots of bases but every basis has the same number 00:34:57.270 --> 00:34:57.950 of vectors. 00:34:57.950 --> 00:35:02.000 Let me write that great fact down. 00:35:02.000 --> 00:35:07.940 Every basis -- we're given a space. 00:35:07.940 --> 00:35:10.680 Given a space. 00:35:13.950 --> 00:35:18.840 R^3 or R^n or some other column space of a matrix or the null 00:35:18.840 --> 00:35:21.790 space of a matrix or some other vector space. 00:35:21.790 --> 00:35:25.200 Then the great fact is that every basis 00:35:25.200 --> 00:35:40.365 for this, for the space has the same number of vectors. 00:35:47.750 --> 00:35:50.770 If one basis has six vectors, then every other basis 00:35:50.770 --> 00:35:52.770 has six vectors. 00:35:52.770 --> 00:35:56.050 So that number six is telling me like 00:35:56.050 --> 00:35:59.250 it's telling me how big is the space. 00:35:59.250 --> 00:36:01.640 It's telling me how many vectors do 00:36:01.640 --> 00:36:04.650 I have to have to have a basis. 00:36:04.650 --> 00:36:08.440 And of course we're seeing it this way. 00:36:08.440 --> 00:36:10.900 That number six, if we had seven vectors, 00:36:10.900 --> 00:36:13.150 then we've got too many. 00:36:13.150 --> 00:36:17.070 If we have five vectors we haven't got enough. 00:36:17.070 --> 00:36:21.860 Sixes are like just right for whatever space that is. 00:36:21.860 --> 00:36:24.600 And what do we call that number? 00:36:24.600 --> 00:36:29.550 That number is -- now I'm ready for the last definition today. 00:36:29.550 --> 00:36:33.650 It's the dimension of that space. 00:36:33.650 --> 00:36:37.410 So every basis for a space has the same number of vectors in 00:36:37.410 --> 00:36:37.910 it. 00:36:37.910 --> 00:36:41.860 Not the same vectors, all sorts of bases -- 00:36:41.860 --> 00:36:44.850 but the same number of vectors is always the same, 00:36:44.850 --> 00:36:47.010 and that number is the dimension. 00:36:47.010 --> 00:36:47.900 This is definitional. 00:36:50.780 --> 00:36:58.245 This number is the dimension of the space. 00:37:03.390 --> 00:37:03.890 OK. 00:37:06.670 --> 00:37:08.417 OK. 00:37:08.417 --> 00:37:09.375 Let's do some examples. 00:37:12.930 --> 00:37:14.410 Because now we've got definitions. 00:37:14.410 --> 00:37:17.550 Let me repeat the four things, the four words that 00:37:17.550 --> 00:37:19.270 have now got defined. 00:37:19.270 --> 00:37:23.320 Independence, that looks at combinations not 00:37:23.320 --> 00:37:24.560 being zero. 00:37:24.560 --> 00:37:27.700 Spanning, that looks at all the combinations. 00:37:27.700 --> 00:37:30.580 Basis, that's the one that combines 00:37:30.580 --> 00:37:32.360 independence and spanning. 00:37:32.360 --> 00:37:36.460 And now we've got the idea of the dimension of a space. 00:37:36.460 --> 00:37:40.470 It's the number of vectors in any basis, 00:37:40.470 --> 00:37:43.160 because all bases have the same number. 00:37:43.160 --> 00:37:44.580 OK. 00:37:44.580 --> 00:37:47.750 Let's take examples. 00:37:47.750 --> 00:37:53.790 Suppose I take, my space is -- examples now -- 00:37:53.790 --> 00:37:58.170 space is the, say, the column space of this matrix. 00:37:58.170 --> 00:38:00.240 Let me write down a matrix. 00:38:00.240 --> 00:38:06.370 1 1 1, 2 1 2, and I'll -- just to make it clear, 00:38:06.370 --> 00:38:12.430 I'll take the sum there, 3 2 3, and let me take the sum of all 00:38:12.430 --> 00:38:15.050 -- oh, let me put in one -- yeah, 00:38:15.050 --> 00:38:18.530 I'll put in one one one again. 00:38:18.530 --> 00:38:20.300 OK. 00:38:20.300 --> 00:38:21.260 So that's four vectors. 00:38:24.710 --> 00:38:29.310 OK, do they span the column space of that matrix? 00:38:29.310 --> 00:38:35.290 Let me repeat, do they span the column space of that matrix? 00:38:35.290 --> 00:38:37.700 By definition, that's what the column space -- 00:38:37.700 --> 00:38:39.280 Yes. where it comes from. 00:38:39.280 --> 00:38:41.470 Are they a basis for the column space? 00:38:41.470 --> 00:38:43.920 Are they independent? 00:38:43.920 --> 00:38:45.860 No, they're not independent. 00:38:45.860 --> 00:38:49.460 There's something in that null space. 00:38:49.460 --> 00:38:55.140 Maybe we can -- so let's look at the null space of the matrix. 00:38:55.140 --> 00:38:58.490 Tell me a vector that's in the null space of that matrix. 00:39:01.090 --> 00:39:05.320 So I'm looking for some vector that combines those columns 00:39:05.320 --> 00:39:08.350 and produces the zero column. 00:39:08.350 --> 00:39:10.680 Or in other words, I'm looking for solutions 00:39:10.680 --> 00:39:12.280 to A X equals zero. 00:39:12.280 --> 00:39:16.930 So tell me a vector in the null space. 00:39:16.930 --> 00:39:21.070 Maybe -- well, this was, this column was that one plus that 00:39:21.070 --> 00:39:25.030 one, so maybe if I have one of those and minus one of those 00:39:25.030 --> 00:39:26.560 that would be a vector in the null 00:39:26.560 --> 00:39:27.060 space. 00:39:29.450 --> 00:39:33.440 So, you've already told me now, are those vectors independent, 00:39:33.440 --> 00:39:38.140 the answer is -- those column vectors, the answer is -- 00:39:38.140 --> 00:39:38.670 no. 00:39:38.670 --> 00:39:39.170 Right? 00:39:39.170 --> 00:39:40.770 They're not independent. 00:39:40.770 --> 00:39:44.060 Because -- you knew they weren't independent. 00:39:44.060 --> 00:39:47.060 Anyway, minus one of this minus one 00:39:47.060 --> 00:39:50.530 of this plus one of this zero of that is the zero vector. 00:39:54.120 --> 00:39:55.660 OK, so they're not independent. 00:39:55.660 --> 00:39:55.720 OK. 00:39:55.720 --> 00:39:57.500 They span, but they're not independent. 00:39:57.500 --> 00:40:04.560 Tell me a basis for that column space. 00:40:04.560 --> 00:40:06.300 What's a basis for the column space? 00:40:06.300 --> 00:40:09.220 These are all the questions that the homework asks, the quizzes 00:40:09.220 --> 00:40:11.800 ask, the final exam will ask. 00:40:11.800 --> 00:40:17.710 Find a basis for the column space of this matrix. 00:40:17.710 --> 00:40:20.280 OK. 00:40:20.280 --> 00:40:22.500 Now there's many answers, but give me 00:40:22.500 --> 00:40:25.590 the most natural answer. 00:40:25.590 --> 00:40:29.670 Columns one and two. 00:40:29.670 --> 00:40:31.170 Columns one and two. 00:40:31.170 --> 00:40:32.410 That's the natural answer. 00:40:32.410 --> 00:40:35.320 Those are the pivot columns, because, I mean, 00:40:35.320 --> 00:40:37.140 we s- we begin systematically. 00:40:37.140 --> 00:40:39.300 We look at the first column, it's OK. 00:40:39.300 --> 00:40:41.330 We can put that in the basis. 00:40:41.330 --> 00:40:43.550 We look at the second column, it's OK. 00:40:43.550 --> 00:40:46.300 We can put that in the basis. 00:40:46.300 --> 00:40:48.960 The third column we can't put in the basis. 00:40:48.960 --> 00:40:54.070 The fourth column we can't, again. 00:40:54.070 --> 00:40:57.570 So the rank of the matrix is -- 00:40:57.570 --> 00:40:59.850 what's the rank of our matrix? 00:40:59.850 --> 00:41:01.210 Two. 00:41:01.210 --> 00:41:02.000 Two. 00:41:02.000 --> 00:41:06.610 And, and now that rank is also -- we also have another word. 00:41:06.610 --> 00:41:08.710 We, we have a great theorem here. 00:41:08.710 --> 00:41:23.060 The rank of A, that rank r, is the number of pivot columns 00:41:23.060 --> 00:41:25.340 and it's also -- 00:41:25.340 --> 00:41:27.890 well, so now please use my new word. 00:41:32.640 --> 00:41:34.700 This, it's the number two, of course, 00:41:34.700 --> 00:41:41.790 two is the rank of my matrix, it's 00:41:41.790 --> 00:41:44.440 the number of pivot columns, those pivot columns form 00:41:44.440 --> 00:41:49.104 a basis, of course, so what's two? 00:41:49.104 --> 00:41:49.895 It's the dimension. 00:41:52.820 --> 00:41:55.580 The rank of A, the number of pivot columns, 00:41:55.580 --> 00:42:02.555 is the dimension of the column space. 00:42:07.450 --> 00:42:08.840 Of course, you say. 00:42:08.840 --> 00:42:10.670 It had to be. 00:42:10.670 --> 00:42:12.000 Right. 00:42:12.000 --> 00:42:17.440 But just watch, look for one moment at the, 00:42:17.440 --> 00:42:20.240 the language, the way the English words 00:42:20.240 --> 00:42:21.810 get involved here. 00:42:21.810 --> 00:42:28.610 I take the rank of a matrix, the rank of a matrix. 00:42:28.610 --> 00:42:34.580 It's a number of columns and it's the dimension of -- 00:42:34.580 --> 00:42:37.810 not the dimension of the matrix, that's what I want to say. 00:42:37.810 --> 00:42:43.890 It's the dimension of a space, a subspace, the column space. 00:42:43.890 --> 00:42:46.800 Do you see, I don't take the dimension of A. 00:42:46.800 --> 00:42:49.600 That's not what I want. 00:42:49.600 --> 00:42:53.190 I'm looking for the dimension of the column space of A. 00:42:53.190 --> 00:42:56.050 If you use those words right, it shows you've got the idea 00:42:56.050 --> 00:42:57.400 right. 00:42:57.400 --> 00:42:59.170 Similarly here. 00:42:59.170 --> 00:43:03.430 I don't talk about the rank of a subspace. 00:43:03.430 --> 00:43:05.440 It's a matrix that has a rank. 00:43:05.440 --> 00:43:08.040 I talk about the rank of a matrix. 00:43:08.040 --> 00:43:11.750 And the beauty is that these definitions just 00:43:11.750 --> 00:43:14.160 merge so that the rank of a matrix 00:43:14.160 --> 00:43:16.170 is the dimension of its column space. 00:43:16.170 --> 00:43:18.840 And in this example it's two. 00:43:18.840 --> 00:43:22.570 And then the further question is, what's a basis? 00:43:22.570 --> 00:43:25.450 And the first two columns are a basis. 00:43:25.450 --> 00:43:27.500 Tell me another basis. 00:43:27.500 --> 00:43:29.710 Another basis for the columns space. 00:43:29.710 --> 00:43:31.390 You see I just keep hammering away. 00:43:31.390 --> 00:43:35.140 I apologize, but it's, I have to be sure you 00:43:35.140 --> 00:43:36.660 have the idea of basis. 00:43:36.660 --> 00:43:40.940 Tell me another basis for the column space. 00:43:40.940 --> 00:43:47.410 Well, you could take columns one and three. 00:43:47.410 --> 00:43:49.930 That would be a basis for the column space. 00:43:49.930 --> 00:43:53.220 Or columns two and three would be a basis. 00:43:53.220 --> 00:43:55.660 Or columns two and four. 00:43:55.660 --> 00:43:58.871 Or tell me another basis that's not made out of those columns 00:43:58.871 --> 00:43:59.370 at all? 00:44:03.290 --> 00:44:07.180 So -- I guess I'm giving you infinitely many possibilities, 00:44:07.180 --> 00:44:11.150 so I can't expect a unanimous answer here. 00:44:11.150 --> 00:44:14.770 I'll tell you -- but let's look at another basis, though. 00:44:14.770 --> 00:44:17.836 I'll just -- because it's only one out of zillions, 00:44:17.836 --> 00:44:19.960 I'm going to put it down and I'm going to erase it. 00:44:19.960 --> 00:44:26.680 Another basis for the column space would be -- 00:44:26.680 --> 00:44:27.879 let's see. 00:44:27.879 --> 00:44:29.670 I'll put in some things that are not there. 00:44:29.670 --> 00:44:33.830 Say, oh well, just to make it -- my life easy, 2 2 2. 00:44:33.830 --> 00:44:37.040 That's in the column space. 00:44:37.040 --> 00:44:40.380 And, that was sort of obvious. 00:44:40.380 --> 00:44:43.458 Let me take the sum of those, say 6 4 6. 00:44:46.810 --> 00:44:52.040 Or the sum of all of the columns, 7 5 7, why not. 00:44:52.040 --> 00:44:54.910 That's in the column space. 00:44:54.910 --> 00:44:59.540 Those are independent and I've got the number right, 00:44:59.540 --> 00:45:01.290 I've got two. 00:45:01.290 --> 00:45:02.940 Actually, this is a key point. 00:45:05.890 --> 00:45:09.170 If you know the dimension of the space you're working with, 00:45:09.170 --> 00:45:14.480 and we know that this column -- we know that the dimension, 00:45:14.480 --> 00:45:19.430 DIM, the dimension of the column space is two. 00:45:23.730 --> 00:45:30.120 If you know the dimension, then -- 00:45:30.120 --> 00:45:33.670 and we have a couple of vectors that are independent, 00:45:33.670 --> 00:45:35.710 they'll automatically be a 00:45:35.710 --> 00:45:36.550 basis. 00:45:36.550 --> 00:45:38.890 If we've got the number of vectors right, 00:45:38.890 --> 00:45:43.760 two vectors in this case, then if they're independent, 00:45:43.760 --> 00:45:47.037 they can't help but span the space. 00:45:47.037 --> 00:45:48.620 Because if they didn't span the space, 00:45:48.620 --> 00:45:52.130 there'd be a third guy to help span the space, 00:45:52.130 --> 00:45:54.160 but it couldn't be independent. 00:45:54.160 --> 00:45:58.190 So, it just has to be independent 00:45:58.190 --> 00:46:01.640 if we've got the numbers right. 00:46:01.640 --> 00:46:02.590 And they span. 00:46:02.590 --> 00:46:03.280 OK. 00:46:03.280 --> 00:46:04.100 Very good. 00:46:04.100 --> 00:46:06.060 So you got the dimension of a space. 00:46:06.060 --> 00:46:08.870 So this was another basis that I just invented. 00:46:08.870 --> 00:46:09.470 OK. 00:46:09.470 --> 00:46:16.270 Now, now I get to ask about the null space. 00:46:16.270 --> 00:46:18.320 What's the dimension of the null space? 00:46:18.320 --> 00:46:20.700 So we, we got a great fact there, 00:46:20.700 --> 00:46:28.560 the dimension of the column space is the rank. 00:46:28.560 --> 00:46:30.760 Now I want to ask you about the null space. 00:46:30.760 --> 00:46:34.190 That's the other part of the lecture, 00:46:34.190 --> 00:46:38.320 and it'll go on to the next lecture. 00:46:38.320 --> 00:46:40.320 OK. 00:46:40.320 --> 00:46:44.230 So we know the dimension of the column space is two, the rank. 00:46:44.230 --> 00:46:46.280 What about the null space? 00:46:46.280 --> 00:46:47.870 This is a vector in the null space. 00:46:47.870 --> 00:46:49.770 Are there other vectors in the null space? 00:46:52.280 --> 00:46:53.530 Yes or no? 00:46:53.530 --> 00:46:54.870 Yes. 00:46:54.870 --> 00:46:58.820 So this isn't a basis because it's doesn't span, right? 00:46:58.820 --> 00:47:01.970 There's more in the null space than we've got so far. 00:47:01.970 --> 00:47:04.570 I need another vector at least. 00:47:04.570 --> 00:47:08.790 So tell me another vector in the null space. 00:47:08.790 --> 00:47:12.290 Well, the natural choice, the choice you naturally think of 00:47:12.290 --> 00:47:16.140 is I'm going on to the fourth column, 00:47:16.140 --> 00:47:20.210 I'm letting that free variable be a one, 00:47:20.210 --> 00:47:23.750 and that free variable be a zero, and I'm asking 00:47:23.750 --> 00:47:26.130 is that fourth column a combination 00:47:26.130 --> 00:47:27.410 of my pivot columns? 00:47:27.410 --> 00:47:28.670 Yes, it is. 00:47:28.670 --> 00:47:31.650 And it's -- that will do. 00:47:35.730 --> 00:47:37.820 So what I've written there are actually the two 00:47:37.820 --> 00:47:39.860 special solutions, right? 00:47:39.860 --> 00:47:44.520 I took the two free variables, free and free. 00:47:44.520 --> 00:47:50.410 I gave them the values 1 0 or 0 I figured out the rest. 00:47:50.410 --> 00:47:53.640 So do you see, let me just say it in words. 00:47:53.640 --> 00:47:58.000 This vector, these vectors in the null space are telling me, 00:47:58.000 --> 00:48:00.190 they're telling me the combinations 00:48:00.190 --> 00:48:02.610 of the columns that give zero. 00:48:02.610 --> 00:48:08.770 They're telling me in what way the, the columns are dependent. 00:48:08.770 --> 00:48:10.770 That's what the null space is doing. 00:48:10.770 --> 00:48:13.890 Have I got enough now? 00:48:13.890 --> 00:48:15.780 And what's the null space now? 00:48:15.780 --> 00:48:18.540 We have to think about the null space. 00:48:18.540 --> 00:48:20.740 These are two vectors in the null space. 00:48:20.740 --> 00:48:21.790 They're independent. 00:48:21.790 --> 00:48:24.620 Are they a basis for the null space? 00:48:24.620 --> 00:48:27.650 What's the dimension of the null space? 00:48:27.650 --> 00:48:30.100 You see that those questions just keep coming up all the 00:48:30.100 --> 00:48:31.120 time. 00:48:31.120 --> 00:48:34.500 Are they a basis for the null space? 00:48:34.500 --> 00:48:36.940 You can tell me the answer even though we haven't 00:48:36.940 --> 00:48:39.460 written out a proof of that. 00:48:39.460 --> 00:48:40.190 Can you? 00:48:40.190 --> 00:48:41.050 Yes or no? 00:48:41.050 --> 00:48:44.510 Do these two special solutions form 00:48:44.510 --> 00:48:46.400 a basis for the null space? 00:48:46.400 --> 00:48:48.790 In other words, does the null space 00:48:48.790 --> 00:48:52.390 consist of all combinations of those two guys? 00:48:52.390 --> 00:48:54.110 Yes or no? 00:48:54.110 --> 00:48:55.180 Yes. 00:48:55.180 --> 00:48:56.880 Yes. 00:48:56.880 --> 00:48:58.970 The null space is two dimensional. 00:48:58.970 --> 00:49:01.500 The null space, the dimension of the null space, 00:49:01.500 --> 00:49:03.530 is the number of free variables. 00:49:03.530 --> 00:49:09.330 So the dimension of the null space 00:49:09.330 --> 00:49:12.065 is the number of free variables. 00:49:17.260 --> 00:49:21.420 And at the last second, give me the formula. 00:49:21.420 --> 00:49:24.490 This is then the key formula that we know. 00:49:24.490 --> 00:49:29.740 How many free variables are there in terms of R, the rank, 00:49:29.740 --> 00:49:35.020 m -- the number of rows, n, the number of columns? 00:49:35.020 --> 00:49:36.740 What do we get? 00:49:36.740 --> 00:49:42.360 We have n columns, r of them are pivot columns, 00:49:42.360 --> 00:49:48.200 so n-r is the number of free columns, free variables. 00:49:48.200 --> 00:49:51.950 And now it's the dimension of the null space. 00:49:51.950 --> 00:49:53.000 OK. 00:49:53.000 --> 00:49:53.950 That's great. 00:49:53.950 --> 00:49:57.280 That's the key spaces, their bases, and their dimensions. 00:49:57.280 --> 00:49:58.830 Thanks.