WEBVTT 00:00:09.510 --> 00:00:10.100 OK. 00:00:10.100 --> 00:00:16.500 Uh this is the review lecture for the first part 00:00:16.500 --> 00:00:21.880 of the course, the Ax=b part of the course. 00:00:21.880 --> 00:00:28.710 And the exam will emphasize chapter three. 00:00:28.710 --> 00:00:31.890 Because those are the --0 chapter 00:00:31.890 --> 00:00:34.940 three was about the rectangular matrices 00:00:34.940 --> 00:00:39.750 where we had null spaces and null spaces of A transpose, 00:00:39.750 --> 00:00:43.450 and ranks, and all the things that 00:00:43.450 --> 00:00:49.890 are so clear when the matrix is square and invertible, 00:00:49.890 --> 00:00:57.080 they became things to think about for rectangular matrices. 00:00:57.080 --> 00:01:01.590 So, and vector spaces and subspaces and above all 00:01:01.590 --> 00:01:04.400 those four subspaces. 00:01:04.400 --> 00:01:07.940 OK, I'm thinking to start at least I'll 00:01:07.940 --> 00:01:13.210 just look at old exams, read out questions, write 00:01:13.210 --> 00:01:15.320 on the board what I need to and we 00:01:15.320 --> 00:01:17.980 can see what the answers are. 00:01:17.980 --> 00:01:22.420 The first one I see is one I can just read out. 00:01:22.420 --> 00:01:23.930 Well, I'll write a little. 00:01:23.930 --> 00:01:32.145 Suppose u, v and w are nonzero vectors in R^7. 00:01:39.230 --> 00:01:44.190 What are the possible -- they span a -- a vector space. 00:01:44.190 --> 00:01:47.510 They span a subspace of R^7, and what are the possible 00:01:47.510 --> 00:01:49.130 dimensions? 00:01:49.130 --> 00:01:50.840 So that's a straightforward question, 00:01:50.840 --> 00:01:55.610 what are the possible dimensions of the subspace spanned 00:01:55.610 --> 00:01:58.040 by u, v and w? 00:01:58.040 --> 00:02:03.430 OK, one, two, or three, right. 00:02:03.430 --> 00:02:05.320 One, two or three. 00:02:05.320 --> 00:02:08.930 Couldn't be more because we've only got three vectors, 00:02:08.930 --> 00:02:12.430 and couldn't be zero because -- 00:02:15.580 --> 00:02:19.030 because I told you the vectors were nonzero. 00:02:19.030 --> 00:02:23.020 Otherwise if I allowed the possibility that those were all 00:02:23.020 --> 00:02:27.319 the zero vector -- then the zero-dimensional subspace would 00:02:27.319 --> 00:02:28.110 have been in there. 00:02:28.110 --> 00:02:29.250 OK. 00:02:29.250 --> 00:02:35.970 Now can I jump to a more serious question? 00:02:35.970 --> 00:02:37.600 OK. 00:02:37.600 --> 00:02:41.850 We have a five by three matrix. 00:02:41.850 --> 00:02:44.860 And I'm calling it U. 00:02:44.860 --> 00:02:47.010 I'm saying it's in echelon form. 00:02:47.010 --> 00:02:50.600 And it has three pivots, r=3. 00:02:50.600 --> 00:02:51.810 Three pivots. 00:02:57.560 --> 00:02:58.140 Ok. 00:02:58.140 --> 00:03:01.820 First question what's the null space? 00:03:01.820 --> 00:03:03.865 What's the null space of this matrix 00:03:03.865 --> 00:03:11.910 U, so this matrix is five by three, and I find it helpful to 00:03:11.910 --> 00:03:13.860 just see 00:03:13.860 --> 00:03:17.750 visually what five by three means, what that shape is. 00:03:17.750 --> 00:03:19.420 Three columns. 00:03:19.420 --> 00:03:20.810 Three columns in U, 00:03:20.810 --> 00:03:23.040 then five rows, 00:03:23.040 --> 00:03:24.430 three pivots, 00:03:24.430 --> 00:03:26.650 and what's the null space? 00:03:26.650 --> 00:03:29.360 The null space of U is -- 00:03:29.360 --> 00:03:33.000 and it asks for a spec-of course I'm 00:03:33.000 --> 00:03:35.390 looking for an answer that isn't just 00:03:35.390 --> 00:03:37.340 the definition of the null space, 00:03:37.340 --> 00:03:42.030 but is the null space of this matrix, with this information. 00:03:42.030 --> 00:03:44.220 And what is it? 00:03:44.220 --> 00:03:46.110 It's only the zero vector. 00:03:46.110 --> 00:03:48.810 Because we're told that the rank is three, 00:03:48.810 --> 00:03:53.840 so those three columns must be independent, no combination -- 00:03:53.840 --> 00:03:58.890 of those columns is the zero vector except -- 00:03:58.890 --> 00:04:01.671 so the only thing in this null space is the zero vector, 00:04:01.671 --> 00:04:02.170 and I -- 00:04:02.170 --> 00:04:05.030 I could even say what that vector is, zero, columns also? 00:04:05.030 --> 00:04:06.200 zero, zero. 00:04:06.200 --> 00:04:07.870 That's OK. 00:04:07.870 --> 00:04:10.020 So that's what's in the null space. 00:04:10.020 --> 00:04:15.015 All right? let me go on with -- this question has multiple 00:04:15.015 --> 00:04:15.515 parts. 00:04:18.089 --> 00:04:24.950 What's the -- oh now it asks you about a ten by three matrix, B, 00:04:24.950 --> 00:04:31.530 which is the matrix U and two U. 00:04:31.530 --> 00:04:37.340 It actually -- I would probably be writing R -- 00:04:37.340 --> 00:04:40.630 and maybe I should be writing R here now. 00:04:40.630 --> 00:04:47.800 This exam goes back a few years when I emphasized U more 00:04:47.800 --> 00:04:49.980 than R. 00:04:49.980 --> 00:04:54.630 Now, what's the echelon form for that matrix? 00:04:54.630 --> 00:04:57.150 So the echelon form, what's the rank, 00:04:57.150 --> 00:04:59.125 Yes. and what's the echelon form? 00:05:04.160 --> 00:05:06.270 Let's suppose this is in reduced echelon form, 00:05:06.270 --> 00:05:06.360 so that I could be using the letter R. 00:05:06.360 --> 00:05:06.480 So I'll ask for the reduced row echelon form so imagine that 00:05:06.480 --> 00:05:06.610 So I'm seeing the same length in b, three, and also in x. 00:05:06.610 --> 00:05:07.110 these are -- 00:05:07.110 --> 00:05:09.880 U is in reduced row echelon form but now 00:05:09.880 --> 00:05:15.500 I've doubled the height of the matrix, 00:05:15.500 --> 00:05:22.720 So the (x)-s that it multiplies have three components, 00:05:22.720 --> 00:05:29.150 what will happen when we do row reduction? 00:05:29.150 --> 00:05:32.950 What row reduction will take us to what matrix here? 00:05:39.170 --> 00:05:42.270 So you start doing elimination. 00:05:42.270 --> 00:05:44.590 You're doing elimination on single rows. 00:05:44.590 --> 00:05:48.050 But of course we're allowed to think of blocks. 00:05:48.050 --> 00:05:51.820 So what, well, what's the answer look like? 00:05:51.820 --> 00:05:53.921 U and z- or R -- 00:05:53.921 --> 00:05:56.170 let's -- I'll stay with this letter U but I'm what was 00:05:56.170 --> 00:05:56.660 the name of that --2 winning a million dollars or really 00:05:56.660 --> 00:06:00.030 thinking it's in reduced form, and zero. 00:06:00.030 --> 00:06:00.710 OK. 00:06:00.710 --> 00:06:01.810 Fine. 00:06:01.810 --> 00:06:10.390 Then it asks oh, further, it asks about this matrix. 00:06:10.390 --> 00:06:15.720 U, U, U, and zero. 00:06:15.720 --> 00:06:18.190 OK, what's the echelon form of this? 00:06:18.190 --> 00:06:20.050 So it's just like practice in thinking 00:06:20.050 --> 00:06:22.560 through what would row elimination, 00:06:22.560 --> 00:06:25.790 what would row reduction do. 00:06:25.790 --> 00:06:29.450 Have I thought this through, so what -- what are we -- 00:06:29.450 --> 00:06:31.920 if we start doing elimination, basically we're going 00:06:31.920 --> 00:06:34.030 to subtract these rows from these -- 00:06:34.030 --> 00:06:36.080 In the column space, because I do 00:06:36.080 --> 00:06:40.190 know that it can so it's going to take us to U, U, zero, 00:06:40.190 --> 00:06:44.090 and minus U, For example, I'm going to ask you for a I guess, 00:06:44.090 --> 00:06:44.590 right? 00:06:44.590 --> 00:06:47.240 But is it a- is it three by three? 00:06:47.240 --> 00:06:53.690 Take the thing all the way to R -- let's suppose U is really R. 00:06:53.690 --> 00:06:56.660 Suppose that we're really going for the reduced row echelon 00:06:56.660 --> 00:06:56.930 doing linear algebra of course -- 00:06:56.930 --> 00:06:58.390 but I didn't have to do the form. 00:06:58.390 --> 00:07:01.090 Then would we stop there? 00:07:01.090 --> 00:07:03.260 No. 00:07:03.260 --> 00:07:07.740 We would clean out, we would -- we could use this to -- 00:07:07.740 --> 00:07:10.770 is that right, can I so I took this row -- 00:07:10.770 --> 00:07:13.540 these rows away from these to get there. 00:07:13.540 --> 00:07:16.885 Now I take these rows away from these, so that gives me zero. 00:07:19.460 --> 00:07:25.030 There. there?1 And now what more would I do if I'm really 00:07:25.030 --> 00:07:29.460 shooting for R, the reduced row echelon form? 00:07:29.460 --> 00:07:33.200 I would -- then I want plus ones in the pivot, 00:07:33.200 --> 00:07:37.750 so I would multiply through by minus one to get plus there. 00:07:37.750 --> 00:07:42.170 So essentially I'm seeing reduced row echelon form 00:07:42.170 --> 00:07:46.180 there and there, and there's just one little twist 00:07:46.180 --> 00:07:47.040 still to go. 00:07:47.040 --> 00:07:51.324 Do you see what that final twist might be? 00:07:51.324 --> 00:07:52.490 It certainly has three rows. 00:07:52.490 --> 00:07:57.790 To have if -- if U is in reduced row echelon form and now 00:07:57.790 --> 00:08:02.490 I'm looking at U, U, there's one little step to take, 00:08:02.490 --> 00:08:05.090 this isn't like a big deal at all, but -- 00:08:05.090 --> 00:08:08.880 but if I really want this to be in reduced form, 00:08:08.880 --> 00:08:11.050 what would I still -- 00:08:11.050 --> 00:08:14.740 might I still have to do? 00:08:14.740 --> 00:08:17.750 I might have some zero rows here, 00:08:17.750 --> 00:08:20.720 I might have some zero rows here that strictly 00:08:20.720 --> 00:08:22.720 should move to the bottom. 00:08:22.720 --> 00:08:26.079 Well, I'm not going to make a project out of that. 00:08:26.079 --> 00:08:26.620 first of all? 00:08:26.620 --> 00:08:27.760 OK. 00:08:27.760 --> 00:08:29.445 What's the rank of that matrix? 00:08:32.510 --> 00:08:37.250 What's the rank of this matrix C? 00:08:37.250 --> 00:08:40.854 Given that I know that the original U has rank three, 00:08:40.854 --> 00:08:42.020 what's the rank of this guy? 00:08:42.020 --> 00:08:42.080 Six, right. 00:08:42.080 --> 00:08:42.289 That has rank six, I can tell. 00:08:42.289 --> 00:08:43.030 What was -- what's the rank of this B, while -- 00:08:43.030 --> 00:08:43.159 while we're at on TV for a few weeks, did you see that, 00:08:43.159 --> 00:08:43.659 it? 00:08:43.659 --> 00:08:56.690 The rank of B, is that six or three? 00:08:56.690 --> 00:08:58.200 So A is a three by three matrix. 00:08:58.200 --> 00:08:59.300 Three is right. 00:08:59.300 --> 00:08:59.940 Three is right. 00:08:59.940 --> 00:09:02.520 Because we actually got it to where we could just see three 00:09:02.520 --> 00:09:03.330 pivots. 00:09:03.330 --> 00:09:08.230 OK, and oh, now finally this easy one, 00:09:08.230 --> 00:09:15.020 what's the dimension be solved exactly when B is in the column 00:09:15.020 --> 00:09:18.410 space, of the null space -- 00:09:18.410 --> 00:09:22.370 of the null space of C transpose? 00:09:22.370 --> 00:09:22.980 Oh, boy. 00:09:22.980 --> 00:09:27.260 OK, so what do I -- if I want the dimension of a null space, 00:09:27.260 --> 00:09:29.240 I want to know the size of the matrix -- 00:09:29.240 --> 00:09:32.450 so what's the size of the matrix C? 00:09:32.450 --> 00:09:38.200 It looks like it's ten by six, is it? 00:09:38.200 --> 00:09:49.280 Ten by six, so C is ten by six, so m is ten, so C has ten rows, 00:09:49.280 --> 00:09:54.190 C transpose has ten columns, so there are ten columns there. 00:09:54.190 --> 00:09:57.950 So how many free variables have I got, once I -- 00:09:57.950 --> 00:10:00.950 There's quite a few trues, shall I take a poll, 00:10:00.950 --> 00:10:03.950 if I start with the ten columns in C transpose, 00:10:03.950 --> 00:10:06.050 that's the m for the original C. 00:10:06.050 --> 00:10:07.640 And what do I subtract off? 00:10:07.640 --> 00:10:08.140 Six. 00:10:08.140 --> 00:10:10.240 Because we said that was the rank. 00:10:10.240 --> 00:10:11.540 So I'm left with four. 00:10:11.540 --> 00:10:12.040 Thanks. 00:10:12.040 --> 00:10:12.540 OK. 00:10:12.540 --> 00:10:16.840 If b is in the -- and what would -- what does the exam say 00:10:16.840 --> 00:10:19.080 So I think that's the right answer -- 00:10:19.080 --> 00:10:23.580 the dimension of the null space of C transpose would be four. 00:10:23.580 --> 00:10:24.870 Right. 00:10:24.870 --> 00:10:25.870 OK. 00:10:25.870 --> 00:10:27.570 Yeah. 00:10:27.570 --> 00:10:28.650 OK. 00:10:28.650 --> 00:10:34.710 So that's one question, at least it brought in some -- 00:10:34.710 --> 00:10:37.130 some of the dimension counts. 00:10:37.130 --> 00:10:37.630 OK. 00:10:37.630 --> 00:10:40.088 Here's another type of question. think about, what's the -- 00:10:40.088 --> 00:10:44.520 what's the shape of the matrix, 00:10:44.520 --> 00:10:52.870 I give you an equation, Ax equals two four two. 00:10:52.870 --> 00:10:54.590 And I give you the complete solution. 00:10:54.590 --> 00:10:57.992 And what's its rank? 00:10:57.992 --> 00:10:59.325 But I don't give you the matrix. 00:11:05.390 --> 00:11:10.810 And another -- there's another vector, zero, zero, one. 00:11:10.810 --> 00:11:11.310 OK. 00:11:15.951 --> 00:11:16.450 All right. 00:11:16.450 --> 00:11:22.660 My first question is what's the dimension of the row space? 00:11:22.660 --> 00:11:24.160 Of the matrix A? 00:11:24.160 --> 00:11:31.890 So the main thing that you want to get from this question 00:11:31.890 --> 00:11:36.130 is that a question could start this way. 00:11:36.130 --> 00:11:38.770 exam, don't just tell me. 00:11:38.770 --> 00:11:40.890 Sort of backward way. 00:11:40.890 --> 00:11:45.130 so I guess I'm asking you what is 00:11:45.130 --> 00:11:50.420 the column space for this basis for the null space. 00:11:50.420 --> 00:11:56.770 By giving you the answer and not telling you what the problem 00:11:56.770 --> 00:12:02.590 homework, so I was watching it, so there were three -- 00:12:02.590 --> 00:12:03.650 the is. 00:12:03.650 --> 00:12:07.890 But we can get a lot of information, 00:12:07.890 --> 00:12:11.590 and sometimes we can get complete information 00:12:11.590 --> 00:12:13.710 about that matrix A. 00:12:13.710 --> 00:12:14.240 OK. 00:12:14.240 --> 00:12:19.530 So what's the dimension of the row space of A? 00:12:19.530 --> 00:12:21.120 What's the rank? 00:12:21.120 --> 00:12:27.470 Tell me about what's the size of the matrix, yeah, just -- 00:12:27.470 --> 00:12:32.230 These are the things we want to matrix b 00:12:32.230 --> 00:12:36.470 and I'll answer them without multiplying it out.2 00:12:36.470 --> 00:12:41.760 Its rank -- tell me something about its null space, 00:12:41.760 --> 00:12:46.000 I heard the right answer for the rank, 00:12:46.000 --> 00:12:49.700 the rank is one in this case. 00:12:49.700 --> 00:12:50.230 Why? 00:12:50.230 --> 00:12:53.940 Because the dimension of the null space, 00:12:53.940 --> 00:12:59.760 so the dimension of this instant?3 the null space of A 00:12:59.760 --> 00:13:05.050 is from knowing that that's the complete solution, it's two. 00:13:05.050 --> 00:13:08.760 I'm seeing two vectors here, and they're 00:13:08.760 --> 00:13:12.460 independent in the null space of A, 00:13:12.460 --> 00:13:16.170 because they have to be in the 00:13:16.170 --> 00:13:20.400 OK, which -- did you watch that quiz, 00:13:20.400 --> 00:13:26.750 there was a quiz program null space of A if I'm allowed 00:13:26.750 --> 00:13:31.650 to throw into the solution any amount of those vectors, 00:13:31.650 --> 00:13:35.620 that tells me that's the null space part then. 00:13:35.620 --> 00:13:38.400 So the dimension of the null space is two, and then I -- 00:13:38.400 --> 00:13:44.010 of course I know the dimensions of all the -- four subspaces. 00:13:44.010 --> 00:13:46.090 Now actually it asks what's the matrix? 00:13:46.090 --> 00:13:52.440 Well, what's the matrix in this case? 00:13:52.440 --> 00:14:01.210 Do we want to -- shall I try to figure that out? 00:14:01.210 --> 00:14:01.790 Sure. 00:14:01.790 --> 00:14:03.290 Let's -- you'd like me to do it, OK. 00:14:03.290 --> 00:14:04.164 Well, what's the ---- 00:14:04.164 --> 00:14:05.590 I actually say in this in the 00:14:05.590 --> 00:14:09.630 So what about the matrix, or let me at least start it, 00:14:09.630 --> 00:14:10.140 OK. 00:14:10.140 --> 00:14:14.720 If A times this x gives two, four, two, 00:14:14.720 --> 00:14:19.300 what does that tell me about the matrix A? 00:14:19.300 --> 00:14:26.210 If A times that x solves that equation then it tells me that 00:14:26.210 --> 00:14:28.900 the first column is -- 00:14:28.900 --> 00:14:35.560 the first column of A is -- one, two, one, right. 00:14:35.560 --> 00:14:37.580 The first column of A has to be one, two, one, 00:14:37.580 --> 00:14:37.650 because if I multiply by x, that's 00:14:37.650 --> 00:14:37.730 going to multiply just matrix? the first column, 00:14:37.730 --> 00:14:37.790 and give me two, four, two. 00:14:37.790 --> 00:14:37.900 And then I've got two more columns to find, 00:14:37.900 --> 00:14:38.000 and what information have I got to find them with? 00:14:38.000 --> 00:14:38.070 A basis for the null space. 00:14:38.070 --> 00:14:38.130 I've got the null space. 00:14:38.130 --> 00:14:38.230 So the fact that this is in the null space, 00:14:38.230 --> 00:14:38.370 what does that tell So what is if b has the form -- 00:14:38.370 --> 00:14:40.411 so I guess I'm asking what's me about the matrix? 00:14:40.411 --> 00:14:59.047 A matrix that has zero, zero, one in its null space? 00:15:05.510 --> 00:15:10.840 That tells me that the last column of the matrix is zeroes. 00:15:10.840 --> 00:15:11.880 so how many think true? 00:15:11.880 --> 00:15:13.820 Because this is in the null space, 00:15:13.820 --> 00:15:14.897 the last column has to be 00:15:14.897 --> 00:15:16.230 When could it be solved? zeroes. 00:15:16.230 --> 00:15:19.510 And because this is in the null space, what's 00:15:19.510 --> 00:15:23.850 the second column? 00:15:23.850 --> 00:15:25.880 Well, this in the null space means 00:15:25.880 --> 00:15:27.910 that if I multiply A by that vector 00:15:27.910 --> 00:15:32.530 I must be getting zeroes, so I think that better be minus one, 00:15:32.530 --> 00:15:44.370 minus two, and minus one. 00:15:44.370 --> 00:15:45.300 OK. 00:15:45.300 --> 00:15:50.340 That's a type of question that just brings out 00:15:50.340 --> 00:15:53.050 the information that's in that complete solution. 00:15:53.050 --> 00:15:53.180 And then actually I go on to ask what vectors -- for what 00:15:53.180 --> 00:15:53.250 OK. vectors B can Ax=b be solved? 00:15:53.250 --> 00:15:53.380 Ax=b can be solved if what -- so I'm looking for a condition 00:15:53.380 --> 00:15:53.460 Definitely not. on b, if any. interesting -- 00:15:53.460 --> 00:15:53.550 the novel point was there were three ways that 00:15:53.550 --> 00:15:53.640 Can it be solved for every right-hand side b? 00:15:53.640 --> 00:15:54.140 No. 00:15:54.140 --> 00:16:07.315 And the answer is -- 00:16:11.410 --> 00:16:25.505 yes or no -- 00:16:29.555 --> 00:16:30.054 right. 00:16:34.097 --> 00:16:34.596 solvable? 00:16:38.700 --> 00:16:44.920 the column space of this matrix, and what is it? 00:16:44.920 --> 00:16:50.810 It's so the column space of that matrix is all multiples b 00:16:50.810 --> 00:16:54.490 Can you tell me something -- 00:16:54.490 --> 00:16:59.170 so I'll ask questions about this -- 00:16:59.170 --> 00:17:04.609 b is a multiple of one, two, one. 00:17:04.609 --> 00:17:05.290 Right? 00:17:05.290 --> 00:17:14.140 I can solve the thing if it's a multiple of one, two, one, 00:17:14.140 --> 00:17:18.220 and of course sure enough -- 00:17:18.220 --> 00:17:24.140 yeah, that was a multiple of one, two, one, 00:17:24.140 --> 00:17:26.829 and so I had a solution. 00:17:26.829 --> 00:17:34.640 So this is a case where we've got lots of null space. 00:17:34.640 --> 00:17:44.760 Let me just recall rank is big, don't forget those cases, 00:17:44.760 --> 00:17:57.390 don't forget the other cases when r is as big as it can be, 00:17:57.390 --> 00:18:00.580 OK. r equal m or r equal n. 00:18:00.580 --> 00:18:04.970 Those are -- we had a full lecture on that, the full rank, 00:18:04.970 --> 00:18:09.480 full lecture, and important -- important case. 00:18:09.480 --> 00:18:13.471 I gave you every chance to think about that. 00:18:13.471 --> 00:18:13.970 OK. 00:18:13.970 --> 00:18:15.760 I'll just move on. 00:18:15.760 --> 00:18:19.800 I think this is the best type of review. 00:18:19.800 --> 00:18:23.380 So I'm going to solve Bx equals zero. 00:18:23.380 --> 00:18:26.080 It's just brings these ideas out. 00:18:26.080 --> 00:18:30.511 Apologies to the camera while I recover glasses and exam. 00:18:30.511 --> 00:18:31.010 OK. 00:18:31.010 --> 00:18:33.700 How about a few true-false ones? 00:18:33.700 --> 00:18:37.100 Actually there won't be a true-false on the quiz. 00:18:37.100 --> 00:18:40.980 But it gives us a moment of quick review. 00:18:40.980 --> 00:18:44.500 True or false, how do you feel about it at 00:18:44.500 --> 00:18:45.190 Here's one. 00:18:45.190 --> 00:18:46.930 If the null space -- 00:18:46.930 --> 00:18:49.400 I have a square matrix. 00:18:49.400 --> 00:18:52.740 If its null space is just the zero vector, 00:18:52.740 --> 00:18:57.750 what about the null space of A transpose? 00:18:57.750 --> 00:19:00.840 If the null space of A is just the zero vector, 00:19:00.840 --> 00:19:03.580 and the matrix is square, what do I 00:19:03.580 --> 00:19:06.430 know about the null space of A transpose? 00:19:06.430 --> 00:19:07.420 Also the zero vector. 00:19:07.420 --> 00:19:07.960 Good. 00:19:07.960 --> 00:19:10.270 And that's a very very important fact. 00:19:10.270 --> 00:19:11.630 OK. 00:19:11.630 --> 00:19:15.000 How about this? 00:19:15.000 --> 00:19:20.450 These -- look at the space of five by five matrices 00:19:20.450 --> 00:19:23.280 as a vector space. 00:19:23.280 --> 00:19:25.690 So it's actually a twenty-five-dimensional vector 00:19:25.690 --> 00:19:26.190 space. 00:19:26.190 --> 00:19:28.990 All five by five matrices. 00:19:28.990 --> 00:19:31.770 Look at the invertible matrices. 00:19:31.770 --> 00:19:35.430 Do they form a subspace? 00:19:35.430 --> 00:19:40.430 So I have this five by -- a space of all five by five 00:19:40.430 --> 00:19:41.580 matrices. 00:19:41.580 --> 00:19:44.180 I can add them, I can multiply by numbers. 00:19:44.180 --> 00:19:48.110 But now I narrow down to the invertible ones. 00:19:48.110 --> 00:19:50.700 And I ask are they a subspace? 00:19:50.700 --> 00:19:54.600 And you -- your answer is -- 00:19:54.600 --> 00:19:57.418 quiet, but nevertheless definite, no. 00:20:00.130 --> 00:20:00.800 Right? 00:20:00.800 --> 00:20:04.040 Because if I add two invertible matrices I have no idea if the 00:20:04.040 --> 00:20:06.570 No. answer is invertible. 00:20:06.570 --> 00:20:09.340 If I multiply that invertible -- well, 00:20:09.340 --> 00:20:11.600 it doesn't even have the zero matrix in it, 00:20:11.600 --> 00:20:13.630 it couldn't be a subspace. 00:20:13.630 --> 00:20:16.020 I have to be able to multiply by zero -- 00:20:16.020 --> 00:20:21.010 and stay in my subspace, and the invertible ones wouldn't work. 00:20:21.010 --> 00:20:24.210 Well, the singular ones wouldn't work either. 00:20:24.210 --> 00:20:27.540 They have zero -- the zero matrix is in the singular 00:20:27.540 --> 00:20:31.720 matrices, but if I add two singular matrices I don't know 00:20:31.720 --> 00:20:33.410 if the answer is singular or not. 00:20:33.410 --> 00:20:34.340 OK. 00:20:34.340 --> 00:20:36.420 So another true-false. 00:20:36.420 --> 00:20:36.510 If b squared equals zero then b equals zero. 00:20:36.510 --> 00:20:36.580 columns then the question is does Ax=b, 00:20:36.580 --> 00:20:37.904 is it always True or false? 00:20:37.904 --> 00:20:40.320 If b squared equals zero, true, false? you could get help, 00:20:40.320 --> 00:20:40.820 right -- 00:20:40.820 --> 00:20:54.590 b squared equals zero, b has to be a square -- square matrix, 00:20:54.590 --> 00:20:57.960 so that I can multiply it by itself, 00:20:57.960 --> 00:21:01.910 does that imply that B is zero? 00:21:06.450 --> 00:21:10.470 Are there matrices whose square could be the zero matrix? 00:21:14.720 --> 00:21:15.630 Yes or no? 00:21:15.630 --> 00:21:17.830 Yes there are. 00:21:17.830 --> 00:21:20.350 There are matrices whose square is the zero matrix. 00:21:20.350 --> 00:21:21.725 So this statement is false. 00:21:24.280 --> 00:21:26.980 If b squared is zero, we don't know that b is zero. 00:21:26.980 --> 00:21:31.290 For example -- the best example is that matrix. 00:21:31.290 --> 00:21:35.510 That matrix is a dangerous matrix. 00:21:35.510 --> 00:21:42.580 It will come up in later parts of this course as an example 00:21:42.580 --> 00:21:44.890 of what can go wrong. 00:21:44.890 --> 00:21:47.720 And here is a real simple -- so this -- 00:21:47.720 --> 00:21:52.010 so if I square that without doing the multiplication 00:21:52.010 --> 00:21:55.940 and finding the matrix b. matrix, 00:21:55.940 --> 00:22:04.770 I do get the zero matrix,and it shows -- 00:22:04.770 --> 00:22:05.270 OK. 00:22:05.270 --> 00:22:16.220 A system of m equations in m unknowns 00:22:16.220 --> 00:22:30.620 is solvable for every right-hand side 00:22:30.620 --> 00:22:36.490 if the columns are independent. 00:22:36.490 --> 00:22:39.870 So can I say that again, I'm -- 00:22:39.870 --> 00:22:41.750 I'll write it down then 00:22:41.750 --> 00:22:42.440 OK. 00:22:42.440 --> 00:22:43.820 for short. 00:22:43.820 --> 00:22:47.270 m by m matrix independent 00:22:47.270 --> 00:22:52.110 So give me a basis for -- 00:22:52.110 --> 00:22:56.250 for the null space of B. 00:22:56.250 --> 00:22:57.630 Let's see. 00:22:57.630 --> 00:23:03.150 but you could only use each way once, 00:23:03.150 --> 00:23:09.370 so you couldn't like use them all the time. 00:23:09.370 --> 00:23:11.440 So remember that? 00:23:11.440 --> 00:23:17.650 You could -- so you could poll the audience, 00:23:17.650 --> 00:23:21.800 and that was a very -- 00:23:21.800 --> 00:23:26.630 that was a hundred percent successful way, 00:23:26.630 --> 00:23:31.460 so I'll poll the audience on this. 00:23:31.460 --> 00:23:34.920 If the other possibility -- 00:23:34.920 --> 00:23:40.440 another possibility you could call your friend, right, 00:23:40.440 --> 00:23:48.030 or he's your friend until he gives you the wrong answer, 00:23:48.030 --> 00:23:49.410 which -- 00:23:49.410 --> 00:23:57.010 that turned out subspaces, then A is some multiple of B. 00:23:57.010 --> 00:24:01.840 to be very unreliable, you know, you'd 00:24:01.840 --> 00:24:05.990 call up your brother or something 00:24:05.990 --> 00:24:12.200 and ask him for the capital of whatever, Bosnia. 00:24:12.200 --> 00:24:15.940 What does he know, he makes some guess, 00:24:15.940 --> 00:24:29.560 So what -- I would just take extreme cases if it was me, 00:24:29.560 --> 00:24:32.602 matrix, independent columns, is Ax=b always solvable? 00:24:32.602 --> 00:24:33.810 Maybe just hands up for that? 00:24:33.810 --> 00:24:34.310 A few. 00:24:39.360 --> 00:24:42.420 And who says no? 00:24:42.420 --> 00:24:46.730 Oh, gosh, this audience is not reliable. 00:24:46.730 --> 00:24:47.230 Fifty fifty. 00:24:47.230 --> 00:24:51.070 I guess I'd say, I'd vote yes. 00:24:51.070 --> 00:24:54.120 Because independent columns, that 00:24:54.120 --> 00:24:57.190 means that the rank is the full size m, 00:24:57.190 --> 00:24:59.890 I have a matrix of rank m. 00:24:59.890 --> 00:25:01.530 That means it's -- 00:25:01.530 --> 00:25:04.710 I mean it's square, so it's an invertible matrix, 00:25:04.710 --> 00:25:07.330 and nothing could go wrong. 00:25:07.330 --> 00:25:08.090 Yes. 00:25:08.090 --> 00:25:12.070 So that's the good case and we always expect it in chapter 00:25:12.070 --> 00:25:18.130 two, but of course chapter three is -- 00:25:18.130 --> 00:25:24.990 only one of the possibilities. 00:25:24.990 --> 00:25:26.360 OK. 00:25:26.360 --> 00:25:41.460 Let me move on to another question from an old quiz. 00:25:41.460 --> 00:25:42.840 OK. 00:25:42.840 --> 00:25:44.210 OK. 00:25:44.210 --> 00:25:46.960 Let's see. 00:25:46.960 --> 00:26:07.550 I'm going to give you a matrix, but I'm going to give it to you 00:26:07.550 --> 00:26:08.920 OK. 00:26:08.920 --> 00:26:19.910 as a product of a couple of matrices, 00:26:19.910 --> 00:26:32.260 one, one, zero, zero, one, zero, one, zero, one, 00:26:32.260 --> 00:26:39.130 times another matrix, one, zero, 00:26:39.130 --> 00:26:52.860 I would like to ask you questions about that matrix 00:26:52.860 --> 00:27:07.960 minus one, two, zero, one, one, minus one, and all zeroes. 00:27:07.960 --> 00:27:09.330 OK. 00:27:09.330 --> 00:27:11.070 Let's see, what dimension I in -- 00:27:11.070 --> 00:27:17.210 the null space of B is a subspace of R. 00:27:17.210 --> 00:27:20.110 What size vectors I looking for here? 00:27:20.110 --> 00:27:24.124 Because if we don't know the size, 00:27:24.124 --> 00:27:26.040 we aren't going to find it, right? the null -- 00:27:26.040 --> 00:27:33.530 this matrix is three by four obviously. 00:27:33.530 --> 00:27:37.950 So if we're looking for the null space we're looking for those 00:27:37.950 --> 00:27:39.870 vectors x in R^4. 00:27:39.870 --> 00:27:40.830 OK. 00:27:40.830 --> 00:27:45.490 So the null space of B is certainly a subspace of R^4. 00:27:45.490 --> 00:27:48.490 What do you think its dimension is? 00:27:48.490 --> 00:27:51.930 Of course once we find the basis we 00:27:51.930 --> 00:27:55.010 would know the dimension immediately, but let's 00:27:55.010 --> 00:28:00.590 stop first, what's the rank of this matrix B? 00:28:00.590 --> 00:28:05.010 Let's see, what -- is that matrix invertible, 00:28:05.010 --> 00:28:06.306 that square one there? 00:28:10.890 --> 00:28:15.110 Let's say sure, I think it is, yes, 00:28:15.110 --> 00:28:17.330 that matrix B looks invertible. 00:28:17.330 --> 00:28:21.230 Is that pretty clear? 00:28:21.230 --> 00:28:23.090 Yeah. 00:28:23.090 --> 00:28:24.180 Yeah. 00:28:24.180 --> 00:28:28.450 So I've gone wrong in this course already, 00:28:28.450 --> 00:28:33.210 but I'll still hope that that matrix is invertible. 00:28:33.210 --> 00:28:36.650 Yeah, yeah, because if I look for a combination of those 00:28:36.650 --> 00:28:41.600 three columns -- well, I couldn't use this middle column 00:28:41.600 --> 00:28:46.890 because it would have a one and in a position that I -- 00:28:46.890 --> 00:28:51.370 column is otherwise all zero, so a combination that gives zero 00:28:51.370 --> 00:28:56.180 can't give us that problem, and then the other two are clearly 00:28:56.180 --> 00:29:00.090 independent sets -- so that matrix is invertible. 00:29:00.090 --> 00:29:03.600 Later we could take a determinant or other things. 00:29:03.600 --> 00:29:04.100 OK. 00:29:04.100 --> 00:29:06.500 What's the setup? 00:29:06.500 --> 00:29:11.850 If I have an invertible matrix, a nice invertible square 00:29:11.850 --> 00:29:16.620 matrix, times this guy, times this second factor, 00:29:16.620 --> 00:29:18.840 and I'm looking for the null space, 00:29:18.840 --> 00:29:20.305 does this have any effect? 00:29:22.960 --> 00:29:26.490 Is the null -- so what I'm asking is is the null space 00:29:26.490 --> 00:29:33.500 of B the same as the null space of just this part? 00:29:33.500 --> 00:29:35.950 I think so. 00:29:35.950 --> 00:29:36.820 I think so. 00:29:36.820 --> 00:29:41.470 If Bx is zero, then multiplying by that guy 00:29:41.470 --> 00:29:42.870 I'll still have zero. 00:29:42.870 --> 00:29:47.970 But also if this times some x give zero, 00:29:47.970 --> 00:29:51.550 I could always multiply on the left by the inverse of that, 00:29:51.550 --> 00:29:55.100 because it is invertible, and I would discover 00:29:55.100 --> 00:29:59.040 that this kind of Bx is zero. 00:29:59.040 --> 00:30:02.700 You want me to write some of that down -- 00:30:02.700 --> 00:30:06.730 if I have a product here, C times -- 00:30:06.730 --> 00:30:14.040 times D, say, and if C is invertible, 00:30:14.040 --> 00:30:18.660 the null space of CD, well, it will the same 00:30:18.660 --> 00:30:20.880 as the null space of D. 00:30:20.880 --> 00:30:23.770 If C is invertible. 00:30:26.600 --> 00:30:29.570 Multiplying by an invertible matrix on the left 00:30:29.570 --> 00:30:32.520 can't change the null space. 00:30:32.520 --> 00:30:33.050 OK. 00:30:33.050 --> 00:30:35.550 So basically I'm asking you for the null space of this. 00:30:35.550 --> 00:30:38.920 so when do I know -- well, I would say suppose the matrix 00:30:38.920 --> 00:30:40.540 I don't have to do the multiplication 00:30:40.540 --> 00:30:43.210 because I have C is invertible. 00:30:43.210 --> 00:30:46.220 That first factor C is invertible. 00:30:46.220 --> 00:30:48.550 It's not going to change the null space. 00:30:48.550 --> 00:30:49.320 OK. 00:30:49.320 --> 00:30:55.430 So can we just write down a basis now for the null space? 00:30:55.430 --> 00:30:59.060 So what's the basis for the null space of -- of that? 00:30:59.060 --> 00:31:06.080 So basis for the null space I'm looking for the two -- 00:31:06.080 --> 00:31:08.330 there are two pivot columns obviously. 00:31:08.330 --> 00:31:10.200 It clearly has rank two. 00:31:10.200 --> 00:31:15.450 If -- so true or false, if A and B have the same four 00:31:15.450 --> 00:31:18.070 I'm looking for the two special solutions. 00:31:18.070 --> 00:31:21.070 They'll come from the third and the fourth. 00:31:21.070 --> 00:31:22.200 The free variables. 00:31:22.200 --> 00:31:25.940 OK. so if the third free variable is a one, 00:31:25.940 --> 00:31:29.690 then I think probably I need a minus one there 00:31:29.690 --> 00:31:32.320 and a one there, it looks like. 00:31:32.320 --> 00:31:36.690 Do you agree that if I then do that multiplication I'll get 00:31:36.690 --> 00:31:37.190 zero? 00:31:37.190 --> 00:31:40.940 And if I have one in the fourth variable, then 00:31:40.940 --> 00:31:46.190 maybe I need a one in the second variable and maybe a minus two 00:31:46.190 --> 00:31:47.970 in the third. 00:31:47.970 --> 00:31:51.310 So I just reasoned that through and then if I look back I see 00:31:51.310 --> 00:31:55.450 sure enough that the free variable part that I sometimes 00:31:55.450 --> 00:31:59.410 call F, that up -- that two by two corner, 00:31:59.410 --> 00:32:06.250 is sitting here with all its signs reversed. 00:32:06.250 --> 00:32:08.720 So that's -- here I'm seeing minus F, 00:32:08.720 --> 00:32:13.110 and here I'm seeing the identity in the null space matrix. 00:32:13.110 --> 00:32:15.460 OK, so that's the null space. 00:32:15.460 --> 00:32:23.980 Another question is solve Bx equal one, zero, one. 00:32:23.980 --> 00:32:25.190 OK. 00:32:25.190 --> 00:32:34.890 So that's one question, now solve complete solutions. 00:32:34.890 --> 00:32:39.090 To Bx equal one, zero, one. 00:32:42.360 --> 00:32:45.200 OK. 00:32:45.200 --> 00:32:49.020 Yeah, so I guess I'm seeing if I wanted 00:32:49.020 --> 00:32:54.790 to get one, zero, one - What's our particular solution? 00:32:54.790 --> 00:32:56.460 So I'm looking for a particular solution 00:32:56.460 --> 00:32:57.720 and then the null space 00:32:57.720 --> 00:32:59.390 part. 00:32:59.390 --> 00:33:00.090 OK. 00:33:00.090 --> 00:33:05.360 I-- actually the first column of B, 00:33:05.360 --> 00:33:08.840 so what's the first column of our matrix B? 00:33:08.840 --> 00:33:10.770 It's the vector one, zero, one. 00:33:10.770 --> 00:33:14.140 The first column of our matrix agrees 00:33:14.140 --> 00:33:15.430 with the right-hand side. 00:33:15.430 --> 00:33:18.670 So I guess I'm thinking x particular 00:33:18.670 --> 00:33:22.570 plus x null space will be the particular solution, 00:33:22.570 --> 00:33:27.814 since the first column of B is exactly right, that's great. 00:33:27.814 --> 00:33:29.980 And then I have C times that first null space vector 00:33:29.980 --> 00:33:37.881 and D times the other null space vector. 00:33:37.881 --> 00:33:38.380 Right? 00:33:38.380 --> 00:33:42.700 The two -- the null space part of the solution, 00:33:42.700 --> 00:33:48.060 as always has the arbitrary constants, 00:33:48.060 --> 00:33:51.380 the particular solution doesn't have any arbitrary constants, 00:33:51.380 --> 00:33:54.840 it's one particular solution, and in this case it'll -- 00:33:54.840 --> 00:33:56.160 that one would do. 00:33:56.160 --> 00:33:57.070 OK. 00:33:57.070 --> 00:33:57.750 Fine. 00:33:57.750 --> 00:34:03.030 so those are questions taken from old quizzes, 00:34:03.030 --> 00:34:09.781 any questions coming to mind? 00:34:09.781 --> 00:34:10.280 Yeah. 00:34:10.280 --> 00:34:10.780 Q: value. 00:34:10.780 --> 00:34:11.300 OK. 00:34:11.300 --> 00:34:15.000 Well, so that particular x particular, 00:34:15.000 --> 00:34:22.330 it says that let's see, when I multiply by this guy, 00:34:22.330 --> 00:34:26.850 I'm going to get the first column of B. 00:34:26.850 --> 00:34:30.050 That -- if that's a solution, I multiply B, 00:34:30.050 --> 00:34:34.060 B times this x will be the first column of B, 00:34:34.060 --> 00:34:39.480 and so I'm saying that the first column of this B agrees with 00:34:39.480 --> 00:34:41.300 the right-hand side. 00:34:41.300 --> 00:34:47.340 So I'm saying that look at the first column of that matrix B. 00:34:47.340 --> 00:34:50.780 If you do the multiplication, it's -- 00:34:50.780 --> 00:34:52.659 so what's the first column of that matrix? 00:34:52.659 --> 00:34:54.370 Is that how you do that multiplication? 00:34:54.370 --> 00:34:56.203 I multiply that matrix by that first column. 00:34:56.203 --> 00:34:59.400 And it picks out one, zero, one. 00:34:59.400 --> 00:35:11.020 So the first column of B is exactly that. 00:35:11.020 --> 00:35:18.650 And therefore a particular solution will be this guy. 00:35:18.650 --> 00:35:19.960 So I'll repeat that question. 00:35:19.960 --> 00:35:21.150 Yeah. 00:35:21.150 --> 00:35:21.650 OK. 00:35:21.650 --> 00:35:22.150 Yes. 00:35:22.150 --> 00:35:23.170 Q: particular solution. 00:35:23.170 --> 00:35:26.320 Any of the solutions can be the particular one 00:35:26.320 --> 00:35:28.180 that we pick out. 00:35:28.180 --> 00:35:31.940 So like this plus -- plus this would be another particular 00:35:31.940 --> 00:35:32.610 OK. 00:35:32.610 --> 00:35:33.110 solution. 00:35:43.290 --> 00:35:48.260 It would be another solution. 00:35:48.260 --> 00:35:53.040 The particular is just telling us only take one. 00:35:53.040 --> 00:35:57.370 But it's not telling us which one we have to take. 00:35:57.370 --> 00:35:58.850 We take the most convenient one. 00:35:58.850 --> 00:36:02.560 I guess in this -- in this problem that was that one. 00:36:02.560 --> 00:36:03.480 Good. 00:36:03.480 --> 00:36:05.750 Other questions? 00:36:05.750 --> 00:36:06.250 Yes. 00:36:06.250 --> 00:36:10.550 And this pattern of particular plus 00:36:10.550 --> 00:36:15.370 null space, of course, that's going throughout mathematics 00:36:15.370 --> 00:36:17.770 of linear systems. 00:36:17.770 --> 00:36:20.970 We're really doing mathematics of linear systems here. 00:36:20.970 --> 00:36:24.070 Our systems are discrete and they're finite-dimensional -- 00:36:24.070 --> 00:36:31.860 and so it's linear algebra, but this particular plus null space 00:36:31.860 --> 00:36:37.850 goes -- that doesn't depend on having finite matrices -- 00:36:37.850 --> 00:36:42.990 that spreads much -- that spreads everywhere. 00:36:42.990 --> 00:36:46.280 OK, I'm going to just like to encourage 00:36:46.280 --> 00:36:48.370 you to take problems out of the book, 00:36:48.370 --> 00:36:50.250 let me do the same myself. 00:36:50.250 --> 00:36:54.160 OK well here's some easy true or falses. 00:36:54.160 --> 00:36:58.440 I don't know why the author put these in here. 00:36:58.440 --> 00:36:59.530 OK. 00:36:59.530 --> 00:37:03.720 If m=n, then the row space equals the column space. 00:37:03.720 --> 00:37:07.390 is invertible -- suppose A is an invertible 00:37:07.390 --> 00:37:09.790 So these are true or falses. 00:37:09.790 --> 00:37:13.230 If m equals n, so that means the matrix is square, 00:37:13.230 --> 00:37:17.410 then the row space equals the column space? 00:37:17.410 --> 00:37:18.760 False, good. 00:37:18.760 --> 00:37:20.260 Good, what is equal there? 00:37:20.260 --> 00:37:25.830 What can I say is equal, if M -- well, 00:37:25.830 --> 00:37:27.440 yeah. 00:37:27.440 --> 00:37:29.610 Yeah it -- so that's definitely false -- 00:37:29.610 --> 00:37:31.160 the row space and the column space, 00:37:31.160 --> 00:37:37.550 and this matrix is like always a good example to consider. 00:37:37.550 --> 00:37:40.530 So there's a square matrix but it's row 00:37:40.530 --> 00:37:44.200 space is the multiples of zero, one, 00:37:44.200 --> 00:37:48.310 and its column space is the multiples of one, zero. 00:37:48.310 --> 00:37:50.240 Very different. 00:37:50.240 --> 00:37:51.680 The row space and the column space 00:37:51.680 --> 00:37:54.230 are totally different for that matrix. 00:37:54.230 --> 00:37:57.730 Now of course if the matrix was symmetric, 00:37:57.730 --> 00:38:01.780 well, then clearly the row space equals the column space. 00:38:01.780 --> 00:38:02.280 OK. 00:38:02.280 --> 00:38:03.930 How about this question? 00:38:03.930 --> 00:38:08.910 The matrices A and minus A share the same four subspaces? 00:38:11.490 --> 00:38:15.230 Do the matrices A and minus A have the same column space, 00:38:15.230 --> 00:38:17.100 do they have the same null space, 00:38:17.100 --> 00:38:18.480 do they have the same row 00:38:18.480 --> 00:38:19.050 space? 00:38:19.050 --> 00:38:20.960 What's the answer on that? 00:38:20.960 --> 00:38:31.120 Yes or no. 00:38:31.120 --> 00:38:31.620 Yes. 00:38:31.620 --> 00:38:32.310 Good. 00:38:32.310 --> 00:38:33.100 OK. 00:38:33.100 --> 00:38:33.900 How about this? 00:38:33.900 --> 00:38:40.020 If A and B have the same four subspaces, 00:38:40.020 --> 00:38:41.470 then A is a multiple of B. 00:38:41.470 --> 00:38:43.261 If -- suppose those subspaces are the same. 00:38:43.261 --> 00:38:44.680 Then is A a multiple of B? 00:38:44.680 --> 00:38:45.510 OK. 00:38:45.510 --> 00:38:53.000 How how do you answer a question like that? 00:38:53.000 --> 00:38:56.500 Of course if you want to answer it yes, then I would -- 00:38:56.500 --> 00:38:59.460 then they'd have to think of a reason why. 00:38:59.460 --> 00:39:02.050 If you want to answer no way, then you would -- 00:39:02.050 --> 00:39:06.410 and I would sort of like first I would try to think no, 00:39:06.410 --> 00:39:10.540 I would say can I come up with an example where it isn't true? 00:39:10.540 --> 00:39:17.680 Let me repeat the question. 00:39:17.680 --> 00:39:24.810 And then write the answer. 00:39:24.810 --> 00:39:30.520 matrix, then what -- 00:39:30.520 --> 00:39:40.510 suppose it's six by six invertible matrix, 00:39:40.510 --> 00:39:59.070 then what's its row space, and its column space is all of R^6, 00:39:59.070 --> 00:40:17.620 and the null space, and the null space of A transpose would be 00:40:17.620 --> 00:40:21.900 the zero vector. 00:40:21.900 --> 00:40:36.170 So every invertible matrix is going to give that answer. 00:40:36.170 --> 00:40:49.020 If I have a six by six invertible matrix, 00:40:49.020 --> 00:40:51.602 I know what those subspaces are. 00:40:51.602 --> 00:40:53.060 Heck, that was back in chapter two, 00:40:53.060 --> 00:40:55.720 when I didn't even know what subspaces were. 00:40:55.720 --> 00:41:01.890 The row space and column space are both all six-dimensional 00:41:01.890 --> 00:41:05.470 space -- the whole space, and the rank is six, 00:41:05.470 --> 00:41:10.280 in other words, and the null spaces have zero dimension. 00:41:10.280 --> 00:41:11.870 So do you see now the answer? 00:41:11.870 --> 00:41:13.060 So A and B could be. 00:41:13.060 --> 00:41:16.970 So A and B could be for example any -- so I'm going to say 00:41:16.970 --> 00:41:17.470 false. 00:41:17.470 --> 00:41:21.070 Because A and B for example -- 00:41:21.070 --> 00:41:34.230 So an example: A and B any invertible six by six, six 00:41:34.230 --> 00:41:34.730 by six. 00:41:34.730 --> 00:41:39.230 So those would have the same four subspaces 00:41:39.230 --> 00:41:40.480 but they wouldn't be the same. 00:41:40.480 --> 00:41:44.890 Of course th- there should be something about those matrices 00:41:44.890 --> 00:41:45.930 that would be the same. 00:41:45.930 --> 00:41:50.070 It's sort of a natural problem, so now actually we're 00:41:50.070 --> 00:41:52.120 getting to a math question. 00:41:52.120 --> 00:41:54.390 The answer is this is not true. 00:41:54.390 --> 00:41:59.300 One matrix doesn't have to be a multiple of the other. 00:41:59.300 --> 00:42:02.420 But there must be something that's true. 00:42:02.420 --> 00:42:07.770 And that would be sort of like a natural question to ask. 00:42:07.770 --> 00:42:15.850 If they have the same subspaces, same four subspaces, 00:42:15.850 --> 00:42:29.700 then what -- what could you -- 00:42:29.700 --> 00:42:32.480 instinct wasn't necessarily right. 00:42:32.480 --> 00:42:37.700 But I hope you now see that the correct answer is false. 00:42:37.700 --> 00:42:40.550 And then you might think OK, well, they certainly 00:42:40.550 --> 00:42:43.530 do have the same rank. 00:42:43.530 --> 00:42:51.860 But do -- obviously if they have the same four subspaces, 00:42:51.860 --> 00:42:54.460 they have the same rank. 00:42:54.460 --> 00:42:58.620 I might say if they have the well, 00:42:58.620 --> 00:43:03.820 I could extend that question and think about other possibilities 00:43:03.820 --> 00:43:08.500 and finally come up with something that was true. 00:43:08.500 --> 00:43:11.620 But I won't press that one. 00:43:11.620 --> 00:43:15.270 Let me keep going with practice questions. 00:43:15.270 --> 00:43:19.430 And these practice questions are quite appropriate I 00:43:19.430 --> 00:43:21.510 think for the exam. 00:43:21.510 --> 00:43:22.030 OK. 00:43:22.030 --> 00:43:23.070 let's see. 00:43:23.070 --> 00:43:29.310 If I exchange two rows of A which subspaces stay the same? 00:43:29.310 --> 00:43:32.950 So I'm trying to take out questions 00:43:32.950 --> 00:43:38.670 that we can answer without you know we can answer quickly. 00:43:38.670 --> 00:43:45.560 If I have a matrix A, and I exchange two of its rows, which 00:43:45.560 --> 00:43:47.570 subspaces stay the same? 00:43:47.570 --> 00:43:50.560 The row space does stay the same. 00:43:50.560 --> 00:43:53.440 And the null space stays the same. 00:43:53.440 --> 00:43:54.270 Good. 00:43:54.270 --> 00:43:55.200 Good. 00:43:55.200 --> 00:43:56.690 Correct. 00:43:56.690 --> 00:43:59.170 Column space would be a wrong answer. 00:43:59.170 --> 00:44:00.280 OK. 00:44:00.280 --> 00:44:03.140 all right, here's a question. 00:44:03.140 --> 00:44:05.730 Oh, this leads into the next chapter. 00:44:05.730 --> 00:44:08.990 Why can the vector one, two, three not 00:44:08.990 --> 00:44:12.280 be a row and also in the null space? 00:44:12.280 --> 00:44:15.040 Fitting we close with this question. 00:44:15.040 --> 00:44:19.860 Close is -- so V equal this one, two, 00:44:19.860 --> 00:44:33.900 three can't be in the null space of a matrix and the row space. 00:44:33.900 --> 00:44:40.690 And my question is why not? 00:44:40.690 --> 00:44:41.378 Why not? 00:44:45.030 --> 00:44:47.950 So this is a question that we can 00:44:47.950 --> 00:44:51.360 because it's sort of asked in a straightforward way, 00:44:51.360 --> 00:44:55.410 we can figure out an answer. 00:44:55.410 --> 00:44:57.460 Well, actually yeah -- 00:44:57.460 --> 00:45:00.550 I'll even pin it down, it can't be in the null space -- 00:45:00.550 --> 00:45:04.420 and be a row. 00:45:04.420 --> 00:45:07.050 I'll even pin it down further. 00:45:07.050 --> 00:45:12.550 Ask it to be a row of A. 00:45:12.550 --> 00:45:15.000 Why not? 00:45:15.000 --> 00:45:18.820 So I'm -- we know the dimensions of these spaces. 00:45:18.820 --> 00:45:26.330 But now I'm asking you sort of like the overlap between -- 00:45:26.330 --> 00:45:30.320 so the null space and the row space, 00:45:30.320 --> 00:45:33.210 those are in the same n-dimensional space. 00:45:33.210 --> 00:45:41.410 Those are -- well, those are both subspaces of n-dimensional 00:45:41.410 --> 00:45:45.060 space, and I'm basically saying they can't overlap. 00:45:45.060 --> 00:45:49.040 I can't have a vector like this, a typical vector, that's 00:45:49.040 --> 00:45:53.110 in the null space and it's also a row of the matrix. 00:45:53.110 --> 00:45:54.580 Why is that? 00:45:54.580 --> 00:45:56.350 So that's a new sort of idea. 00:45:56.350 --> 00:45:58.320 Let's just see what it would mean. 00:45:58.320 --> 00:46:04.860 I mean that A times this V, why can this A times this 00:46:04.860 --> 00:46:07.770 V it can't be zero. 00:46:07.770 --> 00:46:12.410 Oh well, if it's zero, so this is -- 00:46:12.410 --> 00:46:16.830 I'm getting it into the null space here. 00:46:16.830 --> 00:46:20.750 So this is -- now let's put that vector's in the null space, 00:46:20.750 --> 00:46:27.600 why can't the first row of a matrix be one, two, three? 00:46:27.600 --> 00:46:35.470 I can fill out the matrix as I like. 00:46:35.470 --> 00:46:37.010 Why is that impossible? 00:46:37.010 --> 00:46:39.310 Well, you're seeing it's impossible, right? 00:46:39.310 --> 00:46:44.690 That if that was a row of the matrix and in the null space, 00:46:44.690 --> 00:46:48.421 that number would not be zero, it would be fourteen. 00:46:48.421 --> 00:46:48.920 Right. 00:46:48.920 --> 00:46:52.960 So now we actually are beginning to get a more complete picture 00:46:52.960 --> 00:46:55.170 of these four subspaces. 00:46:55.170 --> 00:47:00.220 The two that are over in n-dimensional space, 00:47:00.220 --> 00:47:04.450 they actually only share the zero vector. 00:47:04.450 --> 00:47:08.110 The intersection of the null space and the row space 00:47:08.110 --> 00:47:09.310 is only the zero vector. 00:47:09.310 --> 00:47:14.970 And in fact the null space is perpendicular to the row space. 00:47:14.970 --> 00:47:18.920 That'll be the first topic let's see, 00:47:18.920 --> 00:47:23.940 we have a holiday Monday -- 00:47:23.940 --> 00:47:29.800 and I'll see you Wednesday with perpendiculars. 00:47:29.800 --> 00:47:33.980 And I'll see you Friday. 00:47:33.980 --> 00:47:39.710 So good luck on the quiz.