WEBVTT 00:00:09.870 --> 00:00:14.850 OK, here's linear algebra lecture seven. 00:00:14.850 --> 00:00:19.540 I've been talking about vector spaces 00:00:19.540 --> 00:00:24.050 and specially the null space of a matrix 00:00:24.050 --> 00:00:26.320 and the column space of a matrix. 00:00:26.320 --> 00:00:28.210 What's in those spaces. 00:00:28.210 --> 00:00:32.940 Now I want to actually describe them. 00:00:32.940 --> 00:00:36.050 How do you describe all the vectors 00:00:36.050 --> 00:00:37.200 that are in those spaces? 00:00:37.200 --> 00:00:39.570 How do you compute these things? 00:00:39.570 --> 00:00:46.380 So this is the, turning the idea, the definition, 00:00:46.380 --> 00:00:48.590 into an algorithm. 00:00:48.590 --> 00:00:53.410 What's the algorithm for solving A x =0? 00:00:53.410 --> 00:00:56.920 So that's the null space that I'm interested in. 00:00:56.920 --> 00:01:01.630 So can I take a particular matrix A and describe 00:01:01.630 --> 00:01:06.840 the natural algorithm, and I'll execute it for that matrix -- 00:01:06.840 --> 00:01:08.740 here we go. 00:01:08.740 --> 00:01:12.740 So let me take the matrix as an example. 00:01:12.740 --> 00:01:16.560 So we're definitely talking rectangular matrices in this 00:01:16.560 --> 00:01:17.350 chapter. 00:01:17.350 --> 00:01:22.050 So I'll make, I'll have four columns. 00:01:22.050 --> 00:01:24.190 And three rows. 00:01:24.190 --> 00:01:29.050 Two four six eight and three six eight ten. 00:01:32.960 --> 00:01:33.460 OK. 00:01:36.450 --> 00:01:42.330 If I just look at those columns, and rows, well, 00:01:42.330 --> 00:01:45.010 I notice right away that column two 00:01:45.010 --> 00:01:47.860 is a multiple of column one. 00:01:47.860 --> 00:01:50.710 It's in the same direction as column one. 00:01:50.710 --> 00:01:52.840 It's not independent. 00:01:52.840 --> 00:01:56.380 I'll expect to discover that in the process. 00:01:56.380 --> 00:02:01.360 Actually, with rows, I notice that that row plus this row 00:02:01.360 --> 00:02:03.770 gives the third row. 00:02:03.770 --> 00:02:07.500 So the third row is not independent. 00:02:07.500 --> 00:02:12.600 So, all that should come out of elimination. 00:02:12.600 --> 00:02:14.620 So now what I -- 00:02:14.620 --> 00:02:19.590 my algorithm is elimination, but extended now 00:02:19.590 --> 00:02:23.420 to the rectangular case, where we 00:02:23.420 --> 00:02:30.320 have to continue even if there's zeros in the pivot position, 00:02:30.320 --> 00:02:31.690 we go on. 00:02:31.690 --> 00:02:36.880 OK, so let me execute elimination for that matrix. 00:02:36.880 --> 00:02:42.160 My goal is always, while I'm doing elimination -- 00:02:42.160 --> 00:02:44.650 I'm not changing the null space. 00:02:44.650 --> 00:02:46.520 That's very important, right? 00:02:46.520 --> 00:02:50.840 I'm solving A x equals zero by elimination, 00:02:50.840 --> 00:02:53.960 and when I do these operations that you already know, 00:02:53.960 --> 00:02:56.120 when I subtract a multiple of one 00:02:56.120 --> 00:03:02.330 equation from another equation, I'm not changing the solutions. 00:03:02.330 --> 00:03:04.850 So I'm not changing the null space. 00:03:04.850 --> 00:03:08.850 Actually, I changing the column space, as you'll see. 00:03:08.850 --> 00:03:10.130 So you have to pay attention. 00:03:10.130 --> 00:03:13.270 What does elimination leave unchanged? 00:03:13.270 --> 00:03:18.310 And the answer is the solutions to the system are not changed 00:03:18.310 --> 00:03:20.890 because I'm doing the same thing to -- 00:03:20.890 --> 00:03:24.620 I'm doing a legitimate operations on the equations. 00:03:24.620 --> 00:03:27.130 Of course, on the right hand side it's always zero, 00:03:27.130 --> 00:03:30.860 and I don't plan to write zero all the time. 00:03:30.860 --> 00:03:33.340 OK, so I'm really just working on the left side, 00:03:33.340 --> 00:03:39.330 but the right side is, is keeping up always zeros. 00:03:39.330 --> 00:03:41.750 OK, so what's elimination? 00:03:41.750 --> 00:03:44.470 Well, you know where the first pivot is 00:03:44.470 --> 00:03:46.060 and you know what to do 00:03:46.060 --> 00:03:46.560 with it. 00:03:46.560 --> 00:03:51.580 So can I just take the first step below here? 00:03:51.580 --> 00:03:55.940 So that pivot row is fine. 00:03:55.940 --> 00:04:00.300 I take two times that row away from this one and I get zero 00:04:00.300 --> 00:04:01.140 zero. 00:04:01.140 --> 00:04:03.570 That's signaling a difficulty. 00:04:03.570 --> 00:04:07.660 Two, two twos away from the six leaves me with a two. 00:04:07.660 --> 00:04:10.740 Two twos away from the eight leaves me with a four. 00:04:10.740 --> 00:04:13.040 And now three of those away from here 00:04:13.040 --> 00:04:17.630 is zero, again another zero, three twos away 00:04:17.630 --> 00:04:20.260 from that eight is the two, three twos away 00:04:20.260 --> 00:04:22.200 from that ten is a four. 00:04:22.200 --> 00:04:23.250 OK. 00:04:23.250 --> 00:04:26.780 That's the first stage of elimination. 00:04:26.780 --> 00:04:31.150 I've got the first column straight. 00:04:31.150 --> 00:04:35.530 So of course I move on to the second column. 00:04:35.530 --> 00:04:39.710 I look in that position, I see a zero. 00:04:39.710 --> 00:04:43.500 I look below it, hoping for a non-zero 00:04:43.500 --> 00:04:44.980 that I can do a row exchange. 00:04:44.980 --> 00:04:47.720 But it's zero below. 00:04:47.720 --> 00:04:53.330 So that's telling me that that column is -- well, 00:04:53.330 --> 00:04:56.320 what it's really going to be telling me is that that column 00:04:56.320 --> 00:04:59.840 is a combination of the earlier columns. 00:04:59.840 --> 00:05:04.080 It's that second column is dependent on the earlier 00:05:04.080 --> 00:05:04.670 columns. 00:05:04.670 --> 00:05:09.280 But I don't stop to think here. 00:05:09.280 --> 00:05:11.680 In that column there's nothing to do. 00:05:11.680 --> 00:05:12.970 I go on to the next. 00:05:12.970 --> 00:05:15.240 So here's the next pivot. 00:05:15.240 --> 00:05:18.080 So there's the first pivot and there's the second pivot, 00:05:18.080 --> 00:05:22.660 and I just keep this elimination going downwards. 00:05:22.660 --> 00:05:30.720 So, so the next step keeps the first row, 00:05:30.720 --> 00:05:34.600 keeps the second row with its pivot, 00:05:34.600 --> 00:05:40.510 so I've got my two pivots, and does elimination 00:05:40.510 --> 00:05:44.730 to clear out the column below that pivot. 00:05:44.730 --> 00:05:47.530 So actually you see the multiplier is one. 00:05:47.530 --> 00:05:50.100 It subtracts row two from row three 00:05:50.100 --> 00:05:53.480 and produces a row of zeros. 00:05:53.480 --> 00:05:54.190 OK. 00:05:54.190 --> 00:05:59.050 That I would call that matrix U, right? 00:05:59.050 --> 00:06:01.870 That's our upper -- 00:06:01.870 --> 00:06:05.140 well, I can't quite say upper triangular. 00:06:05.140 --> 00:06:08.260 Maybe upper -- I don't know -- upper something. 00:06:12.680 --> 00:06:17.150 It's in this so-called echelon form. 00:06:17.150 --> 00:06:21.700 The word echelon means, like, staircase form. 00:06:21.700 --> 00:06:25.960 It's the, the non-zeros come in that staircase form. 00:06:25.960 --> 00:06:30.720 If there was another pivot here, then the staircase 00:06:30.720 --> 00:06:32.260 would include that. 00:06:32.260 --> 00:06:37.870 But here's a case where we have two pivots only. 00:06:37.870 --> 00:06:41.320 OK, so actually we've already discovered the most important 00:06:41.320 --> 00:06:44.210 number about this matrix. 00:06:44.210 --> 00:06:48.540 The number of pivots is two. 00:06:48.540 --> 00:06:51.930 That number we will call the rank of the matrix. 00:06:51.930 --> 00:06:54.520 So let me put immediately. 00:06:54.520 --> 00:06:58.696 The rank of A -- 00:06:58.696 --> 00:07:00.070 so I'm telling you what this word 00:07:00.070 --> 00:07:04.280 rank means in the algorithm. 00:07:04.280 --> 00:07:08.650 It's equal to the number of pivots. 00:07:08.650 --> 00:07:12.900 And in this case, two. 00:07:12.900 --> 00:07:17.320 OK, for me that number two is the crucial number. 00:07:17.320 --> 00:07:22.310 OK, now I go to -- you remember I'm always solving A x equals 00:07:22.310 --> 00:07:26.240 zero, but now I can solve U x equals zero, right? 00:07:26.240 --> 00:07:29.650 Same solution, same null space. 00:07:29.650 --> 00:07:30.150 OK. 00:07:30.150 --> 00:07:32.830 So I could stop here -- 00:07:32.830 --> 00:07:36.810 why don't I stop here and do the back substitution. 00:07:36.810 --> 00:07:42.750 So now I have to ask you, how do I describe the solutions? 00:07:42.750 --> 00:07:45.880 There are solutions, right, to A x equals zero. 00:07:45.880 --> 00:07:47.070 I knew there would be. 00:07:47.070 --> 00:07:50.110 I had three equations in four unknowns. 00:07:50.110 --> 00:07:52.850 I certainly expected some solutions. 00:07:52.850 --> 00:07:55.240 Now I want to see what are they. 00:07:55.240 --> 00:07:56.800 OK, here's the critical step. 00:07:59.920 --> 00:08:04.860 I refer to it up here as separating out 00:08:04.860 --> 00:08:09.990 the pivot variables, the pivot columns, which are these two. 00:08:09.990 --> 00:08:13.080 Here I have two pivot columns. 00:08:13.080 --> 00:08:15.530 Those, obviously, they're the columns with the pivots. 00:08:15.530 --> 00:08:18.736 So I have two pivot columns. 00:08:22.110 --> 00:08:27.750 And I have the other columns, I'll call free. 00:08:27.750 --> 00:08:30.235 These are free columns, OK. 00:08:32.950 --> 00:08:35.470 Why do I use those words? 00:08:35.470 --> 00:08:38.110 Why do I use that word free? 00:08:38.110 --> 00:08:43.289 Because now I want to write, I want to find the solutions to U 00:08:43.289 --> 00:08:45.210 x equals zero. 00:08:45.210 --> 00:08:48.090 Here is the way I do it. 00:08:48.090 --> 00:08:58.040 These free columns I can assign any number freely to those -- 00:08:58.040 --> 00:09:03.900 to the variables x2 and x4, the ones that multiply 00:09:03.900 --> 00:09:06.570 columns two and four. 00:09:06.570 --> 00:09:10.620 So I can assign anything I like to x2 and x4. 00:09:10.620 --> 00:09:17.430 And then I can solve the equations for x1 and x3. 00:09:17.430 --> 00:09:19.450 Let me say that again. 00:09:19.450 --> 00:09:22.390 If I give -- let me, let me assign. 00:09:22.390 --> 00:09:29.320 So, so one particular x is to assign, say, the value one 00:09:29.320 --> 00:09:33.640 to the, to x2, and zero to x4. 00:09:33.640 --> 00:09:35.850 Those are -- that was a free choice, 00:09:35.850 --> 00:09:38.031 but it's a convenient choice. 00:09:38.031 --> 00:09:38.530 OK. 00:09:38.530 --> 00:09:41.140 Now I want to solve U x equals zero 00:09:41.140 --> 00:09:47.780 and find numbers one and three, complete the solution. 00:09:47.780 --> 00:09:49.481 Can I write down -- 00:09:49.481 --> 00:09:49.980 let's see. 00:09:49.980 --> 00:09:50.480 OK. 00:09:50.480 --> 00:09:53.220 Shall we just remember what U x equals zero represents? 00:09:53.220 --> 00:09:54.730 What are my equations? 00:09:54.730 --> 00:10:00.900 That first equation is x1 plus just -- 00:10:00.900 --> 00:10:05.770 I'm just saying what are these matrices meaning. 00:10:05.770 --> 00:10:07.580 That's the first equation. 00:10:07.580 --> 00:10:14.040 And the second equation was 2x3 + 4x4=0. 00:10:14.040 --> 00:10:16.330 Those are my two equations. 00:10:16.330 --> 00:10:17.020 OK. 00:10:17.020 --> 00:10:24.600 Now the point is I can find x1 and x3 by back substitution. 00:10:24.600 --> 00:10:27.500 So we're building on what we already know. 00:10:27.500 --> 00:10:31.970 The new thing is that there were some free variables that I 00:10:31.970 --> 00:10:34.120 could give any numbers to. 00:10:34.120 --> 00:10:38.260 And I'm systematically going to make a choice like this, Now 00:10:38.260 --> 00:10:38.760 what is x3? 00:10:38.760 --> 00:10:39.920 1 and 0. 00:10:39.920 --> 00:10:42.970 Let's, let's go backwards, back up. 00:10:42.970 --> 00:10:45.360 I look at the last equation. 00:10:45.360 --> 00:10:50.740 x3 is zero, from the last equation, 00:10:50.740 --> 00:10:55.790 because, because x4 we've set x4 to zero, 00:10:55.790 --> 00:10:58.180 and then we get x3 as zero. 00:10:58.180 --> 00:10:58.680 OK. 00:10:58.680 --> 00:11:03.400 Now we set x2 to be one, so what is x1? 00:11:03.400 --> 00:11:06.050 Negative two, right? 00:11:06.050 --> 00:11:10.150 So then I have negative two plus two, zero and zero, 00:11:10.150 --> 00:11:11.400 correctly giving zero. 00:11:11.400 --> 00:11:14.810 There is a vector in the null space. 00:11:14.810 --> 00:11:17.530 There is a solution to A x equals zero. 00:11:17.530 --> 00:11:19.740 In fact, what solution is that? 00:11:19.740 --> 00:11:24.360 That simply says that minus two times the first column 00:11:24.360 --> 00:11:28.502 plus one times the second column is the zero column. 00:11:28.502 --> 00:11:29.460 Of course that's right. 00:11:29.460 --> 00:11:33.310 Minus two of that column plus one of that, or minus two 00:11:33.310 --> 00:11:35.580 of that plus one of that. 00:11:35.580 --> 00:11:39.620 That solution is -- that, that's just what we saw immediately, 00:11:39.620 --> 00:11:43.690 that the second column is twice as big as the first column. 00:11:43.690 --> 00:11:48.400 OK, tell me some more vectors in the null space. 00:11:48.400 --> 00:11:51.580 I found one. 00:11:51.580 --> 00:11:58.030 Tell me, how to get a bunch more immediately out of that one. 00:11:58.030 --> 00:11:59.620 Just take multiples of it. 00:11:59.620 --> 00:12:01.440 Any multiple of -- 00:12:01.440 --> 00:12:04.200 I could multiply this by anything. 00:12:04.200 --> 00:12:08.430 I might as well call it, I could say, C, some multiple of this. 00:12:08.430 --> 00:12:10.380 So let me -- 00:12:10.380 --> 00:12:14.890 so X could be any multiple of this one. 00:12:14.890 --> 00:12:17.700 OK, that, that describes now a line, 00:12:17.700 --> 00:12:23.430 an infinitely long line in four dimensional space. 00:12:23.430 --> 00:12:26.720 But -- which is in the null space. 00:12:26.720 --> 00:12:29.390 Is that the whole null space? 00:12:29.390 --> 00:12:30.460 No. 00:12:30.460 --> 00:12:34.220 I've got two free variables here. 00:12:34.220 --> 00:12:37.330 I made this choice, one and zero, for the free variables, 00:12:37.330 --> 00:12:40.090 but I could have made another choice. 00:12:40.090 --> 00:12:44.420 Let me make the other choice zero and one. 00:12:44.420 --> 00:12:47.140 You see my system. 00:12:47.140 --> 00:12:48.580 So let me repeat the system. 00:12:48.580 --> 00:12:54.087 This is the algorithm that you, you just learned to do. 00:12:56.890 --> 00:12:59.100 Do elimination. 00:12:59.100 --> 00:13:01.060 Decide which are the pivot columns 00:13:01.060 --> 00:13:02.420 and which are the free columns. 00:13:02.420 --> 00:13:05.600 That tells you that, that variables one and three 00:13:05.600 --> 00:13:09.760 are pivot variables, two and four are free variables. 00:13:09.760 --> 00:13:14.800 Then those free variables, you assign them -- 00:13:14.800 --> 00:13:17.870 you give one of them the value one and the others the value 00:13:17.870 --> 00:13:22.070 zero -- in this case, we had a one and a zero -- 00:13:22.070 --> 00:13:24.080 and complete the solution. 00:13:24.080 --> 00:13:28.550 And you do -- you give the other one the value one and zero. 00:13:28.550 --> 00:13:30.160 And now complete the solution. 00:13:30.160 --> 00:13:32.310 So let's complete that solution. 00:13:32.310 --> 00:13:35.790 I'm looking for a vector in the null space, 00:13:35.790 --> 00:13:38.550 and it's absolutely going to be different from this guy, 00:13:38.550 --> 00:13:42.350 because, because any multiple of that zero 00:13:42.350 --> 00:13:43.920 is never going to give the one. 00:13:43.920 --> 00:13:46.050 So I have somebody new in the null space, 00:13:46.050 --> 00:13:46.930 and let me finish it 00:13:46.930 --> 00:13:47.430 out. 00:13:47.430 --> 00:13:50.140 What's x3 here? 00:13:50.140 --> 00:13:52.360 So we're going by back substitution, 00:13:52.360 --> 00:13:53.790 looking at this equation. 00:13:53.790 --> 00:13:59.680 Now x4 we've changed, we're doing the other possibility, 00:13:59.680 --> 00:14:02.870 where x2 is zero and x4 is one. 00:14:02.870 --> 00:14:06.590 So x3 will happen to be minus two. 00:14:06.590 --> 00:14:10.510 And now what do I get for that first equation? 00:14:10.510 --> 00:14:12.420 x1 -- let's see. 00:14:12.420 --> 00:14:19.430 Two x3s is minus four plus two -- do I get a two there? 00:14:19.430 --> 00:14:20.270 Perhaps, yeah. 00:14:20.270 --> 00:14:25.500 Two for x1, minus four, and two. 00:14:25.500 --> 00:14:26.020 OK. 00:14:26.020 --> 00:14:28.360 That's in the null space. 00:14:28.360 --> 00:14:32.220 What does that say about the columns? 00:14:32.220 --> 00:14:37.320 That says that two of this column minus two 00:14:37.320 --> 00:14:41.900 of this column plus this column gives zero, which it does. 00:14:41.900 --> 00:14:46.090 Two of that minus two of that and one of that 00:14:46.090 --> 00:14:47.520 gives the zero column. 00:14:47.520 --> 00:14:52.420 OK, now, now I've found another vector in the null space. 00:14:52.420 --> 00:14:55.880 Now we're ready to tell me the whole null space. 00:14:55.880 --> 00:15:00.160 What are all the solutions to Ax=0? 00:15:00.160 --> 00:15:05.660 I've got this guy and when I have him, 00:15:05.660 --> 00:15:10.660 what else is, goes into the null space along with that? 00:15:10.660 --> 00:15:13.760 These are my two special solutions. 00:15:13.760 --> 00:15:14.710 I call them special -- 00:15:14.710 --> 00:15:16.250 I just invented that name. 00:15:16.250 --> 00:15:18.110 Special solutions. 00:15:18.110 --> 00:15:21.180 What's special about them is the special numbers 00:15:21.180 --> 00:15:27.360 I gave to the free variables, the values zero and one 00:15:27.360 --> 00:15:29.300 for the free variables. 00:15:29.300 --> 00:15:32.560 I could have given the free variables any values 00:15:32.560 --> 00:15:35.440 and got vectors in the null space. 00:15:35.440 --> 00:15:40.210 But this was a good way to be sure I got t- got everybody. 00:15:40.210 --> 00:15:45.480 OK, so once I have him, I also have any multiple, right? 00:15:45.480 --> 00:15:48.700 So I could take any multiple of that 00:15:48.700 --> 00:15:50.910 and it's in the null space. 00:15:50.910 --> 00:15:52.150 And now what else -- 00:15:52.150 --> 00:15:55.090 I left a little space for what? 00:15:55.090 --> 00:15:58.910 What -- a plus sign. 00:15:58.910 --> 00:16:00.580 I can take any combination. 00:16:00.580 --> 00:16:04.050 Here is a line of vectors in the null space. 00:16:04.050 --> 00:16:06.110 A bunch of solutions. 00:16:06.110 --> 00:16:09.540 Would you rather I say in the null space or would you rather 00:16:09.540 --> 00:16:13.330 I say, OK, I'm solving Ax=0? 00:16:13.330 --> 00:16:15.545 Well, really I'm solving Ux=0. 00:16:19.540 --> 00:16:23.290 Well, OK, let me put in that crucial plus sign. 00:16:23.290 --> 00:16:29.420 I'm taking all the combinations of my two special solutions. 00:16:29.420 --> 00:16:32.140 That's my conclusion there. 00:16:32.140 --> 00:16:37.300 The null space contains, contains exactly 00:16:37.300 --> 00:16:42.270 all the combinations of the special solutions. 00:16:42.270 --> 00:16:46.020 And how many special solutions are there? 00:16:46.020 --> 00:16:49.650 There's one for every free variable. 00:16:49.650 --> 00:16:51.330 And how many free variables are there? 00:16:51.330 --> 00:16:54.860 Oh, I mean, we can see all the whole picture now. 00:16:54.860 --> 00:16:59.830 If the rank R was two, this is the, 00:16:59.830 --> 00:17:04.260 this is the number of pivot variables, right, 00:17:04.260 --> 00:17:05.569 because it counted the pivots. 00:17:08.800 --> 00:17:11.960 So how many free variables? 00:17:11.960 --> 00:17:14.970 Well, you know it's two, right? 00:17:14.970 --> 00:17:20.260 What is it in -- for a matrix that's m rows, n columns, 00:17:20.260 --> 00:17:25.010 n variables that means, with rank r? 00:17:25.010 --> 00:17:28.420 How many free variables have we got left? 00:17:28.420 --> 00:17:34.440 If r of the variables are pivot variables, we have n-r -- 00:17:34.440 --> 00:17:38.400 in this case four minus two -- free variables. 00:17:38.400 --> 00:17:53.390 Do you see that first of all we get clean answers here? 00:17:53.390 --> 00:17:59.300 We get r pivot variables -- so there really were r equations 00:17:59.300 --> 00:18:00.170 here. 00:18:00.170 --> 00:18:01.990 There looked like three equations, 00:18:01.990 --> 00:18:05.730 but there were really only two independent equations. 00:18:05.730 --> 00:18:11.820 And there were n-r variables that we could choose freely, 00:18:11.820 --> 00:18:16.400 and we gave them those special zero one values, 00:18:16.400 --> 00:18:19.180 and we got the special solutions. 00:18:19.180 --> 00:18:19.680 OK. 00:18:22.260 --> 00:18:25.710 For me -- we could stop at that point. 00:18:25.710 --> 00:18:28.580 That gives you a complete algorithm 00:18:28.580 --> 00:18:34.690 for finding all the solutions to A x equals zero. 00:18:34.690 --> 00:18:35.190 OK. 00:18:38.300 --> 00:18:41.720 Again, you do elimination -- 00:18:41.720 --> 00:18:46.050 going onward when a column, when there's nothing 00:18:46.050 --> 00:18:50.570 to be done on one column, you just continue. 00:18:50.570 --> 00:18:55.360 There's this number r, the number of pivots, is crucial, 00:18:55.360 --> 00:19:01.150 and it leaves n-r free variables, which you 00:19:01.150 --> 00:19:03.100 give values zero and one to. 00:19:03.100 --> 00:19:05.390 I would like to take one more step. 00:19:08.210 --> 00:19:12.020 I would like to clean up this matrix even more. 00:19:12.020 --> 00:19:14.930 So now I'm going to go to -- this is in its, 00:19:14.930 --> 00:19:20.420 this is in echelon form, upper triangular if you like. 00:19:20.420 --> 00:19:25.660 I want to go one more step to make it as good as it can be. 00:19:25.660 --> 00:19:31.310 OK, so now I'm going to speak about the reduced row echelon 00:19:31.310 --> 00:19:32.200 form. 00:19:32.200 --> 00:19:35.540 OK, so now I'm going to speak about the matrix R, which is 00:19:35.540 --> 00:19:44.680 the reduced row echelon form. 00:19:44.680 --> 00:19:46.700 So what does that mean? 00:19:46.700 --> 00:19:49.470 That means I just -- 00:19:49.470 --> 00:19:52.370 I can, I can work harder on U. 00:19:52.370 --> 00:19:55.310 So let me start, let me suppose I got as far 00:19:55.310 --> 00:19:58.125 as U, which was good. 00:20:08.230 --> 00:20:11.450 Notice how that row of zeros appeared. 00:20:11.450 --> 00:20:15.360 I didn't comment on that, but I should have. 00:20:15.360 --> 00:20:20.630 That row of zeros up here is because row three was 00:20:20.630 --> 00:20:23.060 a combination of rows one and two, 00:20:23.060 --> 00:20:27.390 and elimination discovered that fact. 00:20:27.390 --> 00:20:32.610 When we get a row of zeros, that's telling us that the -- 00:20:32.610 --> 00:20:38.580 original row that was there was a combination of other rows, 00:20:38.580 --> 00:20:42.630 and elimination knocked it out. 00:20:42.630 --> 00:20:44.710 OK, so we got this far. 00:20:44.710 --> 00:20:47.260 Now how can I clean that up further? 00:20:47.260 --> 00:20:51.430 I can do, elimination upwards. 00:20:51.430 --> 00:20:54.390 I can get zero above the pivots. 00:20:54.390 --> 00:20:58.740 So this reduced row echelon form has zeros 00:20:58.740 --> 00:21:05.745 above and below the pivots. 00:21:08.530 --> 00:21:11.460 So let me do that. 00:21:11.460 --> 00:21:14.940 So now I'll subtract one of this from the row above. 00:21:14.940 --> 00:21:20.150 That will leave a zero and a minus two in there. 00:21:20.150 --> 00:21:22.540 And that's good. 00:21:26.670 --> 00:21:30.710 OK, and I can clean it up even one more step. 00:21:30.710 --> 00:21:33.400 I can make the pivots -- 00:21:33.400 --> 00:21:37.300 the pivots I'm going to make equal to one, because I can 00:21:37.300 --> 00:21:41.380 divide equation two by the pivot. 00:21:41.380 --> 00:21:44.110 That won't change the solutions. 00:21:44.110 --> 00:21:45.940 So let me do that. 00:21:45.940 --> 00:21:46.870 And then I really -- 00:21:46.870 --> 00:21:47.910 I'm ready to stop. 00:21:47.910 --> 00:21:53.230 One, two, zero, minus two, zero, zero, one, two. 00:21:53.230 --> 00:21:58.000 I divided the second equation by two, 00:21:58.000 --> 00:22:05.560 because now I have a one in the pivot and zeros below. 00:22:05.560 --> 00:22:06.060 OK. 00:22:06.060 --> 00:22:14.790 This is my matrix R. 00:22:14.790 --> 00:22:17.810 I guess I'm hoping that you could now 00:22:17.810 --> 00:22:21.710 execute the whole algorithm. 00:22:21.710 --> 00:22:27.518 Matlab will do it immediately with the command -- 00:22:30.870 --> 00:22:34.490 reduced row echelon form of A. 00:22:34.490 --> 00:22:37.630 So if I input that original matrix A 00:22:37.630 --> 00:22:43.360 and then I write, then I type that command, press return, 00:22:43.360 --> 00:22:46.210 that matrix will appear. 00:22:46.210 --> 00:22:49.140 That's the reduced row echelon form, 00:22:49.140 --> 00:23:00.140 and it's got all the information as clear as can be. 00:23:00.140 --> 00:23:01.890 What, what information has it got? 00:23:01.890 --> 00:23:04.000 Well, of course it immediately tells me 00:23:04.000 --> 00:23:08.530 the pivot rows, pivot rows, one and two, 00:23:08.530 --> 00:23:11.490 pivot columns, one and three. 00:23:11.490 --> 00:23:15.240 And in fact it's got the identity matrix in there, 00:23:15.240 --> 00:23:18.820 It's, it's got zeros above and below the pivots, right? 00:23:18.820 --> 00:23:22.460 and the pivots are one, so it's, so it's got a -- 00:23:22.460 --> 00:23:30.180 so notice the two by two identity matrix that's sitting 00:23:30.180 --> 00:23:33.370 in the pivot rows and pivot columns. 00:23:33.370 --> 00:23:42.710 it's I in the pivot rows and columns. 00:23:46.590 --> 00:23:48.930 It's got zero rows below. 00:23:52.120 --> 00:23:56.580 Those are always indicating that original rows were, 00:23:56.580 --> 00:23:58.740 were combinations of other rows. 00:23:58.740 --> 00:24:02.100 So we really only had two rows there. 00:24:02.100 --> 00:24:05.740 And now it also -- so there's the identity. 00:24:05.740 --> 00:24:10.740 Now it's also got its free columns. 00:24:10.740 --> 00:24:17.160 And, they're cleaned up as much as possible. 00:24:17.160 --> 00:24:21.720 Actually, actually it's now so cleaned up that the special 00:24:21.720 --> 00:24:26.070 solutions, I can read off -- you remember I went through 00:24:26.070 --> 00:24:30.220 the steps of computing this -- 00:24:30.220 --> 00:24:32.760 doing back substitution? 00:24:32.760 --> 00:24:37.390 Let me, let me, instead of that system, 00:24:37.390 --> 00:24:39.600 let me take this improved system. 00:24:39.600 --> 00:24:43.850 So I'm going to use these numbers, right. 00:24:43.850 --> 00:24:45.850 In these equations, what did I do? 00:24:45.850 --> 00:24:52.820 I divided this equation by two and, oh yeah 00:24:52.820 --> 00:24:54.540 and I had subtracted two of this, 00:24:54.540 --> 00:24:58.290 which knocked out this guy and made that a minus sign. 00:24:58.290 --> 00:25:01.500 Is that what -- 00:25:01.500 --> 00:25:04.570 I've now written Rx equals zero. 00:25:10.350 --> 00:25:14.570 Now I guess I'm hoping everybody in this room understands 00:25:14.570 --> 00:25:19.150 the solutions to the original A x equals zero, 00:25:19.150 --> 00:25:23.050 the midway, halfway, U x equals zero, 00:25:23.050 --> 00:25:27.980 and the final R x equals zero are all the same. 00:25:27.980 --> 00:25:30.900 Because going from one of those to another one 00:25:30.900 --> 00:25:33.330 I didn't mess up. 00:25:33.330 --> 00:25:36.280 I just multiplied equations and subtracted 00:25:36.280 --> 00:25:39.840 from other equations, which I'm allowed to do. 00:25:39.840 --> 00:25:40.340 OK. 00:25:40.340 --> 00:25:47.530 But my point is that now if I do this free variables 00:25:47.530 --> 00:25:52.410 and back substitution, it's just, the numbers are there. 00:25:52.410 --> 00:26:01.040 When I let x -- so in this guy, I let x2 be one and x4 be zero. 00:26:01.040 --> 00:26:04.380 I, I guess, what I seeing here? 00:26:04.380 --> 00:26:07.070 Let me, let me sort of separate this out here. 00:26:07.070 --> 00:26:12.100 I'm seeing in the pivot, in the pivot columns, 00:26:12.100 --> 00:26:16.910 if I, if I put the pivot columns here, I'm seeing those. 00:26:16.910 --> 00:26:21.970 And I'm -- in the free columns I'm seeing -- 00:26:21.970 --> 00:26:23.570 what I seeing in the free columns? 00:26:23.570 --> 00:26:29.030 A two, zero in that first free column, the x2 column, 00:26:29.030 --> 00:26:33.920 and a minus two, two in the fourth column, 00:26:33.920 --> 00:26:36.010 the other free column. 00:26:36.010 --> 00:26:41.750 And the row of zeros below, which of course have no -- 00:26:41.750 --> 00:26:43.860 that equation is zero equals zero. 00:26:43.860 --> 00:26:46.080 That's satisfied. 00:26:46.080 --> 00:26:48.530 Here's my point. 00:26:48.530 --> 00:26:51.220 That when I do back substitution, 00:26:51.220 --> 00:26:55.700 these numbers are exactly what shows up -- 00:26:55.700 --> 00:26:58.480 oh, their signs get switched. 00:26:58.480 --> 00:27:01.550 I was going to say those numbers, two, minus two, zero, 00:27:01.550 --> 00:27:06.240 two, can I just circle the -- this is the free part 00:27:06.240 --> 00:27:07.120 of the matrix. 00:27:07.120 --> 00:27:10.220 This is the identity part. 00:27:10.220 --> 00:27:14.710 This is the free part, maybe I'll call it F. 00:27:14.710 --> 00:27:18.650 This, of course, I call I, because it's the identity. 00:27:18.650 --> 00:27:25.020 The free part is a, I mean, I'm just doing back substitution. 00:27:25.020 --> 00:27:29.340 And those free numbers will show up, with a minus sign, 00:27:29.340 --> 00:27:32.040 because they pop onto the other side of the equation -- 00:27:32.040 --> 00:27:35.870 so I see minus two, zero, and I see two, minus two. 00:27:39.670 --> 00:27:41.310 So that wasn't magic. 00:27:41.310 --> 00:27:43.540 It had to happen. 00:27:43.540 --> 00:27:48.240 Let me, show you clearly why it happened. 00:27:48.240 --> 00:27:50.340 OK, so that's -- 00:27:50.340 --> 00:27:53.290 this is what I'm interested in here. 00:27:53.290 --> 00:27:59.410 And now let me, let me just, like, do it, do it for -- 00:27:59.410 --> 00:28:04.530 let's suppose we've, we've got to -- 00:28:07.750 --> 00:28:11.940 let's suppose we've got this system already in, 00:28:11.940 --> 00:28:18.200 in rref form. 00:28:18.200 --> 00:28:22.580 So my matrix R is -- what does it look like? 00:28:22.580 --> 00:28:25.130 OK, and I'll -- 00:28:25.130 --> 00:28:30.570 let me pretend that the pivot columns come first 00:28:30.570 --> 00:28:33.220 and then whatever's in the free columns. 00:28:33.220 --> 00:28:37.970 And there might be some zero rows below. 00:28:37.970 --> 00:28:40.600 There's a typical -- 00:28:40.600 --> 00:28:45.750 a pretty typical reduced row echelon form. 00:28:49.450 --> 00:28:51.510 You see what's typical. 00:28:51.510 --> 00:28:55.430 It's got -- this is r by r. 00:28:55.430 --> 00:28:57.950 This is r pivot rows. 00:29:02.610 --> 00:29:06.050 This is r pivot columns. 00:29:09.780 --> 00:29:16.290 And here are n-r free columns. 00:29:16.290 --> 00:29:17.480 OK. 00:29:17.480 --> 00:29:21.650 Tell me what are the special solutions? 00:29:21.650 --> 00:29:23.350 What are the -- 00:29:23.350 --> 00:29:24.060 what's x? 00:29:24.060 --> 00:29:27.440 If I want to solve R x equals zero -- 00:29:27.440 --> 00:29:31.170 in fact, let me -- 00:29:31.170 --> 00:29:36.110 I'm really going to, do the whole -- since these -- 00:29:36.110 --> 00:29:39.030 this is now block matrices, I might as well do all 00:29:39.030 --> 00:29:41.300 of the special solutions at once. 00:29:41.300 --> 00:29:44.740 So I want to solve R x equals zero, 00:29:44.740 --> 00:29:49.660 and I'll have some special solutions. 00:29:49.660 --> 00:29:52.830 Let me, actually -- 00:29:52.830 --> 00:29:54.780 can I do them all at once? 00:29:54.780 --> 00:30:00.750 I'm going to create a null space matrix, OK. 00:30:00.750 --> 00:30:01.790 A matrix. 00:30:04.970 --> 00:30:13.130 Its, its, its columns are the special -- 00:30:13.130 --> 00:30:15.085 the columns are the special solutions. 00:30:19.080 --> 00:30:21.070 This is, I'm making it sound harder, 00:30:21.070 --> 00:30:22.670 it's going to be totally easy. 00:30:22.670 --> 00:30:25.800 N will be this null space matrix. 00:30:25.800 --> 00:30:31.070 I want R N to be the zero matrix. 00:30:31.070 --> 00:30:33.830 These columns of N are supposed to multipl- 00:30:33.830 --> 00:30:37.300 to get multiplied by R and give zero columns. 00:30:37.300 --> 00:30:40.100 So what N will do the job? 00:30:40.100 --> 00:30:41.450 Let me put -- 00:30:41.450 --> 00:30:45.290 I'm going to put the identity in the free variable part 00:30:45.290 --> 00:30:54.020 and then minus F will show up in the pivot variables, just 00:30:54.020 --> 00:30:55.980 the way it did in that example. 00:30:55.980 --> 00:30:58.970 There we had the identity and F. 00:30:58.970 --> 00:31:02.160 Here -- in the special solution. 00:31:02.160 --> 00:31:05.400 So these columns are -- there's the matrix of special 00:31:05.400 --> 00:31:06.430 solutions. 00:31:06.430 --> 00:31:09.460 And actually, there -- so there's a Matlab command 00:31:09.460 --> 00:31:14.360 or a teaching code command, NULL -- 00:31:14.360 --> 00:31:19.460 N equal, so this is the -- 00:31:19.460 --> 00:31:24.710 produces the null basis, the null space matrix, NULL of A, 00:31:24.710 --> 00:31:26.230 and there it is. 00:31:30.450 --> 00:31:33.630 And how does that command actually work? 00:31:33.630 --> 00:31:38.590 It uses Matlab to compute R, then 00:31:38.590 --> 00:31:43.100 it picks out the pivot variables, the free variables, 00:31:43.100 --> 00:31:47.980 puts, puts ones and zeros in for the free variables, 00:31:47.980 --> 00:31:51.850 and copies out the pivot variables. 00:31:51.850 --> 00:31:54.760 It, it does back substitution, but back substitution 00:31:54.760 --> 00:31:57.180 for this system is totally simple. 00:31:57.180 --> 00:32:00.070 What is this system? 00:32:00.070 --> 00:32:03.030 R x equals zero. 00:32:03.030 --> 00:32:11.620 So this is R is I F, and x is the pivot variables 00:32:11.620 --> 00:32:18.240 and the free variables, and it's supposed to give zero. 00:32:18.240 --> 00:32:20.240 So what does that mean? 00:32:20.240 --> 00:32:24.960 That means that the pivot variables plus F 00:32:24.960 --> 00:32:28.640 times the free variables give zero. 00:32:28.640 --> 00:32:31.950 So let me put F times the free variables on the other side. 00:32:31.950 --> 00:32:37.950 I get minus F times the free variables. 00:32:37.950 --> 00:32:43.530 There's my, equation, as simple as it can be. 00:32:43.530 --> 00:32:45.840 That's what back substitution comes 00:32:45.840 --> 00:32:49.280 to when I've reduced and reduced and reduced this system 00:32:49.280 --> 00:32:52.090 to the, to the best form, OK. 00:32:52.090 --> 00:32:56.680 And, then if the free variables, I 00:32:56.680 --> 00:33:00.070 make this special choice of the identity, 00:33:00.070 --> 00:33:02.570 then the pivot variables are 00:33:02.570 --> 00:33:08.710 minus F. OK, can I do, another example? 00:33:08.710 --> 00:33:10.180 Could you do another example? 00:33:10.180 --> 00:33:12.300 Can I -- let me just take another matrix 00:33:12.300 --> 00:33:17.330 and, and let's go through this algorithm once more, OK. 00:33:17.330 --> 00:33:19.180 Here we go. 00:33:19.180 --> 00:33:25.100 Here's a blackboard for another matrix, OK. 00:33:25.100 --> 00:33:31.180 So I'll call the matrix A again, but let me make it -- 00:33:31.180 --> 00:33:33.680 yeah, how big shall we make it this time? 00:33:36.480 --> 00:33:38.620 Why don't I do this? 00:33:38.620 --> 00:33:39.920 Just for the heck of it. 00:33:39.920 --> 00:33:44.860 Let me take the transpose of this A and see what happens to 00:33:44.860 --> 00:33:45.670 that. 00:33:45.670 --> 00:33:55.820 Two four six eight and three six eight ten. 00:34:00.250 --> 00:34:06.380 Before we do the calculations, tell me what's coming? 00:34:06.380 --> 00:34:12.170 How many pivot variables do you expect here? 00:34:12.170 --> 00:34:16.900 How many columns are going to have pivots? 00:34:16.900 --> 00:34:22.060 How many -- we have three columns in that matrix, 00:34:22.060 --> 00:34:25.739 but are we going to, are we going to have three pivots? 00:34:25.739 --> 00:34:31.429 No, because this third columns is the sum of the first two 00:34:31.429 --> 00:34:32.100 columns. 00:34:32.100 --> 00:34:38.350 I'm totally expecting, totally expecting that these will be 00:34:38.350 --> 00:34:41.000 pivot columns -- 00:34:41.000 --> 00:34:45.949 because they're independent, but that this third guy, 00:34:45.949 --> 00:34:49.659 the third column, which is dependent on the first two, 00:34:49.659 --> 00:34:52.219 is going to be a free column. 00:34:52.219 --> 00:34:54.889 Elimination better discover that. 00:34:54.889 --> 00:34:58.270 And elimination will also straighten out 00:34:58.270 --> 00:35:05.170 the rows, dependent rows and some independent rows. 00:35:05.170 --> 00:35:09.510 What's the, what's the row reduced echelon form for this? 00:35:09.510 --> 00:35:11.970 Let's just do it, OK. 00:35:11.970 --> 00:35:16.710 So, so that's the first pivot. 00:35:16.710 --> 00:35:20.360 Two times that away from that gives me a row of zeros. 00:35:20.360 --> 00:35:24.970 Two times that away from that gives me a zero two two. 00:35:24.970 --> 00:35:28.865 And two times that away from that gives me a zero four four. 00:35:32.300 --> 00:35:35.750 OK, first column is straight. 00:35:35.750 --> 00:35:37.780 First variable is a pivot variable. 00:35:37.780 --> 00:35:39.010 No problem. 00:35:39.010 --> 00:35:40.370 On to the second column. 00:35:40.370 --> 00:35:43.880 I look at the second pivot, it's a zero. 00:35:43.880 --> 00:35:45.430 I look below it. 00:35:45.430 --> 00:35:46.740 There's a two. 00:35:46.740 --> 00:35:47.890 OK, I do a row exchange. 00:35:50.530 --> 00:35:53.630 So this zero is now there. 00:35:53.630 --> 00:35:58.710 I now have a perfectly good pivot, and I use it. 00:35:58.710 --> 00:36:03.030 OK, and I subtract two of that row away from this row. 00:36:03.030 --> 00:36:05.630 All right if I do it like that? 00:36:05.630 --> 00:36:08.310 I've got to the form U now. 00:36:08.310 --> 00:36:10.460 This was my A. 00:36:10.460 --> 00:36:17.260 Now there's my U. I can see now -- 00:36:17.260 --> 00:36:18.630 I have to stop, right? 00:36:18.630 --> 00:36:20.340 I would go on to the third column. 00:36:20.340 --> 00:36:22.650 I should have tried. 00:36:22.650 --> 00:36:24.120 I quit, but without trying. 00:36:24.120 --> 00:36:25.440 I shouldn't have done that. 00:36:25.440 --> 00:36:28.810 On to the third column, look at the pivot position. 00:36:28.810 --> 00:36:30.100 It's got a zero in it. 00:36:30.100 --> 00:36:32.300 Look below, all zeros. 00:36:32.300 --> 00:36:35.660 Now I'm entitled to stop, OK. 00:36:35.660 --> 00:36:37.445 So the rank is two again. 00:36:45.860 --> 00:36:48.440 What about the null space? 00:36:48.440 --> 00:36:52.450 How many special solutions are there this time? 00:36:52.450 --> 00:36:56.990 How many special solutions for this matrix? 00:36:56.990 --> 00:36:59.600 I've got -- and which are the free variables and which are 00:36:59.600 --> 00:37:01.400 the pivot variables and so on? 00:37:01.400 --> 00:37:04.610 Pivot columns, I've got two pivot columns, 00:37:04.610 --> 00:37:06.890 and that's no accident. 00:37:06.890 --> 00:37:11.200 The number of pivot columns for a matrix A, that's 00:37:11.200 --> 00:37:16.120 a great fact, that the number of pivot columns for A 00:37:16.120 --> 00:37:19.250 and A transpose are the same. 00:37:19.250 --> 00:37:21.680 And then I have a free column. 00:37:21.680 --> 00:37:24.240 There's a free column. 00:37:24.240 --> 00:37:30.330 One free column, because the count is three minus two. 00:37:30.330 --> 00:37:33.990 Three minus two gives me one free column. 00:37:37.760 --> 00:37:47.410 OK, so now let me solve, what's in the null space. 00:37:47.410 --> 00:37:49.390 OK, so how do I -- 00:37:49.390 --> 00:37:50.530 let's see. 00:37:50.530 --> 00:37:52.800 These vectors have length three. 00:37:52.800 --> 00:37:54.480 They only have three components. 00:37:54.480 --> 00:37:58.200 I'm making too much space for the, to write x. 00:37:58.200 --> 00:38:03.320 x has just got three components, and what are they? 00:38:03.320 --> 00:38:06.340 I'm looking for the null space. 00:38:09.660 --> 00:38:12.430 OK, so how do I start? 00:38:12.430 --> 00:38:17.710 I give the free variable some convenient value. 00:38:17.710 --> 00:38:20.290 And what's that? 00:38:20.290 --> 00:38:23.000 I set it to one. 00:38:23.000 --> 00:38:24.970 I set the free variable to one. 00:38:24.970 --> 00:38:28.520 If I set the free variable to zero and solve 00:38:28.520 --> 00:38:33.300 for the pivot variables, I'll get all zeros: no progress. 00:38:33.300 --> 00:38:36.430 But by setting the free variable to one -- 00:38:36.430 --> 00:38:39.680 you see w- my two equations now are -- 00:38:39.680 --> 00:38:45.230 my equations are x1+ 2x2+ 3 x3 is zero, 00:38:45.230 --> 00:38:47.200 that's my first equation. 00:38:47.200 --> 00:38:51.130 And my second equation is now 2x2+2x3 equals zero. 00:38:51.130 --> 00:38:56.110 And, OK. 00:38:56.110 --> 00:39:02.400 So if x3 is one, then x2 is minus one. 00:39:02.400 --> 00:39:07.430 And if x3 is one and x2 is minus one, then maybe x1 00:39:07.430 --> 00:39:08.510 is minus one. 00:39:12.080 --> 00:39:14.360 And actually I go back to check now. 00:39:14.360 --> 00:39:17.410 I don't, like -- 00:39:17.410 --> 00:39:19.410 I did a quick calculation mentally. 00:39:19.410 --> 00:39:21.570 Can I mentally do a quick check? 00:39:21.570 --> 00:39:24.180 That matrix, that solution x says 00:39:24.180 --> 00:39:28.710 that minus this column minus this column plus this one 00:39:28.710 --> 00:39:31.560 is the zero column. 00:39:31.560 --> 00:39:32.860 And it is. 00:39:32.860 --> 00:39:35.510 Minus that minus that plus that is zero. 00:39:35.510 --> 00:39:37.720 So that's in the null space. 00:39:37.720 --> 00:39:40.790 And now you can tell me what else is in the null space. 00:39:40.790 --> 00:39:44.080 What's, what's the whole null space now? 00:39:44.080 --> 00:39:46.570 I multiply by C, right. 00:39:46.570 --> 00:39:51.720 The whole null space is a line. 00:39:51.720 --> 00:39:52.970 So that's the description. 00:39:52.970 --> 00:39:57.520 You know, if I ask you on a homework or a quiz or the final 00:39:57.520 --> 00:40:01.630 what -- give me, describe, tell me the null space, 00:40:01.630 --> 00:40:04.410 find the null space of this matrix, 00:40:04.410 --> 00:40:07.340 you can take those steps. 00:40:07.340 --> 00:40:10.790 And that's the answer I'm looking for. 00:40:10.790 --> 00:40:15.190 And I'm looking for that C too, because that's telling me 00:40:15.190 --> 00:40:18.360 that you're remembering that it's a whole space and not 00:40:18.360 --> 00:40:20.640 just one vector. 00:40:20.640 --> 00:40:23.960 Later I will ask you for a basis for the null space. 00:40:23.960 --> 00:40:26.850 Then I just want this vector. 00:40:26.850 --> 00:40:28.840 But if I ask for the whole null space, 00:40:28.840 --> 00:40:31.510 then there's the whole line through that vector. 00:40:31.510 --> 00:40:36.340 OK, now one more natural thing to do with this example, right, 00:40:36.340 --> 00:40:43.300 is keep going to the reduced matrix, R. 00:40:43.300 --> 00:40:46.180 So can I push onwards to R? 00:40:46.180 --> 00:40:49.340 That should be quick, but let's just practice. 00:40:49.340 --> 00:40:54.000 Let me keep going to R. OK, so what do I do here? 00:40:54.000 --> 00:40:55.970 I subtract -- 00:40:55.970 --> 00:40:57.830 I clear out above the pivot, so I 00:40:57.830 --> 00:41:02.870 subtract that from that, that's one zero one is left, right? 00:41:02.870 --> 00:41:04.740 When I subtracted this row from this 00:41:04.740 --> 00:41:07.390 it produced a zero above this pivot. 00:41:07.390 --> 00:41:12.070 And now I want that pivot to be a one. 00:41:12.070 --> 00:41:18.160 So for the R matrix, I'll divide this equation by two, 00:41:18.160 --> 00:41:23.270 and of course these zero, zeros are great, they don't change. 00:41:23.270 --> 00:41:24.700 There's R. 00:41:24.700 --> 00:41:27.370 That's R. 00:41:27.370 --> 00:41:28.430 You see what R is? 00:41:28.430 --> 00:41:32.100 You see the identity matrix sitting up here? 00:41:32.100 --> 00:41:36.370 You see the free part F, the F part here? 00:41:36.370 --> 00:41:38.370 And you see the zeros below. 00:41:38.370 --> 00:41:42.460 This is I F zero zero. 00:41:42.460 --> 00:41:45.180 And what's the x? 00:41:45.180 --> 00:41:48.660 The x has the identity -- 00:41:48.660 --> 00:41:51.460 well, it's only a single number one, 00:41:51.460 --> 00:41:56.090 but it's the identity matrix in the free, in the free part. 00:41:56.090 --> 00:42:00.770 And what does it have in the pivot variables? 00:42:00.770 --> 00:42:03.620 What did back substitution give? 00:42:03.620 --> 00:42:06.910 It gave minus these guys. 00:42:06.910 --> 00:42:11.160 You see that what this is is any multiple of -- 00:42:11.160 --> 00:42:14.360 this is the identity there, and this is minus F here. 00:42:19.190 --> 00:42:24.710 This is our null space matrix N for this. 00:42:24.710 --> 00:42:28.490 Our, our null space matrix is the guy whose columns 00:42:28.490 --> 00:42:30.680 are the special solutions. 00:42:30.680 --> 00:42:33.570 So their free variables have the special values 00:42:33.570 --> 00:42:39.510 one and, pivot variables have minus F. 00:42:39.510 --> 00:42:42.210 So do you see, though, how the minus F just 00:42:42.210 --> 00:42:45.540 automatically shows up in the special solutions. 00:42:49.390 --> 00:42:50.389 That's it really. 00:42:50.389 --> 00:42:51.930 I don't think there's anything more I 00:42:51.930 --> 00:42:55.980 can say about A x equals zero. 00:42:55.980 --> 00:42:59.640 There's more I can say about A x equal b, 00:42:59.640 --> 00:43:03.080 but that'll be on Friday. 00:43:03.080 --> 00:43:05.370 OK, so that's, that's the null space. 00:43:05.370 --> 00:43:06.920 Thanks.