WEBVTT 00:00:05.870 --> 00:00:08.750 OK, when the camera says, we'll start. 00:00:08.750 --> 00:00:13.320 You want to give me a signal? 00:00:13.320 --> 00:00:17.290 OK, this is lecture eight in linear algebra, 00:00:17.290 --> 00:00:20.020 and this is the lecture where we completely 00:00:20.020 --> 00:00:22.820 solve linear equations. 00:00:22.820 --> 00:00:23.810 So Ax=b. 00:00:27.080 --> 00:00:29.040 That's our goal. 00:00:29.040 --> 00:00:32.159 If it has a solution. 00:00:32.159 --> 00:00:35.990 It certainly can happen that there is no solution. 00:00:35.990 --> 00:00:40.370 We have to identify that possibility by elimination. 00:00:40.370 --> 00:00:45.110 And then if there is a solution we want to find out is there 00:00:45.110 --> 00:00:48.650 only one solution or are -- is there a whole family 00:00:48.650 --> 00:00:51.190 of solutions, and then find them all. 00:00:51.190 --> 00:00:52.370 OK. 00:00:52.370 --> 00:00:57.420 Can I use as an example the same matrix 00:00:57.420 --> 00:01:01.010 that I had last time when we were 00:01:01.010 --> 00:01:02.890 looking for the null space. 00:01:02.890 --> 00:01:10.060 So the, the matrix has rows 1 2 2 2, 2 4 6 8, 00:01:10.060 --> 00:01:13.150 and the third row -- you remember the main point was 00:01:13.150 --> 00:01:21.890 the third row, 3 6 8 10, is the sum of row one plus row two. 00:01:21.890 --> 00:01:25.710 In other words, if I add those left-hand sides, 00:01:25.710 --> 00:01:28.710 I get the third left-hand side. 00:01:28.710 --> 00:01:31.530 So you can tell me right away what 00:01:31.530 --> 00:01:35.460 elimination is going to discover about the right-hand sides. 00:01:35.460 --> 00:01:41.570 What's -- there is a condition on b1, b2, 00:01:41.570 --> 00:01:44.690 and b3 for this system to have a solution. 00:01:44.690 --> 00:01:50.240 Most cases -- if I took these numbers to be one -- 00:01:50.240 --> 00:01:53.950 5, and 17, there would not be a solution. 00:01:53.950 --> 00:01:58.700 In fact, if I took those first numbers to be 1 and 5, 00:01:58.700 --> 00:02:03.600 what is the only b3 that would be OK? 00:02:03.600 --> 00:02:05.560 Six. 00:02:05.560 --> 00:02:09.860 If the left-hand -- if these left-hand sides add up to that, 00:02:09.860 --> 00:02:10.729 then B -- 00:02:10.729 --> 00:02:15.000 I need b1 plus b2 to equal b3. 00:02:15.000 --> 00:02:19.470 Let's just see how elimination discovers that. 00:02:19.470 --> 00:02:23.430 But we can see it coming, right? 00:02:23.430 --> 00:02:26.450 That if -- let me say it in other words. 00:02:26.450 --> 00:02:29.300 If some combination on the left-hand side 00:02:29.300 --> 00:02:33.090 gives all 0s then the same combination 00:02:33.090 --> 00:02:35.580 on the right-hand side must give 0. 00:02:35.580 --> 00:02:36.080 OK. 00:02:36.080 --> 00:02:43.670 So let me take that example and write down 00:02:43.670 --> 00:02:46.670 instead of copying out all the plus signs, 00:02:46.670 --> 00:02:49.830 let me write down the matrix. 00:02:49.830 --> 00:02:59.130 1 2 2 2, 2 4 6 8, and that 6 3 8 10, 00:02:59.130 --> 00:03:02.930 where the third row is the sum of the first two rows. 00:03:02.930 --> 00:03:06.700 Now how do we deal with the right-hand side? 00:03:06.700 --> 00:03:09.840 That's -- we want to do the same thing to the right-hand side 00:03:09.840 --> 00:03:12.700 that we're doing to these rows on the left side, 00:03:12.700 --> 00:03:17.920 so we just tack on the right-hand side as another 00:03:17.920 --> 00:03:20.660 vector, another column. 00:03:20.660 --> 00:03:26.220 So this is the augmented matrix. 00:03:29.640 --> 00:03:35.840 It's, it's the matrix A with the vector b tacked on. 00:03:35.840 --> 00:03:38.440 In Matlab, that's all you would need to type. 00:03:38.440 --> 00:03:39.050 OK. 00:03:39.050 --> 00:03:41.470 So we do elimination on that. 00:03:41.470 --> 00:03:43.580 Can we just do elimination quickly? 00:03:43.580 --> 00:03:46.790 The first pivot is fine, I subtract two of this 00:03:46.790 --> 00:03:49.470 away from this, three of this away from this, 00:03:49.470 --> 00:03:54.870 so I have 1 2 2 2 b1. 00:03:54.870 --> 00:04:00.840 Two of those away will give me 0 0 2 and 4, 00:04:00.840 --> 00:04:03.570 and that was b2 minus two b1. 00:04:03.570 --> 00:04:07.130 I, I have to do the same thing to that third, 00:04:07.130 --> 00:04:08.630 that last column. 00:04:08.630 --> 00:04:10.700 And then three of these away from this 00:04:10.700 --> 00:04:17.980 gave me 0 0 2 4 b3 minus three b1s. 00:04:17.980 --> 00:04:21.660 So that's the, that's elimination 00:04:21.660 --> 00:04:25.830 with the first column completed. 00:04:25.830 --> 00:04:26.750 We move on. 00:04:26.750 --> 00:04:29.570 There's the first pivot still. 00:04:29.570 --> 00:04:31.500 Here is the second pivot. 00:04:31.500 --> 00:04:34.660 We're always remembering, now, these are then 00:04:34.660 --> 00:04:36.650 going to be the pivot columns. 00:04:41.810 --> 00:04:47.640 And let me get the final result -- well, let me -- 00:04:47.640 --> 00:04:51.615 can I do it by eraser? 00:04:55.550 --> 00:05:00.680 We're capable of subtracting this row from this row, 00:05:00.680 --> 00:05:05.280 just by -- that'll knock this out completely and give me 00:05:05.280 --> 00:05:08.090 the row of 0s, and on the right-hand side, 00:05:08.090 --> 00:05:12.660 when I subtract this away from this, what do I have? 00:05:16.260 --> 00:05:22.550 I think I have b3 minus a b2, and I had minus three b1s. 00:05:22.550 --> 00:05:25.380 This is going to, it's going to be a minus a b1. 00:05:25.380 --> 00:05:28.000 Oh yeah that's exactly what I expect. 00:05:31.490 --> 00:05:34.430 So now the -- what's the last equation? 00:05:34.430 --> 00:05:38.930 The last equation, this represented by this zero row, 00:05:38.930 --> 00:05:45.750 that last equation is, says 0 equals b3 minus b2 minus b1. 00:05:45.750 --> 00:05:51.519 So that's the condition for solvability. 00:05:51.519 --> 00:05:53.310 That's the condition on the right-hand side 00:05:53.310 --> 00:05:54.450 that we expected. 00:05:54.450 --> 00:05:57.900 It says that b1+b2 has to match b3, 00:05:57.900 --> 00:06:02.350 and if our numbers happen to have been 1, 5, and 6 -- 00:06:02.350 --> 00:06:07.220 so let me take, suppose b is 1 5 6. 00:06:07.220 --> 00:06:09.920 That's an OK b. 00:06:09.920 --> 00:06:13.120 And when I do this elimination, what will I have? 00:06:13.120 --> 00:06:16.390 The b1 will still be a 1. 00:06:16.390 --> 00:06:18.950 b2 would be 5 minus 2, this would be a 3. 00:06:18.950 --> 00:06:24.860 5 -- my 6 minus 5 minus 1, this will be -- 00:06:24.860 --> 00:06:29.800 this is the main point -- this will be a 0, thanks. 00:06:29.800 --> 00:06:30.300 OK. 00:06:30.300 --> 00:06:32.720 So the last equation is OK now. 00:06:37.080 --> 00:06:41.880 And I can proceed to solve the two equations that are really 00:06:41.880 --> 00:06:44.160 there with four unknowns. 00:06:44.160 --> 00:06:48.660 OK, I, I, I want to do that, so this, this b is OK. 00:06:48.660 --> 00:06:51.700 It allows a solution. 00:06:51.700 --> 00:06:56.850 We're going to be, naturally, interested 00:06:56.850 --> 00:07:04.200 to keep track what are the conditions on b that 00:07:04.200 --> 00:07:06.550 make the equation solvable. 00:07:06.550 --> 00:07:11.020 So let me write down what we already see 00:07:11.020 --> 00:07:14.490 before I continue to solve it. 00:07:14.490 --> 00:07:17.286 Let me first -- solvability, solvability. 00:07:23.450 --> 00:07:31.015 So which -- so this is the condition on the right-hand 00:07:31.015 --> 00:07:31.515 sides. 00:07:34.170 --> 00:07:36.010 And what is that condition? 00:07:36.010 --> 00:07:39.440 This is solvability always of Ax=b. 00:07:39.440 --> 00:07:45.150 So Ax=b is solvable -- 00:07:45.150 --> 00:07:50.870 well, actually, we had an answer in the language of the column 00:07:50.870 --> 00:07:52.640 space. 00:07:52.640 --> 00:07:54.530 Can you remind me what that answer is? 00:07:54.530 --> 00:07:58.510 That, that was like our answer from earlier lecture. 00:07:58.510 --> 00:08:00.980 b had to be in the column space. 00:08:00.980 --> 00:08:10.140 Solvable if -- when -- exactly when b is in the column space 00:08:10.140 --> 00:08:13.560 of A. 00:08:13.560 --> 00:08:14.110 Right? 00:08:14.110 --> 00:08:17.840 That just says that b has to be a combination of the columns, 00:08:17.840 --> 00:08:21.890 and of course that's exactly what the equation is looking 00:08:21.890 --> 00:08:22.660 for. 00:08:22.660 --> 00:08:25.800 So that -- now I want to answer it -- 00:08:25.800 --> 00:08:30.050 the same answer but in different language. 00:08:30.050 --> 00:08:33.990 Another way to answer this -- 00:08:33.990 --> 00:08:52.980 if a combination of the rows of A gives the zero row, 00:08:52.980 --> 00:08:57.010 and this was an example where it happened, 00:08:57.010 --> 00:09:00.940 some combination of the rows of A produced the zero row -- 00:09:00.940 --> 00:09:04.560 then what's the requirement on b? 00:09:04.560 --> 00:09:07.130 Since we're going to do the same thing to both sides of all 00:09:07.130 --> 00:09:08.520 equations -- 00:09:08.520 --> 00:09:16.190 the same combination of the components of b has to give 0. 00:09:16.190 --> 00:09:16.740 Right? 00:09:16.740 --> 00:09:19.590 That's -- so if there's a combination of the rows that 00:09:19.590 --> 00:09:29.920 gives the zero row, then the same combination of the entries 00:09:29.920 --> 00:09:34.700 of b must give 0. 00:09:37.950 --> 00:09:40.475 And this isn't the zero row, that's the zero number. 00:09:43.650 --> 00:09:47.860 Tthis is another way of saying -- and it is not immediate, 00:09:47.860 --> 00:09:54.582 OK. right, that these two statements are equivalent. 00:09:54.582 --> 00:09:56.290 But somehow they must be, because they're 00:09:56.290 --> 00:09:59.961 both equivalent to the solvability of the system. 00:09:59.961 --> 00:10:00.460 OK. 00:10:00.460 --> 00:10:05.030 So we've got this, this sort of -- like question zero is, 00:10:05.030 --> 00:10:08.180 does the system have a solution? 00:10:08.180 --> 00:10:12.670 OK, I'll come back to discuss that further. 00:10:12.670 --> 00:10:17.310 Let's go forward when it does. 00:10:17.310 --> 00:10:19.470 When there is a solution. 00:10:19.470 --> 00:10:22.684 And so what's our job now? 00:10:22.684 --> 00:10:24.600 Abstractly we sit back and we say, OK, there's 00:10:24.600 --> 00:10:26.660 a solution, finished. 00:10:26.660 --> 00:10:27.570 It exists. 00:10:27.570 --> 00:10:29.450 But we want to construct it. 00:10:29.450 --> 00:10:34.500 So what's the algorithm, the sequence 00:10:34.500 --> 00:10:37.330 of steps to find the solution? 00:10:37.330 --> 00:10:38.670 That's what I -- 00:10:38.670 --> 00:10:42.130 and of course the quiz and the final, 00:10:42.130 --> 00:10:45.440 I'm going to give you a system Ax=b and I'm going to ask you 00:10:45.440 --> 00:10:48.350 for the solution, if there is one. 00:10:48.350 --> 00:10:54.430 And so this algorithm that you want to follow. 00:10:54.430 --> 00:10:58.290 OK, let's see. 00:10:58.290 --> 00:11:13.190 So what's the -- so now to find the complete solution to Ax=b. 00:11:13.190 --> 00:11:14.000 OK. 00:11:14.000 --> 00:11:17.150 Let me start by finding one solution, 00:11:17.150 --> 00:11:19.410 one particular solution. 00:11:22.030 --> 00:11:26.060 I'm expecting that I can, because my system of equations 00:11:26.060 --> 00:11:30.730 now, that last equation is zero equals zero, 00:11:30.730 --> 00:11:33.850 so that's all fine. 00:11:33.850 --> 00:11:36.720 I really have two equations -- 00:11:36.720 --> 00:11:38.910 actually I've got four unknowns, so I'm 00:11:38.910 --> 00:11:41.410 expecting to find not only a solution 00:11:41.410 --> 00:11:44.230 but a whole bunch of them. 00:11:44.230 --> 00:11:46.020 But let's just find one. 00:11:46.020 --> 00:11:50.930 So step one, a particular solution, x particular. 00:11:54.430 --> 00:11:57.010 How do I find one particular solution? 00:11:57.010 --> 00:12:00.740 Well, let me tell you how I, how I find it. 00:12:00.740 --> 00:12:02.160 So this is -- 00:12:02.160 --> 00:12:04.080 since there are lots of solutions, 00:12:04.080 --> 00:12:07.100 you could have your own way to find a particular one. 00:12:07.100 --> 00:12:10.780 But this is a pretty natural way. 00:12:10.780 --> 00:12:20.880 Set all free variables to zero. 00:12:20.880 --> 00:12:25.810 Since those free variables are the guys that can be anything, 00:12:25.810 --> 00:12:28.790 the most convenient choice is zero. 00:12:28.790 --> 00:12:37.975 And then solve Ax=b for the pivot variables. 00:12:41.170 --> 00:12:44.240 So what does that mean in this example? 00:12:44.240 --> 00:12:46.490 Which are the free variables? 00:12:46.490 --> 00:12:49.500 Which, which are the variables that we can assign freely 00:12:49.500 --> 00:12:52.500 and then there's one and only one way 00:12:52.500 --> 00:12:55.070 to find the pivot variables? 00:12:55.070 --> 00:13:01.120 They're x2 and -- so x2 is zero, because that's in a column 00:13:01.120 --> 00:13:04.280 without a pivot, the second column has no pivot. 00:13:04.280 --> 00:13:08.010 And the -- what's the other one? 00:13:08.010 --> 00:13:11.620 The fourth, x4 is zero. 00:13:11.620 --> 00:13:16.330 Because that, those are the, the free ones. 00:13:16.330 --> 00:13:18.870 Those are in the columns with no pivots. 00:13:18.870 --> 00:13:21.300 So you see what my -- so when I knock -- 00:13:21.300 --> 00:13:28.660 when x2 and x4 are zero, I'm left with the -- 00:13:28.660 --> 00:13:31.010 what I left with here? 00:13:31.010 --> 00:13:33.460 I'm just left with -- 00:13:33.460 --> 00:13:36.500 see, now I'm not using the two free 00:13:36.500 --> 00:13:37.020 columns. 00:13:37.020 --> 00:13:39.380 I'm only using the pivot columns. 00:13:39.380 --> 00:13:42.370 So I'm really left with x1 -- 00:13:42.370 --> 00:13:45.360 the first equation is just x1 and two 00:13:45.360 --> 00:13:48.720 x3s should be the right-hand side, which 00:13:48.720 --> 00:13:50.700 we picked to be a one. 00:13:50.700 --> 00:13:54.130 And the second equation is two x3s, 00:13:54.130 --> 00:13:57.735 as it happened, turned out to be, three. 00:14:02.090 --> 00:14:06.680 I just write it again here with the x2 and the x4 00:14:06.680 --> 00:14:09.420 knocked out, since we're set them to zero. 00:14:09.420 --> 00:14:14.150 And you see that we're back in the normal case of having back 00:14:14.150 --> 00:14:16.030 -- where back substitution will do it. 00:14:16.030 --> 00:14:21.640 So x3 is three halves, and then we go back up 00:14:21.640 --> 00:14:25.490 and x1 is one minus two x3. 00:14:25.490 --> 00:14:29.270 That's probably minus two. 00:14:29.270 --> 00:14:30.400 Good. 00:14:30.400 --> 00:14:34.210 So now we have the solution, x particular 00:14:34.210 --> 00:14:41.940 is the vector minus two zero three halves zero. 00:14:44.710 --> 00:14:46.790 OK, good. 00:14:46.790 --> 00:14:52.200 That's one particular solution, and we should and could plug it 00:14:52.200 --> 00:14:54.600 into the original system. 00:14:54.600 --> 00:14:57.010 Really if -- on the quiz, please, 00:14:57.010 --> 00:14:59.230 it's a good thing to do. 00:14:59.230 --> 00:15:03.650 So we did all this, these, row operations, 00:15:03.650 --> 00:15:06.960 but this is supposed to solve the original system, 00:15:06.960 --> 00:15:09.430 and I think it does. 00:15:09.430 --> 00:15:10.120 OK. 00:15:10.120 --> 00:15:14.810 So that's x particular which we've got. 00:15:14.810 --> 00:15:19.320 So that's like what's new today. 00:15:19.320 --> 00:15:23.780 The particular solution comes -- first you check that you have 00:15:23.780 --> 00:15:26.700 zero equals zero, so you're OK on the last 00:15:26.700 --> 00:15:27.920 equations. 00:15:27.920 --> 00:15:31.980 And then you set the free variables to zero, 00:15:31.980 --> 00:15:34.830 solve for the pivot variables, and you've 00:15:34.830 --> 00:15:38.500 got a particular solution, the particular solution that 00:15:38.500 --> 00:15:41.150 has zero free variables. 00:15:41.150 --> 00:15:42.040 OK. 00:15:42.040 --> 00:15:45.000 Now -- but that's only one solution, 00:15:45.000 --> 00:15:46.270 and now I'm looking for all. 00:15:49.020 --> 00:15:51.480 So how do I find the rest? 00:15:51.480 --> 00:15:58.770 The point is I can add on x -- anything out of the null space. 00:16:03.320 --> 00:16:06.190 We know how to find the vectors in the null space -- 00:16:06.190 --> 00:16:08.950 because we did it last time, but I'll remind you what we 00:16:08.950 --> 00:16:09.710 got. 00:16:09.710 --> 00:16:12.620 And then I'll add. 00:16:15.630 --> 00:16:20.650 So the final result will be that the complete solution -- 00:16:20.650 --> 00:16:23.620 this is now the complete guy -- 00:16:23.620 --> 00:16:27.980 the complete solution is this one particular solution 00:16:27.980 --> 00:16:34.930 plus any, any vector, all different vectors out 00:16:34.930 --> 00:16:37.600 of the null space. 00:16:37.600 --> 00:16:39.050 xn, OK. 00:16:39.050 --> 00:16:42.630 Well why, why this pattern, because this pattern shows up 00:16:42.630 --> 00:16:46.560 through all of mathematics, because it shows up everywhere 00:16:46.560 --> 00:16:48.690 we have linear equations. 00:16:48.690 --> 00:16:52.050 Let me just put here the, the reason. 00:16:52.050 --> 00:17:01.830 A xp, so that's x particular, so what does Ax particular give? 00:17:01.830 --> 00:17:05.410 That gives the correct right-hand side b. 00:17:05.410 --> 00:17:10.520 And what does A times an x in the null space give? 00:17:10.520 --> 00:17:11.710 Zero. 00:17:11.710 --> 00:17:17.920 So I add, and I put in parentheses. 00:17:17.920 --> 00:17:25.420 So xp plus xn is b plus zero, which is b. 00:17:25.420 --> 00:17:27.540 So -- oh, what I saying? 00:17:27.540 --> 00:17:30.450 Let me just say it in words. 00:17:30.450 --> 00:17:36.800 If I have one solution, I can add on anything 00:17:36.800 --> 00:17:40.600 in the null space, because anything in the null space 00:17:40.600 --> 00:17:43.910 has a zero right-hand side, and I still 00:17:43.910 --> 00:17:46.670 have the correct right-hand side B. 00:17:46.670 --> 00:17:47.850 So that's my system. 00:17:47.850 --> 00:17:50.290 That's my complete solution. 00:17:50.290 --> 00:17:54.620 Now let me write out what that will be for this example. 00:17:54.620 --> 00:18:02.070 So in this example, x general, x complete, 00:18:02.070 --> 00:18:07.440 the complete solution, is x particular, 00:18:07.440 --> 00:18:12.230 which is minus two zero three halves zero, 00:18:12.230 --> 00:18:15.900 with those zeroes in the free variable, plus -- 00:18:15.900 --> 00:18:18.410 you remember there were the special solutions in the null 00:18:18.410 --> 00:18:21.680 space that had a one in the free variables -- 00:18:21.680 --> 00:18:24.220 or one and zero in the free variables, 00:18:24.220 --> 00:18:29.880 and then we filled in to find I've forgotten what they were, 00:18:29.880 --> 00:18:32.020 but maybe it was that. the others? 00:18:32.020 --> 00:18:34.010 That was a special solution, and then 00:18:34.010 --> 00:18:36.950 there was another special solution that 00:18:36.950 --> 00:18:41.820 had that free variable zero and this free variable equal one, 00:18:41.820 --> 00:18:46.260 and I have to fill those in. 00:18:46.260 --> 00:18:48.420 Let's see, can I remember how those fill in? 00:18:48.420 --> 00:18:51.570 Maybe this was a minus two and this was a two, 00:18:51.570 --> 00:18:53.070 possibly? 00:18:53.070 --> 00:18:57.700 I think probably that's right. 00:18:57.700 --> 00:18:59.290 I'm not -- yeah. 00:18:59.290 --> 00:19:05.230 Does that look write to you? 00:19:05.230 --> 00:19:08.480 I would have to remember what are my equations. 00:19:08.480 --> 00:19:11.930 Can I, rather than go way back to that board, 00:19:11.930 --> 00:19:14.680 let me remember the first equation was 00:19:14.680 --> 00:19:19.520 two x3 plus two x4 equaling zero now, 00:19:19.520 --> 00:19:22.450 because I'm looking for the guys in the null space. 00:19:22.450 --> 00:19:28.510 So I set x4 to be one and the second equation, 00:19:28.510 --> 00:19:32.850 that I didn't copy again, gave me minus two for this and then 00:19:32.850 --> 00:19:35.090 -- yeah, so I think that's right. 00:19:35.090 --> 00:19:40.131 Two minus four and two gives zero, check. 00:19:40.131 --> 00:19:40.630 OK. 00:19:40.630 --> 00:19:43.830 Those were the special solutions. 00:19:43.830 --> 00:19:46.060 What do we do to get the complete solution? 00:19:49.860 --> 00:19:52.270 How do I get the complete solution now? 00:19:52.270 --> 00:19:57.020 I multiply this by anything, c1, say, 00:19:57.020 --> 00:19:58.930 and I multiply this by anything -- 00:19:58.930 --> 00:20:00.960 I take any combination. 00:20:00.960 --> 00:20:04.140 Remember that's how we described the null space? 00:20:04.140 --> 00:20:09.060 The null space consists of all combinations of -- 00:20:09.060 --> 00:20:10.760 so this is xn -- 00:20:10.760 --> 00:20:15.430 all combinations of the special solutions. 00:20:15.430 --> 00:20:18.410 There were two special solutions because there 00:20:18.410 --> 00:20:20.730 were two free variables. 00:20:20.730 --> 00:20:24.560 And we want to make that count -- 00:20:24.560 --> 00:20:26.660 carefully now. 00:20:26.660 --> 00:20:27.857 Just while I'm up here. 00:20:27.857 --> 00:20:30.190 So there's, that's what the -- that's the kind of answer 00:20:30.190 --> 00:20:31.230 I'm looking for. 00:20:31.230 --> 00:20:34.810 Is there a constant multiplying this guy? 00:20:34.810 --> 00:20:38.790 Is there a free constant that multiplies x particular? 00:20:38.790 --> 00:20:40.180 No way. 00:20:40.180 --> 00:20:44.430 Right? x particular solves A xp=b. 00:20:44.430 --> 00:20:47.450 I'm not allowed to multiply that by three. 00:20:47.450 --> 00:20:51.120 But Axn, I'm allowed to multiply xn by three, 00:20:51.120 --> 00:20:56.790 or add to another xn, because I keep getting zero on the right. 00:20:56.790 --> 00:20:57.290 OK. 00:20:57.290 --> 00:21:02.250 So, so again, xp is one particular guy. 00:21:02.250 --> 00:21:04.630 xn is a whole subspace. 00:21:04.630 --> 00:21:05.590 Right? 00:21:05.590 --> 00:21:09.380 It's one guy plus, plus anything from a subspace. 00:21:09.380 --> 00:21:11.120 Let me draw it. 00:21:11.120 --> 00:21:14.740 Let me try to -- oh. 00:21:14.740 --> 00:21:19.950 I want to draw, I want to graph all this -- 00:21:19.950 --> 00:21:25.910 I want to, I want to plot all solutions. 00:21:25.910 --> 00:21:29.020 Now x. 00:21:29.020 --> 00:21:32.850 So what dimension I in? 00:21:32.850 --> 00:21:35.950 This is a unfortunate point. 00:21:35.950 --> 00:21:38.650 How many components does x have? 00:21:38.650 --> 00:21:39.150 Four. 00:21:39.150 --> 00:21:40.280 There are four unknowns. 00:21:40.280 --> 00:21:47.150 So I have to draw a four dimensional picture on this MIT 00:21:47.150 --> 00:21:48.450 cheap blackboard. 00:21:48.450 --> 00:21:48.950 OK. 00:21:48.950 --> 00:21:50.970 So here we go. 00:21:50.970 --> 00:21:58.080 x1 -- Einstein could do it, but, this, this is -- 00:21:58.080 --> 00:22:06.090 those are four perpendicular axes in -- 00:22:06.090 --> 00:22:08.740 representing four dimensional space. 00:22:08.740 --> 00:22:09.720 OK. 00:22:09.720 --> 00:22:12.520 Where are my solutions? 00:22:12.520 --> 00:22:16.580 Do my solutions form a subspace? 00:22:16.580 --> 00:22:20.470 Does the set of solutions to Ax=b form a subspace? 00:22:20.470 --> 00:22:21.900 No way. 00:22:21.900 --> 00:22:23.890 What does it actually look like, though? 00:22:23.890 --> 00:22:26.530 A subspace is in this picture. 00:22:26.530 --> 00:22:30.250 This part is a subspace, right? 00:22:30.250 --> 00:22:33.060 That part is some, like, two dimensional, 00:22:33.060 --> 00:22:35.860 because I've got two parameters, so it's -- 00:22:35.860 --> 00:22:41.120 I'm thinking of this null space as a two dimensional subspace 00:22:41.120 --> 00:22:42.890 inside R^4. 00:22:42.890 --> 00:22:46.410 Now I have to tell you and will tell you next time, 00:22:46.410 --> 00:22:49.760 what does it mean to say a subspace, what's the dimension 00:22:49.760 --> 00:22:50.580 of a subspace. 00:22:50.580 --> 00:22:52.680 But you see what it's going to be. 00:22:52.680 --> 00:22:58.180 It's the number of free independent constants 00:22:58.180 --> 00:22:59.750 that we can choose. 00:22:59.750 --> 00:23:03.560 So somehow there'll be a two dimensional subspace, not 00:23:03.560 --> 00:23:07.860 a line, and not a three dimensional plane, but only 00:23:07.860 --> 00:23:10.110 a two dimensional guy. 00:23:10.110 --> 00:23:12.710 But it's doesn't go through the origin 00:23:12.710 --> 00:23:15.600 because it goes through this point. 00:23:15.600 --> 00:23:17.350 So there's x particular. 00:23:17.350 --> 00:23:19.970 x particular is somewhere here. 00:23:19.970 --> 00:23:21.580 x particular. 00:23:21.580 --> 00:23:25.750 So it's somehow a subspace -- can I try to draw it that way? 00:23:28.950 --> 00:23:36.070 It's a two dimensional subspace that goes through x particular 00:23:36.070 --> 00:23:39.970 and then onwards by -- so there's x particular, 00:23:39.970 --> 00:23:44.090 and I added on xn, and there's x. 00:23:44.090 --> 00:23:46.420 There's x=xp+xn. 00:23:46.420 --> 00:23:51.430 But the xn was anywhere in this subspace, 00:23:51.430 --> 00:23:56.700 so that filled out a plane. 00:23:56.700 --> 00:23:58.930 It's a subspace -- 00:23:58.930 --> 00:24:02.320 it's not a subspace, what I saying? 00:24:02.320 --> 00:24:05.220 It's like a flat thing, it's like a subspace, 00:24:05.220 --> 00:24:08.450 but it's been shifted, away from the origin. 00:24:08.450 --> 00:24:11.450 It doesn't contain zero. 00:24:11.450 --> 00:24:12.200 Thanks. 00:24:12.200 --> 00:24:13.150 OK. 00:24:13.150 --> 00:24:16.340 That's the picture, and that's the algorithm. 00:24:16.340 --> 00:24:20.940 So the algorithm is just go through elimination 00:24:20.940 --> 00:24:25.150 and, find the particular solution, 00:24:25.150 --> 00:24:27.290 and then find those special solutions. 00:24:27.290 --> 00:24:30.010 You can do that. 00:24:30.010 --> 00:24:34.620 Let me take our time here in the lecture to think, 00:24:34.620 --> 00:24:39.240 about the bigger picture. 00:24:39.240 --> 00:24:43.030 So let me think about -- 00:24:43.030 --> 00:24:46.050 so this is my pattern. 00:24:46.050 --> 00:24:47.040 Now I want to think -- 00:24:47.040 --> 00:24:54.410 I want to ask you about a question -- 00:24:54.410 --> 00:24:57.260 I want to ask you some questions. 00:24:57.260 --> 00:25:01.070 So when I mean think bigger, I mean I'll think about an m 00:25:01.070 --> 00:25:09.495 by n matrix A of rank r. 00:25:12.640 --> 00:25:13.140 OK. 00:25:15.820 --> 00:25:17.630 What's our definition of rank? 00:25:17.630 --> 00:25:22.910 Our current definition of rank is number of pivots. 00:25:22.910 --> 00:25:23.410 OK. 00:25:23.410 --> 00:25:26.400 First of all, how are these numbers related? 00:25:26.400 --> 00:25:31.050 Can you tell me a relation between r and m? 00:25:31.050 --> 00:25:35.860 If I have m rows in the matrix and R pivots, -- 00:25:35.860 --> 00:25:42.240 then I certainly know, always -- 00:25:42.240 --> 00:25:46.890 what relation do I know between r and m? 00:25:46.890 --> 00:25:49.810 r is less or equal, right? 00:25:49.810 --> 00:25:53.620 Because I've got m rows, I can't have more than m pivots, 00:25:53.620 --> 00:25:56.720 I might have m and I might have fewer. 00:25:56.720 --> 00:26:01.980 Also, I've got n columns. 00:26:01.980 --> 00:26:04.630 So what's the relation between r and n? 00:26:04.630 --> 00:26:10.480 It's the same, less or equal, because a column 00:26:10.480 --> 00:26:14.150 can't have more than one pivot. 00:26:14.150 --> 00:26:17.450 So I can't have more than n pivots altogether. 00:26:17.450 --> 00:26:18.740 OK, OK. 00:26:18.740 --> 00:26:22.360 So I have an m by n matrix of rank r. 00:26:22.360 --> 00:26:25.420 And I always know r less than or equal to m, r less than 00:26:25.420 --> 00:26:26.560 or equal to n. 00:26:26.560 --> 00:26:29.870 Now I'm specially interested in the case 00:26:29.870 --> 00:26:35.540 of full rank, when the rank r is as big as it can be. 00:26:35.540 --> 00:26:40.840 Well, I guess I've got two separate possibilities here, 00:26:40.840 --> 00:26:44.310 depending on what these numbers m and n are. 00:26:44.310 --> 00:26:49.970 So let me talk about the case of full column rank. 00:26:53.200 --> 00:26:54.958 And by that I mean r=n. 00:27:02.330 --> 00:27:11.960 And I want to ask you, what does that imply about our solutions? 00:27:11.960 --> 00:27:16.190 What does that tell us about the null space? 00:27:16.190 --> 00:27:21.240 What does that tell us about, the complete solution? 00:27:21.240 --> 00:27:22.860 OK, so what does that mean? 00:27:22.860 --> 00:27:28.520 So I want to ask you, well, OK, if the rank is 00:27:28.520 --> 00:27:31.580 n, what does that mean? 00:27:31.580 --> 00:27:35.570 That means there's a pivot in every column. 00:27:35.570 --> 00:27:39.280 So how many pivot variables are there? 00:27:39.280 --> 00:27:41.160 n. 00:27:41.160 --> 00:27:45.150 All the columns have pivots in this case. 00:27:45.150 --> 00:27:48.150 So how many free variables are there? 00:27:48.150 --> 00:27:50.460 None at all. 00:27:50.460 --> 00:27:52.770 So no free variables. 00:27:52.770 --> 00:27:54.770 r=n, no free variables. 00:27:57.820 --> 00:28:00.270 So what does that tell us about what's 00:28:00.270 --> 00:28:04.610 going to happen then in our, in our little algorithms? 00:28:04.610 --> 00:28:07.440 What will be in the null space? 00:28:07.440 --> 00:28:13.570 The null space of A has got what in it? 00:28:13.570 --> 00:28:15.990 Only the zero vector. 00:28:15.990 --> 00:28:20.590 There are no free variables to give other values to. 00:28:20.590 --> 00:28:23.780 So the null space is only the zero vector. 00:28:29.770 --> 00:28:33.580 And what about our solution to Ax=b? 00:28:33.580 --> 00:28:38.510 Solution to Ax=b? 00:28:38.510 --> 00:28:41.790 What, what's the story on that one? 00:28:41.790 --> 00:28:43.950 So now that's coming from today's lecture. 00:28:47.270 --> 00:28:51.020 The solution x is -- 00:28:51.020 --> 00:28:52.290 what's the complete solution? 00:28:55.920 --> 00:28:59.570 It's just x particular, right? 00:28:59.570 --> 00:29:02.830 If, if, if there is an x, if there is a solution. 00:29:02.830 --> 00:29:05.190 It's x equal x particular. 00:29:05.190 --> 00:29:08.900 There's nothing -- you know, there's just one solution. 00:29:08.900 --> 00:29:11.050 If there's one at all. 00:29:11.050 --> 00:29:13.920 So it's unique solution -- 00:29:13.920 --> 00:29:16.900 unique means only one -- 00:29:16.900 --> 00:29:22.945 unique solution if it exists, if it exists. 00:29:26.430 --> 00:29:29.880 In other words, I would say -- let me put it a different way. 00:29:29.880 --> 00:29:32.910 There're either zero or one solutions. 00:29:38.940 --> 00:29:40.920 This is all in this case r=n. 00:29:45.140 --> 00:29:50.400 So I'm -- because many, many applications in reality, 00:29:50.400 --> 00:29:55.693 the columns will be what I'll later call independent. 00:29:58.710 --> 00:30:04.340 And we'll have, nothing to look for in the null space, 00:30:04.340 --> 00:30:06.611 and we'll only have particular solutions. 00:30:06.611 --> 00:30:07.110 OK. 00:30:09.970 --> 00:30:13.500 Everybody see that possibility? 00:30:13.500 --> 00:30:15.920 But I need an example, right? 00:30:15.920 --> 00:30:18.590 So let me create an example. 00:30:18.590 --> 00:30:23.170 What sort of a matrix -- what's the shape of a matrix that has 00:30:23.170 --> 00:30:25.390 full column rank? 00:30:25.390 --> 00:30:30.140 So can I squeeze in an, an example here? 00:30:30.140 --> 00:30:35.080 If it exists. 00:30:35.080 --> 00:30:38.150 Let me put in an example, and it's just the right space 00:30:38.150 --> 00:30:41.140 to put in an example. 00:30:41.140 --> 00:30:45.330 Because the example will be like tall and thin. 00:30:45.330 --> 00:30:47.940 It will have -- 00:30:47.940 --> 00:30:54.140 well, I mean, here's an example, one two six five, three one 00:30:54.140 --> 00:30:55.130 one one. 00:30:55.130 --> 00:30:56.610 Brilliant example. 00:30:56.610 --> 00:30:57.590 OK. 00:30:57.590 --> 00:31:06.039 So there's a matrix A, and what's its rank? 00:31:06.039 --> 00:31:07.330 What's the rank of that matrix? 00:31:10.000 --> 00:31:12.235 How many pivots will I find if I do elimination? 00:31:14.810 --> 00:31:15.810 Two, right? 00:31:15.810 --> 00:31:16.990 Two. 00:31:16.990 --> 00:31:20.320 I see a pivot there -- 00:31:20.320 --> 00:31:23.730 oh certainly those two columns are headed off 00:31:23.730 --> 00:31:26.810 in different directions. 00:31:26.810 --> 00:31:29.870 When I do elimination, I'll certainly get another pivot 00:31:29.870 --> 00:31:35.270 here, fine, and I can use those to clean out below and above. 00:31:35.270 --> 00:31:43.960 So the -- actually, tell me what its row reduced row echelon 00:31:43.960 --> 00:31:45.360 form would be. 00:31:45.360 --> 00:31:49.410 Can you carry that, that elimination 00:31:49.410 --> 00:31:52.360 process to the bitter end? 00:31:52.360 --> 00:31:54.590 So what do, what does that mean? 00:31:54.590 --> 00:31:57.610 I subtract a multiple of this row from these rows. 00:31:57.610 --> 00:32:00.910 So I clean up, all zeros there. 00:32:00.910 --> 00:32:02.540 Then I've got some pivot here. 00:32:02.540 --> 00:32:04.140 What do I do with that? 00:32:04.140 --> 00:32:07.330 I go subtract it below and above, 00:32:07.330 --> 00:32:11.860 and then I divide through, and what's R for that example? 00:32:11.860 --> 00:32:14.450 Maybe I can -- you'll allow me to put that just here 00:32:14.450 --> 00:32:16.580 in the next board. 00:32:16.580 --> 00:32:21.150 What's the row reduced echelon form, just out of practice, 00:32:21.150 --> 00:32:25.300 for that matrix? 00:32:25.300 --> 00:32:28.800 It's got ones in the pivots. 00:32:28.800 --> 00:32:31.750 It's got the identity matrix, a little two by two identity 00:32:31.750 --> 00:32:34.270 matrix, and below it all zeros. 00:32:37.850 --> 00:32:43.940 That's a matrix that really has two independent rows, 00:32:43.940 --> 00:32:45.510 and they're the first two, actually. 00:32:45.510 --> 00:32:47.280 The first two rows are independent. 00:32:47.280 --> 00:32:49.190 They're not in the same direction. 00:32:49.190 --> 00:32:52.660 But the other rows are combinations of the first two, 00:32:52.660 --> 00:32:55.970 so -- 00:32:55.970 --> 00:32:59.850 is there always a solution to Ax=b? 00:32:59.850 --> 00:33:02.050 Tell me what's the picture here? 00:33:02.050 --> 00:33:06.880 For this matrix A, this is a case of full column rank. 00:33:06.880 --> 00:33:11.320 The two columns are -- give two pivots. 00:33:11.320 --> 00:33:13.090 There's nothing in the null space. 00:33:13.090 --> 00:33:15.540 There's no combination of those columns 00:33:15.540 --> 00:33:19.001 that gives the zero column except the zero zero 00:33:19.001 --> 00:33:19.500 combination. 00:33:22.430 --> 00:33:25.400 So there's nothing in the null space. 00:33:25.400 --> 00:33:29.830 But is there always a solution to A X equal B? 00:33:29.830 --> 00:33:31.530 What's up with A X equal B? 00:33:34.390 --> 00:33:38.220 I've got four, four equations here, and only two Xs. 00:33:40.840 --> 00:33:42.460 So the answer is certainly no. 00:33:42.460 --> 00:33:45.240 There's not always a solution. 00:33:45.240 --> 00:33:49.660 I may have zero solutions, and if I make a random choice, 00:33:49.660 --> 00:33:51.590 I'll have zero solutions. 00:33:51.590 --> 00:33:55.690 Or if I make a great particular choice of the right-hand side, 00:33:55.690 --> 00:33:59.160 which just happens to be a combination of those two guys 00:33:59.160 --> 00:34:01.500 -- like tell me one right-hand side that would have 00:34:01.500 --> 00:34:03.540 a solution. 00:34:03.540 --> 00:34:07.190 Tell me a right-hand side that would have a solution. 00:34:07.190 --> 00:34:09.800 Well, 0 0 0 0, OK. 00:34:09.800 --> 00:34:12.880 No prize for that one. 00:34:12.880 --> 00:34:14.250 Tell me another one. 00:34:14.250 --> 00:34:18.850 Another right-hand side that has a solution would be 4 3 7 6. 00:34:18.850 --> 00:34:21.900 I could add the two columns. 00:34:21.900 --> 00:34:25.070 What would be the total complete solution if the 00:34:25.070 --> 00:34:28.360 Right? right-hand side was 4 3 7 6? 00:34:28.360 --> 00:34:31.489 There would be the particular solution one 00:34:31.489 --> 00:34:34.067 one, one of that column plus one of that, 00:34:34.067 --> 00:34:35.150 and that would be the only 00:34:35.150 --> 00:34:36.429 solution. 00:34:36.429 --> 00:34:39.770 So there would be -- x particular would be one one 00:34:39.770 --> 00:34:43.560 in the case when the right side is the sum of those two 00:34:43.560 --> 00:34:46.850 columns, and that's it. 00:34:46.850 --> 00:34:50.670 So that would be a case with one solution. 00:34:50.670 --> 00:34:51.250 OK. 00:34:51.250 --> 00:34:55.469 That, this is the typical setup with full column rank. 00:34:55.469 --> 00:35:00.000 Now I go to full row rank. 00:35:00.000 --> 00:35:04.260 You see the sort of natural symmetry of this discussion. 00:35:04.260 --> 00:35:14.523 Full row rank means r=m. 00:35:17.390 --> 00:35:19.940 So this is what I'm interested in now, r=m. 00:35:23.420 --> 00:35:24.710 OK, what's up with that? 00:35:29.830 --> 00:35:31.010 How many pivots? 00:35:31.010 --> 00:35:33.000 m. 00:35:33.000 --> 00:35:40.060 So what happens when we do elimination in that case? 00:35:40.060 --> 00:35:42.520 I'm going to get m pivots. 00:35:42.520 --> 00:35:47.520 So every row has a pivot, right? 00:35:47.520 --> 00:35:48.855 Every row has a pivot. 00:35:52.120 --> 00:35:55.950 Then what about solvability? 00:35:55.950 --> 00:35:59.880 What about this business of -- for which right-hand sides can 00:35:59.880 --> 00:36:01.120 I solve it? 00:36:01.120 --> 00:36:02.970 So that's my question. 00:36:02.970 --> 00:36:14.180 I can solve Ax=b for which right-hand sides? 00:36:14.180 --> 00:36:18.450 Do you see what's coming? 00:36:18.450 --> 00:36:23.990 I do elimination, I don't get any zero rows. 00:36:23.990 --> 00:36:26.890 So there aren't any requirements on b. 00:36:26.890 --> 00:36:29.730 I can solve Ax=b for every b. 00:36:36.450 --> 00:36:39.990 I can solve Ax=b for every right-hand side. 00:36:39.990 --> 00:36:46.705 So this is the existence, exists a solution. 00:36:49.260 --> 00:36:57.180 Now tell me, so the, u- u- so every row has a pivot in it. 00:36:57.180 --> 00:37:00.820 So how many free variables are there? 00:37:00.820 --> 00:37:04.780 How many free variables in this case? 00:37:04.780 --> 00:37:07.560 If I had n variables to start with, 00:37:07.560 --> 00:37:11.180 how many are used up by pivot variables? 00:37:11.180 --> 00:37:13.890 r, which is m. 00:37:13.890 --> 00:37:25.600 So I'm left with, left with n-r free variables. 00:37:31.050 --> 00:37:31.910 OK. 00:37:31.910 --> 00:37:37.220 So this case of full row rank I can always solve, 00:37:37.220 --> 00:37:41.440 and then this tells me how many variables are free, 00:37:41.440 --> 00:37:43.400 and this is of course n-m. 00:37:43.400 --> 00:37:48.040 This is n-m free variables. 00:37:48.040 --> 00:37:48.980 Can I do an example? 00:37:52.310 --> 00:37:54.750 You know, the best way for me to do an example is just 00:37:54.750 --> 00:37:58.140 to transpose that example. 00:37:58.140 --> 00:38:01.950 So let me take, let me take that matrix that had column one 00:38:01.950 --> 00:38:05.970 two six five and make it a row. 00:38:05.970 --> 00:38:11.470 And let me take three one one one as the second row. 00:38:11.470 --> 00:38:18.700 And let me ask you, this is my matrix A, what's its rank? 00:38:18.700 --> 00:38:20.230 What's the rank of that matrix? 00:38:20.230 --> 00:38:24.560 Sorry to ask, but not sorry really, 00:38:24.560 --> 00:38:27.130 because we're just getting the idea of rank. 00:38:27.130 --> 00:38:29.770 What's the rank of that matrix? 00:38:29.770 --> 00:38:32.260 Two, exactly, two. 00:38:32.260 --> 00:38:33.770 There will be two pivots. 00:38:33.770 --> 00:38:36.770 What will the row reduced echelon form be? 00:38:36.770 --> 00:38:38.850 Anybody know that one? 00:38:38.850 --> 00:38:42.230 Actually, tell me not only -- you have to tell me not only 00:38:42.230 --> 00:38:45.110 the, there'll be two pivots but which will be the pivot 00:38:45.110 --> 00:38:46.550 columns. 00:38:46.550 --> 00:38:50.060 Which columns of this matrix will be pivot columns? 00:38:50.060 --> 00:38:53.140 So the first column is fine, and then 00:38:53.140 --> 00:38:55.720 I go on to the next column, and what do I get? 00:38:55.720 --> 00:38:57.640 Do I get a second pivot out of -- 00:38:57.640 --> 00:39:00.410 will I get a second pivot in this position? 00:39:00.410 --> 00:39:01.300 Yes. 00:39:01.300 --> 00:39:07.200 So the pivots, when I get all the way to R, will be there. 00:39:07.200 --> 00:39:13.860 And here will be some numbers. 00:39:13.860 --> 00:39:18.670 This is the part that I previously called F. 00:39:18.670 --> 00:39:23.720 This is the part that -- the pivot columns in R will be 00:39:23.720 --> 00:39:25.680 the identity matrix. 00:39:25.680 --> 00:39:31.300 There are no zero rows, no zero rows, because the rank is two. 00:39:31.300 --> 00:39:34.630 But there'll be stuff over here. 00:39:34.630 --> 00:39:42.530 And that will, enter the special solutions and the null space. 00:39:42.530 --> 00:39:43.230 OK. 00:39:43.230 --> 00:39:51.840 So this is a typical matrix with r=m smaller than n. 00:39:51.840 --> 00:39:56.135 Now finally I've got a space here for r=m=n. 00:40:01.540 --> 00:40:06.190 I'm off in the corner here with the most important case of all. 00:40:06.190 --> 00:40:08.910 So what's up with this matrix? 00:40:08.910 --> 00:40:10.970 So let me give an example. 00:40:10.970 --> 00:40:15.065 OK, brilliant example, 1 2 3 1. 00:40:19.860 --> 00:40:23.628 Tell me what -- how do I describe a matrix that has rank 00:40:23.628 --> 00:40:24.127 r=m=n? 00:40:26.930 --> 00:40:32.560 So the matrix is square, right, it's a square matrix. 00:40:32.560 --> 00:40:36.350 And if I know its rank is -- it's full rank, now. 00:40:36.350 --> 00:40:39.290 I don't have to say full column rank or full row rank -- 00:40:39.290 --> 00:40:43.800 I just say full rank, because the count, column count 00:40:43.800 --> 00:40:47.130 and the row count are the same, and the rank 00:40:47.130 --> 00:40:49.040 is as big as it can be. 00:40:49.040 --> 00:40:51.090 And what kind of a matrix have I got? 00:40:53.920 --> 00:40:56.670 It's invertible. 00:40:56.670 --> 00:41:01.510 So that's exactly the invertible matrices. 00:41:01.510 --> 00:41:06.310 r=m=n means the -- what's the row echelon form, 00:41:06.310 --> 00:41:10.920 the reduced row echelon form, for an invertible matrix? 00:41:10.920 --> 00:41:14.630 For a square, nice, square, invertible matrix? 00:41:14.630 --> 00:41:17.320 It's I. 00:41:17.320 --> 00:41:18.880 Right. 00:41:18.880 --> 00:41:25.530 So you see that the, the good matrices 00:41:25.530 --> 00:41:31.580 are the ones that kind of come out trivially in R. 00:41:31.580 --> 00:41:34.270 You reduce them all the way to the identity matrix. 00:41:34.270 --> 00:41:37.900 What's the null space for this, for this matrix? 00:41:37.900 --> 00:41:41.170 Can I just hammer away with questions? 00:41:41.170 --> 00:41:43.160 What's the null space for this matrix? 00:41:45.910 --> 00:41:51.570 The null space of that matrix is the zero vector only. 00:41:51.570 --> 00:41:54.660 The zero vector only. 00:41:54.660 --> 00:41:58.530 What are the conditions to solve Ax=b? 00:41:58.530 --> 00:42:01.950 Which right-hand sides b are OK? 00:42:01.950 --> 00:42:07.560 If I want to solve Ax=b for this example, so A is this, 00:42:07.560 --> 00:42:14.140 b is b1 b2, what are the conditions on b1 and b2? 00:42:14.140 --> 00:42:17.170 None at all, right. 00:42:17.170 --> 00:42:21.850 So this is the case, this is the case where I can solve -- 00:42:21.850 --> 00:42:25.660 so I've coming back here, I can -- since the rank equals m, 00:42:25.660 --> 00:42:28.460 I can solve for every b. 00:42:28.460 --> 00:42:33.840 And since the rank is also n, there's a unique solution. 00:42:33.840 --> 00:42:36.640 Let me summarize the whole picture here. 00:42:39.210 --> 00:42:41.210 Here's the whole picture. 00:42:41.210 --> 00:42:44.780 I could have r=m=n. 00:42:44.780 --> 00:42:51.190 This is the case where this is the identity matrix. 00:42:51.190 --> 00:42:53.595 And this is the case where there is one solution. 00:42:56.790 --> 00:43:02.300 That's the square invertible chapter two case. 00:43:02.300 --> 00:43:04.420 Now we're into chapter three. 00:43:04.420 --> 00:43:08.330 We could have r=m smaller than n. 00:43:11.920 --> 00:43:13.730 Now that's what we had over there, 00:43:13.730 --> 00:43:17.610 and the row echelon form looked like the identity 00:43:17.610 --> 00:43:18.960 with some zero rows. 00:43:21.960 --> 00:43:27.565 And that was the case where there are zero or one solution. 00:43:31.420 --> 00:43:34.420 When I say solution I mean to Ax=b. 00:43:37.540 --> 00:43:39.590 So this case, there's always one. 00:43:39.590 --> 00:43:42.130 This case there's zero or one. 00:43:42.130 --> 00:43:45.640 And now let me take the case of full column rank, 00:43:45.640 --> 00:43:55.860 but some, extra rows. 00:43:55.860 --> 00:44:00.000 So now R has -- 00:44:00.000 --> 00:44:04.700 well, the identity -- 00:44:04.700 --> 00:44:08.280 I'm almost tempted to write the identity matrix and then F, 00:44:08.280 --> 00:44:10.130 but that isn't necessarily right. 00:44:15.970 --> 00:44:20.360 I have -- is that right? 00:44:20.360 --> 00:44:24.310 Am I getting this correct here? 00:44:24.310 --> 00:44:25.430 Oh, I'm not! 00:44:25.430 --> 00:44:26.390 My God! 00:44:26.390 --> 00:44:33.330 This is the case R equals n, the columns, the columns are, 00:44:33.330 --> 00:44:34.390 are OK. 00:44:34.390 --> 00:44:38.770 That's the case that was on that board, r=n, full column rank. 00:44:38.770 --> 00:44:43.240 Now I want the case where m is smaller than n 00:44:43.240 --> 00:44:46.550 and I've got extra columns. 00:44:46.550 --> 00:44:47.050 OK. 00:44:47.050 --> 00:44:47.810 There we go. 00:44:52.030 --> 00:44:57.140 So this is now the case of full row rank, 00:44:57.140 --> 00:45:02.070 and it looks like I F except that I 00:45:02.070 --> 00:45:08.430 can't be sure that the pivot columns are the first columns. 00:45:08.430 --> 00:45:14.690 So the I and the F, could be partly mixed into the I. 00:45:14.690 --> 00:45:18.670 Can I write that with just like that? 00:45:18.670 --> 00:45:23.830 So the F could be sort of partly into the I 00:45:23.830 --> 00:45:28.450 if the first columns weren't the pivot columns. 00:45:28.450 --> 00:45:31.760 Now how many solutions in this case? 00:45:31.760 --> 00:45:34.330 There's always a solution. 00:45:34.330 --> 00:45:35.850 This is the existence case. 00:45:35.850 --> 00:45:37.020 There's always a solution. 00:45:37.020 --> 00:45:39.160 We're not getting any zero rows. 00:45:39.160 --> 00:45:41.430 There are no zero rows here. 00:45:41.430 --> 00:45:45.635 So there's always either one or infinitely many solutions. 00:45:50.061 --> 00:45:50.560 OK. 00:45:53.230 --> 00:45:55.600 Actually, I guess there's always an infinite number, 00:45:55.600 --> 00:46:02.760 because we always have some null space to deal with. 00:46:02.760 --> 00:46:06.150 Then the final case is where r is smaller than m 00:46:06.150 --> 00:46:08.880 and smaller than n. 00:46:08.880 --> 00:46:09.380 OK. 00:46:09.380 --> 00:46:14.830 Now that's the case where R is the identity 00:46:14.830 --> 00:46:19.920 with some free stuff but with some zero rows too. 00:46:19.920 --> 00:46:23.580 And that's the case where there's either no solution -- 00:46:23.580 --> 00:46:29.200 because we didn't get a zero equals zero for some bs, 00:46:29.200 --> 00:46:32.330 or infinitely many solutions. 00:46:37.100 --> 00:46:38.010 OK. 00:46:38.010 --> 00:46:44.310 Do you -- this board really summarizes the lecture, 00:46:44.310 --> 00:46:47.370 and this sentence summarizes the lecture. 00:46:47.370 --> 00:46:55.070 The rank tells you everything about the number of solutions. 00:46:55.070 --> 00:46:57.010 That number, the rank r, tells you 00:46:57.010 --> 00:47:01.620 all the information except the exact entries in the solutions. 00:47:01.620 --> 00:47:04.140 For that you go to the matrix. 00:47:04.140 --> 00:47:05.150 OK, good. 00:47:05.150 --> 00:47:08.770 Have a great weekend, and I'll see you on Monday.