WEBVTT 00:00:00.000 --> 00:00:12.050 2 times this equation, Okay. 00:00:12.050 --> 00:00:13.510 This is it. 00:00:13.510 --> 00:00:17.360 The second lecture in linear algebra, 00:00:17.360 --> 00:00:25.860 and I've put below my main topics for today. 00:00:25.860 --> 00:00:28.270 I put right there a system of equations 00:00:28.270 --> 00:00:30.780 that's going to be our example to work with. 00:00:30.780 --> 00:00:35.150 But what are we going to do with it? 00:00:35.150 --> 00:00:37.540 We're going to solve it. 00:00:37.540 --> 00:00:41.900 And the method of solution will not be determinants. 00:00:41.900 --> 00:00:44.560 Determinants are something that will come later. 00:00:44.560 --> 00:00:47.930 The method we'll use is called elimination. 00:00:47.930 --> 00:00:53.630 And it's the way every software package solves equations. 00:00:57.960 --> 00:01:03.360 And elimination, well, if it succeeds, it gets the answer. 00:01:03.360 --> 00:01:05.870 And normally it does succeed. 00:01:05.870 --> 00:01:09.790 If the matrix A that's coming into that system 00:01:09.790 --> 00:01:12.460 is a good matrix, and I think this one is, 00:01:12.460 --> 00:01:14.770 then elimination will work. 00:01:14.770 --> 00:01:18.290 We'll get the answer in an efficient way. 00:01:18.290 --> 00:01:21.310 But why don't we, as long as we're sort of seeing how 00:01:21.310 --> 00:01:25.120 elimination works -- it's always good to ask how could it 00:01:25.120 --> 00:01:26.010 fail? 00:01:26.010 --> 00:01:29.180 So at the same time, we'll see how 00:01:29.180 --> 00:01:32.570 elimination decides whether the matrix is a good one 00:01:32.570 --> 00:01:35.040 or has problems. 00:01:35.040 --> 00:01:37.030 Then to complete the answer, there's 00:01:37.030 --> 00:01:40.780 an obvious step of back substitution. 00:01:40.780 --> 00:01:44.600 In fact, the idea of elimination is -- 00:01:44.600 --> 00:01:47.750 you would have thought of it, right? 00:01:47.750 --> 00:01:51.090 I mean Gauss thought of it before we did, 00:01:51.090 --> 00:01:54.140 but only because he was born earlier. 00:01:54.140 --> 00:01:58.070 It's a natural idea... 00:01:58.070 --> 00:02:00.020 and died earlier, too. 00:02:00.020 --> 00:02:07.020 Okay, and you've seen the idea. 00:02:07.020 --> 00:02:11.870 But now, the part that I want to show you is elimination 00:02:11.870 --> 00:02:16.040 expressed in matrix language, because the whole course -- 00:02:16.040 --> 00:02:21.740 all the key ideas get expressed as matrix operations, not as 00:02:21.740 --> 00:02:23.420 words. 00:02:23.420 --> 00:02:27.080 And one of the operations, of course, that we'll meet 00:02:27.080 --> 00:02:30.500 is how do we multiply matrices and why? 00:02:30.500 --> 00:02:34.150 Okay, so there's a system of equations. 00:02:34.150 --> 00:02:38.270 Three equations and three unknowns. 00:02:38.270 --> 00:02:42.700 And there's the matrix, the three by three matrix -- 00:02:42.700 --> 00:02:47.860 so this is the system Ax = b. 00:02:47.860 --> 00:02:51.260 This is our system to solve, Ax equal -- 00:02:51.260 --> 00:02:55.260 and the right-hand side is that vector 2, 12, 2. 00:02:55.260 --> 00:02:56.970 Okay. 00:02:56.970 --> 00:03:00.640 Now, when I describe elimination -- 00:03:00.640 --> 00:03:04.300 it gets to be a pain to keep writing the equal signs 00:03:04.300 --> 00:03:06.330 and the pluses and so on. 00:03:06.330 --> 00:03:10.220 It's that matrix that totally matters. 00:03:10.220 --> 00:03:12.340 Everything is in that matrix. 00:03:12.340 --> 00:03:14.820 But behind it is those equations. 00:03:14.820 --> 00:03:17.384 So what does elimination do? 00:03:17.384 --> 00:03:18.925 What's the first step of elimination? 00:03:22.680 --> 00:03:26.110 We accept the first equation, it's okay. 00:03:26.110 --> 00:03:29.220 I'm going to multiply that equation by the right number, 00:03:29.220 --> 00:03:32.980 the right multiplier and I'm going to subtract it 00:03:32.980 --> 00:03:36.220 from the second equation. 00:03:36.220 --> 00:03:37.400 With what purpose? 00:03:37.400 --> 00:03:41.530 So that will decide what the multiplier should be. 00:03:41.530 --> 00:03:50.180 Our purpose is to knock out the x part of equation two. 00:03:50.180 --> 00:03:54.980 So our purpose is to eliminate x. 00:03:54.980 --> 00:03:57.820 So what do I multiply -- 00:03:57.820 --> 00:03:59.650 and again, I'll do it with this matrix, 00:03:59.650 --> 00:04:01.590 because I can do it short. 00:04:01.590 --> 00:04:03.340 What's the multiplier here? 00:04:03.340 --> 00:04:06.770 What do I multiply -- equation one and subtract. 00:04:06.770 --> 00:04:08.810 Notice I'm saying that word subtract. 00:04:08.810 --> 00:04:10.870 I'd like to stick to that convention. 00:04:10.870 --> 00:04:13.890 I'll do a subtraction. 00:04:13.890 --> 00:04:19.790 First of all this is the key number that I'm starting with. 00:04:19.790 --> 00:04:22.170 And that's called the pivot. 00:04:22.170 --> 00:04:25.270 I'll put a box around it and write its name down. 00:04:25.270 --> 00:04:27.800 That's the first pivot. 00:04:27.800 --> 00:04:30.641 The first pivot. 00:04:30.641 --> 00:04:31.140 Okay. 00:04:31.140 --> 00:04:33.130 So I'm going to use -- 00:04:33.130 --> 00:04:35.710 that's sort of like the key number in that equation. 00:04:35.710 --> 00:04:37.260 And now what's the multiplier? 00:04:37.260 --> 00:04:39.720 So I'm going to -- 00:04:39.720 --> 00:04:45.950 my first row won't change, that's the pivot row. 00:04:45.950 --> 00:04:48.680 But I'm going to use it -- 00:04:48.680 --> 00:04:51.700 and now, finally, let me ask you what the multiplier is. 00:04:51.700 --> 00:04:52.680 Yes? 00:04:52.680 --> 00:04:58.170 3 times that first equation will knock out that 3. 00:04:58.170 --> 00:04:59.110 Okay. 00:04:59.110 --> 00:05:00.220 So what will it leave? 00:05:00.220 --> 00:05:02.000 So the multiplier is 3. 00:05:02.000 --> 00:05:05.220 3 times that will make that 0. 00:05:05.220 --> 00:05:06.720 That was our purpose. 00:05:06.720 --> 00:05:11.490 3 2s away from the 8 will leave a 2 and three 1s away from 1 00:05:11.490 --> 00:05:12.960 will leave a minus 2. 00:05:12.960 --> 00:05:15.400 And this guy didn't change. 00:05:17.980 --> 00:05:22.530 Now the next step -- this is forward elimination and that 00:05:22.530 --> 00:05:23.802 Okay. step's completed. 00:05:23.802 --> 00:05:26.260 Oh, well, you could say wait a minute, what about the right 00:05:26.260 --> 00:05:27.280 hand side? 00:05:27.280 --> 00:05:32.610 Shall I carry -- the right-hand side gets carried along. 00:05:32.610 --> 00:05:36.410 Actually MatLab finishes up with the left side before -- 00:05:36.410 --> 00:05:38.330 and then just goes back to do the right side. 00:05:38.330 --> 00:05:42.340 Maybe I'll be MatLab for a moment and do that. 00:05:42.340 --> 00:05:44.610 Okay. 00:05:44.610 --> 00:05:49.470 I'm leaving a room for a column of b, the right-hand side. 00:05:49.470 --> 00:05:51.310 But I'll fill it in later. 00:05:51.310 --> 00:05:52.360 Okay. 00:05:52.360 --> 00:05:54.680 Now the next step of elimination is what? 00:05:54.680 --> 00:05:56.850 Well, strictly speaking... 00:05:56.850 --> 00:06:05.250 this position that I cleaned up was like the 2, 1 position, 00:06:05.250 --> 00:06:07.470 row 2, column 1. 00:06:07.470 --> 00:06:10.310 So I got a 0 in the 2, 1 position. 00:06:10.310 --> 00:06:13.990 I'll use 2,1 as the index of that step. 00:06:13.990 --> 00:06:17.050 The next step should be to finish the column 00:06:17.050 --> 00:06:20.770 and get a 0 in that position. 00:06:20.770 --> 00:06:24.350 So the next step is really the 3,1 step, row three, 00:06:24.350 --> 00:06:25.330 column one. 00:06:25.330 --> 00:06:30.390 But of course, I already have 0. 00:06:30.390 --> 00:06:31.070 Okay. 00:06:31.070 --> 00:06:32.930 So the multiplier is 0. 00:06:32.930 --> 00:06:39.390 I take 0 of this equation away from this one and I'm all set. 00:06:39.390 --> 00:06:43.940 So I won't repeat that, but there was a step there which, 00:06:43.940 --> 00:06:48.750 MatLab would have to look -- it would look at this number 00:06:48.750 --> 00:06:53.370 and, do that step, unless you told it in advance that it was 00:06:53.370 --> 00:06:53.870 0. 00:06:53.870 --> 00:06:54.590 Okay. 00:06:54.590 --> 00:06:56.530 Now what? 00:06:56.530 --> 00:07:00.920 Now we can see the second pivot, which is what? 00:07:00.920 --> 00:07:04.700 The second pivot -- see, we've eliminated -- 00:07:04.700 --> 00:07:06.870 x is now gone from this equation, right? 00:07:06.870 --> 00:07:12.570 We're down to two equations in y and z. 00:07:12.570 --> 00:07:14.260 And so now I just do it again. 00:07:14.260 --> 00:07:17.270 Like, everything's very cursive at this -- this is like -- 00:07:17.270 --> 00:07:20.240 such a basic algorithm and you've seen it, 00:07:20.240 --> 00:07:25.670 but carry me through one last step. 00:07:25.670 --> 00:07:28.490 So this is still the first pivot. 00:07:28.490 --> 00:07:32.060 Now the second pivot is this guy, who has appeared there. 00:07:32.060 --> 00:07:36.620 And what's the multiplier, the appropriate multiplier now? 00:07:36.620 --> 00:07:38.130 And what's my purpose? 00:07:38.130 --> 00:07:44.560 Is it to wipe out the 3, 2 position, right? 00:07:44.560 --> 00:07:47.640 This was the 2, 1 step. 00:07:47.640 --> 00:07:53.010 And now I'm going to take the 3, 2 step. 00:07:53.010 --> 00:08:00.350 So this all stays the same, 1 2 1, 0 2 -1 00:08:00.350 --> 00:08:03.340 and the pivots are there. 00:08:03.340 --> 00:08:05.510 Now I'm using this pivot, so what's the multiplier? 00:08:10.840 --> 00:08:14.630 this row, gets subtracted from this row and makes that a 0. 00:08:14.630 --> 00:08:20.770 So it's 0, 0 and is it a 5? 00:08:20.770 --> 00:08:24.200 Yeah, I guess it's a 5, is that right? 00:08:24.200 --> 00:08:26.560 Because I have a one there and I'm 00:08:26.560 --> 00:08:32.330 subtracting twice of twice this, so I think it's a 5 there. 00:08:32.330 --> 00:08:33.370 There's the third pivot. 00:08:33.370 --> 00:08:35.874 So let me put a box around all three pivots. 00:08:41.370 --> 00:08:48.900 Is there a -- oh, did I just invent a negative one? 00:08:48.900 --> 00:08:53.800 I'm sorry that the tape can't, correct that 00:08:53.800 --> 00:08:55.220 as easily as I can. 00:08:55.220 --> 00:08:55.750 Okay. 00:08:55.750 --> 00:08:57.683 Thank you very much. 00:09:00.860 --> 00:09:03.610 You get an A in the course now. 00:09:03.610 --> 00:09:04.570 Is that correct? 00:09:07.600 --> 00:09:09.790 Is it correct now? 00:09:09.790 --> 00:09:10.810 Okay. 00:09:10.810 --> 00:09:15.830 So the three pivots are there -- 00:09:15.830 --> 00:09:19.840 I know right away a lot about this matrix. 00:09:19.840 --> 00:09:24.400 This elimination step from A -- this matrix I'm going to call 00:09:24.400 --> 00:09:27.830 U. U for upper triangular. 00:09:27.830 --> 00:09:29.690 So the whole purpose of elimination 00:09:29.690 --> 00:09:32.060 was to get from A to U. 00:09:32.060 --> 00:09:35.830 And, literally, that's the most common calculation 00:09:35.830 --> 00:09:38.580 in scientific computing. 00:09:38.580 --> 00:09:40.790 And people think of how could I do that faster? 00:09:40.790 --> 00:09:44.210 Because it's a major, major thing. 00:09:44.210 --> 00:09:47.560 But we're doing it the straightforward way. 00:09:47.560 --> 00:09:52.000 We found three pivots, and by the way, I didn't say this, 00:09:52.000 --> 00:09:54.410 pivots can't be 0. 00:09:54.410 --> 00:09:57.160 I don't accept 0 as a pivot. 00:09:57.160 --> 00:09:58.750 And I didn't get 0. 00:09:58.750 --> 00:10:00.650 So this matrix is great. 00:10:00.650 --> 00:10:03.150 It gave me three pivots, I didn't 00:10:03.150 --> 00:10:07.450 have to do anything special, I just followed the rules and, 00:10:07.450 --> 00:10:11.130 and the pivots are 1, 2 and 5. 00:10:11.130 --> 00:10:14.880 By the way, just because I always anticipate stuff from 00:10:14.880 --> 00:10:20.120 a later day, if I wanted to know the determinant of this matrix 00:10:20.120 --> 00:10:22.010 -- which I never do want to know, 00:10:22.010 --> 00:10:23.990 but I would just multiply the pivots. 00:10:23.990 --> 00:10:25.540 The determinant is 10. 00:10:25.540 --> 00:10:32.820 So even things like the determinant are here. 00:10:32.820 --> 00:10:34.960 Okay. 00:10:34.960 --> 00:10:38.410 Now -- oh, let me talk about failure for a moment, 00:10:38.410 --> 00:10:40.510 and then -- 00:10:40.510 --> 00:10:44.370 and then come back to success. 00:10:44.370 --> 00:10:47.700 How could this have failed? 00:10:47.700 --> 00:10:54.525 How could -- by fail, I mean to come up with three pivots. 00:10:58.570 --> 00:11:00.800 I mean, there are a couple of points. 00:11:00.800 --> 00:11:02.530 I would have already been in trouble 00:11:02.530 --> 00:11:06.520 if this very first number here was 0. 00:11:06.520 --> 00:11:10.610 If it was a 0 there -- suppose that had been a 0, 00:11:10.610 --> 00:11:14.760 there were no Xs in that equation -- first equation. 00:11:14.760 --> 00:11:17.040 Does that mean I can't solve the problem? 00:11:17.040 --> 00:11:19.180 Does that mean I quit? 00:11:19.180 --> 00:11:21.700 No. 00:11:21.700 --> 00:11:22.860 What do I do? 00:11:22.860 --> 00:11:24.570 I switch rows. 00:11:24.570 --> 00:11:26.010 I exchange rows. 00:11:26.010 --> 00:11:30.210 So in case of a 0, I will not say 0 pivot. 00:11:30.210 --> 00:11:33.680 I will never be heard to utter those words, 0 pivot. 00:11:33.680 --> 00:11:37.270 But if there's a 0 in the pivot position, 00:11:37.270 --> 00:11:39.930 maybe I can say that, I would try 00:11:39.930 --> 00:11:45.400 to exchange for a lower equation and get a proper pivot up 00:11:45.400 --> 00:11:46.140 there. 00:11:46.140 --> 00:11:46.640 Okay. 00:11:46.640 --> 00:11:51.020 Now, for example, this second pivot came out two. 00:11:51.020 --> 00:11:52.510 Could it have come out 0? 00:11:52.510 --> 00:11:56.110 What -- actually, if I change that 8 a little bit, 00:11:56.110 --> 00:12:01.650 I would have got a little trouble. 00:12:01.650 --> 00:12:08.210 What should I change that 8 to so that I run into trouble? 00:12:08.210 --> 00:12:09.070 A 6. 00:12:09.070 --> 00:12:14.840 If that had been a 6, then this would have been 0 00:12:14.840 --> 00:12:17.320 and I couldn't have used that as the pivot. 00:12:17.320 --> 00:12:20.230 But I could have exchanged again. 00:12:20.230 --> 00:12:21.960 In this case. 00:12:21.960 --> 00:12:26.840 In this case, because when can I get out of trouble? 00:12:26.840 --> 00:12:28.490 I can get out of trouble if there's 00:12:28.490 --> 00:12:32.580 a non-0 below this troublesome 0. 00:12:32.580 --> 00:12:34.350 And there is here. 00:12:34.350 --> 00:12:36.890 So I would be okay in this case. 00:12:36.890 --> 00:12:40.080 If this was a 6, I would survive by a row 00:12:40.080 --> 00:12:41.290 exchange. 00:12:41.290 --> 00:12:46.430 Now -- of course, it might have happened that I couldn't do 00:12:46.430 --> 00:12:49.280 the row, that -- that there was 0s below it, 00:12:49.280 --> 00:12:50.780 but here there wasn't. 00:12:50.780 --> 00:12:55.370 Now, I could also have got in trouble if this number 1 was 00:12:55.370 --> 00:12:58.540 a little different. 00:12:58.540 --> 00:13:02.260 See, that 1 became a 5, I guess, by the end. 00:13:05.040 --> 00:13:08.940 So can you see what number there would 00:13:08.940 --> 00:13:14.130 have got me trouble that I really couldn't get out of? 00:13:14.130 --> 00:13:15.710 Trouble that I couldn't get out of 00:13:15.710 --> 00:13:22.810 would mean if 0 is in the pivot position 00:13:22.810 --> 00:13:26.270 and I've got no place to exchange. 00:13:26.270 --> 00:13:31.970 So there must be some number which if I had had here 00:13:31.970 --> 00:13:34.460 it would have meant failure. 00:13:34.460 --> 00:13:36.590 Negative 4, good. 00:13:36.590 --> 00:13:39.670 If it was a negative 4 here -- if it happened to be a negative 00:13:39.670 --> 00:13:43.900 4, I'll temporarily put it up here. 00:13:43.900 --> 00:13:47.800 If this had been a negative 4 z, then I 00:13:47.800 --> 00:13:50.264 would have gone through the same steps. 00:13:50.264 --> 00:13:52.680 This would have been a minus 4, it still would have been a 00:13:52.680 --> 00:13:53.750 minus 4. 00:13:53.750 --> 00:13:58.610 But at the last minute it would have become 0. 00:13:58.610 --> 00:14:02.080 And there wouldn't have been a third pivot. 00:14:02.080 --> 00:14:04.790 The matrix would have not been invertible. 00:14:04.790 --> 00:14:08.860 Well, of course, the inverse of a matrix is coming next week, 00:14:08.860 --> 00:14:11.850 but, you've heard these words 00:14:11.850 --> 00:14:12.590 before. 00:14:12.590 --> 00:14:16.940 So, that's how we identify failure. 00:14:16.940 --> 00:14:20.680 There's temporary failure when we can do a row exchange -- 00:14:20.680 --> 00:14:24.350 and get out of it, or there's complete failure when we get 00:14:24.350 --> 00:14:26.020 a 0 and -- 00:14:26.020 --> 00:14:28.280 and there's nothing below that we can use. 00:14:28.280 --> 00:14:29.400 Okay. 00:14:29.400 --> 00:14:31.240 Let's stay with -- 00:14:31.240 --> 00:14:34.380 back to success now. 00:14:34.380 --> 00:14:39.020 In fact, I guess the next topic is back substitution. 00:14:39.020 --> 00:14:40.430 So what's back substitution? 00:14:40.430 --> 00:14:46.690 Well, now I'd better bring the right-hand side in. 00:14:46.690 --> 00:14:52.040 So what would MatLab do and what should we do? 00:14:52.040 --> 00:14:55.540 Let me bring in the right-hand side as an extra column. 00:14:55.540 --> 00:14:56.720 So there comes B. 00:14:56.720 --> 00:15:05.650 So it's 2, 12, I would call this the augmented matrix. 00:15:05.650 --> 00:15:08.270 "Augment" means you've tacked something on. 00:15:08.270 --> 00:15:11.810 I've tacked on this extra column. 00:15:11.810 --> 00:15:14.870 Because, when I'm working with equations, 00:15:14.870 --> 00:15:17.990 I do the same thing to both sides. 00:15:17.990 --> 00:15:22.940 So, at this step, I subtracted 2 of the first equation away from 00:15:22.940 --> 00:15:26.910 the second equation so that this augmented -- 00:15:26.910 --> 00:15:31.960 I even brought some colored chalk, but I don't know if it 00:15:31.960 --> 00:15:32.550 shows up. 00:15:32.550 --> 00:15:35.920 So this is like the augmented -- 00:15:35.920 --> 00:15:36.580 no! 00:15:36.580 --> 00:15:39.060 Damn, circled the wrong thing. 00:15:39.060 --> 00:15:41.020 Okay. 00:15:41.020 --> 00:15:44.020 Here is b. 00:15:44.020 --> 00:15:46.090 Okay, that's the extra column. 00:15:46.090 --> 00:15:46.590 Okay. 00:15:46.590 --> 00:15:49.380 So what happened to that extra column, 00:15:49.380 --> 00:15:51.170 the right-hand side of the equations, 00:15:51.170 --> 00:15:53.160 when I did the first step? 00:15:53.160 --> 00:15:57.320 So that was 3 of this away from this, so it took -- 00:15:57.320 --> 00:16:02.210 the 2 stayed the same, but three 2s got taken away from 12, 00:16:02.210 --> 00:16:04.900 leaving 6, and that 2 stayed the same. 00:16:04.900 --> 00:16:09.050 So this is how it's looking halfway along. 00:16:09.050 --> 00:16:11.310 And let me just carry to the end. 00:16:11.310 --> 00:16:16.350 The 2 and the 6 stay the same, but -- 00:16:16.350 --> 00:16:19.110 what do I have here? 00:16:19.110 --> 00:16:19.790 Oh, gosh. 00:16:23.090 --> 00:16:23.880 Help me out, now. 00:16:23.880 --> 00:16:26.410 What -- so now I'm -- 00:16:26.410 --> 00:16:29.560 This is still like forward elimination. 00:16:29.560 --> 00:16:32.430 I got to this point, which I think is right, 00:16:32.430 --> 00:16:34.420 and now what did I do at this step? 00:16:34.420 --> 00:16:38.250 I multiplied that pivot by 2 or that whole equation by 2 00:16:38.250 --> 00:16:40.430 and subtracted from that, so I think 00:16:40.430 --> 00:16:43.620 I take two 6s, which is 12, away from the 2. 00:16:43.620 --> 00:16:49.630 Do you think minus 10 is my final right-hand side -- 00:16:49.630 --> 00:16:51.940 the right-hand side that goes with U, and let me 00:16:51.940 --> 00:16:56.820 call that once and forever the vector c. 00:16:56.820 --> 00:17:04.060 So c is what happens to b, and U is what happens to A. 00:17:04.060 --> 00:17:04.640 Okay. 00:17:04.640 --> 00:17:09.050 There you've seen elimination clean. 00:17:09.050 --> 00:17:11.319 Okay. 00:17:11.319 --> 00:17:14.300 Oh, what's back substitution? 00:17:14.300 --> 00:17:16.810 So what are my final equations, then? 00:17:16.810 --> 00:17:20.200 Can I copy these equations? 00:17:20.200 --> 00:17:34.650 x+2y+z=2 is still there and 2y-2z=6 is there, and 5z=-10. 00:17:34.650 --> 00:17:36.250 Okay. 00:17:36.250 --> 00:17:40.120 Those are the equations that these numbers 00:17:40.120 --> 00:17:42.000 are telling me about. 00:17:42.000 --> 00:17:46.960 Those are the equations U x equals c. 00:17:46.960 --> 00:17:50.390 Okay, how do I solve them? 00:17:50.390 --> 00:17:52.780 What one do I solve for first? 00:17:52.780 --> 00:17:54.280 z. 00:17:54.280 --> 00:17:59.550 I see immediately that the correct value of z is negative 00:17:59.550 --> 00:18:02.860 And what do I do next? 00:18:02.860 --> 00:18:04.020 I go back upwards. 00:18:04.020 --> 00:18:07.100 I now know z here. 00:18:07.100 --> 00:18:12.360 So, if z is negative 2, that's 4 there, is that right? 00:18:12.360 --> 00:18:17.910 And so 2 y plus a 4 is 6, maybe y is 1. 00:18:17.910 --> 00:18:21.070 Going -- this is back substitution. 00:18:21.070 --> 00:18:23.660 We're doing it on the fly because it's so easy. 00:18:23.660 --> 00:18:27.120 And then x is -- 00:18:27.120 --> 00:18:31.810 so x -- 2y is 2 minus 2, maybe x is 2? 00:18:38.540 --> 00:18:41.580 So you see what back substitution is. 00:18:41.580 --> 00:18:46.160 It's the simple step solving the equations in reverse order 00:18:46.160 --> 00:18:49.350 because the system is triangular. 00:18:49.350 --> 00:18:50.340 Okay. 00:18:50.340 --> 00:18:52.010 Good. 00:18:52.010 --> 00:18:54.860 So that's elimination and back substitution, 00:18:54.860 --> 00:18:57.620 and I kept the right-hand side along. 00:18:57.620 --> 00:19:00.360 Okay, now what do I -- 00:19:00.360 --> 00:19:04.060 that, like, is first piece of the 00:19:04.060 --> 00:19:05.330 lecture. 00:19:05.330 --> 00:19:08.780 What's the second piece? 00:19:08.780 --> 00:19:11.530 Matrices are going to get in. 00:19:11.530 --> 00:19:17.640 So I wrote stuff with x, y-s and z-s in there, then I really, 00:19:17.640 --> 00:19:24.060 got the right shorthand, just writing the matrix entries, 00:19:24.060 --> 00:19:27.980 and now I want to write the operations 00:19:27.980 --> 00:19:32.080 that I did in matrices, right? 00:19:32.080 --> 00:19:34.190 I've carried the matrices along, but I 00:19:34.190 --> 00:19:42.860 haven't said the operation those elimination steps, 00:19:42.860 --> 00:19:45.251 I now want to express as matrices. 00:19:45.251 --> 00:19:45.750 Okay. 00:19:45.750 --> 00:19:46.460 Here they come. 00:19:49.570 --> 00:19:51.466 So now this is elimination matrices. 00:19:55.280 --> 00:19:57.500 Okay. 00:19:57.500 --> 00:20:02.700 Let me take that first step, which took me from 1 2 1 3 8 1 00:20:02.700 --> 00:20:06.140 0 4 1. 00:20:09.970 --> 00:20:12.310 I want to operate on that -- 00:20:12.310 --> 00:20:16.260 I want to do elimination on that. 00:20:16.260 --> 00:20:17.380 Okay. 00:20:17.380 --> 00:20:21.170 Okay, now I'm remembering a point 00:20:21.170 --> 00:20:29.390 I want to single out as especially important. 00:20:29.390 --> 00:20:32.953 Let me move the board up for that. 00:20:36.600 --> 00:20:40.020 Because when we do matrix operations, we've got to, like, 00:20:40.020 --> 00:20:41.860 be able to see the big picture. 00:20:41.860 --> 00:20:42.360 Okay. 00:20:42.360 --> 00:20:46.290 Last time, I spoke about the big picture of -- 00:20:46.290 --> 00:20:50.380 when I multiply a matrix by a right-hand side. 00:20:50.380 --> 00:20:54.690 If I have some matrix there and I multiply it by 3 4 5, 00:20:54.690 --> 00:20:55.920 let's say -- 00:20:55.920 --> 00:20:59.520 so here's a matrix -- 00:20:59.520 --> 00:21:02.120 what did I say -- well, I guess I only said it 00:21:02.120 --> 00:21:06.500 on the videotape, but -- do you remember how I look at that 00:21:06.500 --> 00:21:08.480 matrix multiplication? 00:21:08.480 --> 00:21:13.410 The result of multiplying a matrix by some vector 00:21:13.410 --> 00:21:21.830 is a combination of the columns of the matrix. 00:21:21.830 --> 00:21:24.640 It's 3 times the first column. 00:21:24.640 --> 00:21:33.630 It's 3 times column one plus 4 times column two plus 5 times 00:21:33.630 --> 00:21:34.450 column three. 00:21:39.760 --> 00:21:40.370 Okay. 00:21:40.370 --> 00:21:43.970 I'm going to come back to that multiple times. 00:21:43.970 --> 00:21:51.430 What I wanted to do now was to emphasize the parallel thing 00:21:51.430 --> 00:21:54.390 with rows. 00:21:54.390 --> 00:21:55.050 Why? 00:21:55.050 --> 00:21:59.620 Because all our operations here for this two weeks 00:21:59.620 --> 00:22:04.150 of the course are row operations. 00:22:04.150 --> 00:22:10.270 So this isn't what I need for row operations. 00:22:10.270 --> 00:22:12.040 Let me do a row operation. 00:22:12.040 --> 00:22:20.380 Suppose I have my matrix again and suppose I multiply 00:22:20.380 --> 00:22:23.520 on the left by some -- let's say 1 2 7. 00:22:28.680 --> 00:22:33.860 Again, I'm just, like, saying what the result is. 00:22:33.860 --> 00:22:38.240 And then we'll say how matrix multiplication works 00:22:38.240 --> 00:22:40.470 and we'll see that it's true. 00:22:40.470 --> 00:22:41.260 Okay. 00:22:41.260 --> 00:22:45.600 But maybe already I'm making -- 00:22:45.600 --> 00:22:49.960 I'm sort of bringing up -- the central idea of linear algebra 00:22:49.960 --> 00:22:55.490 is how these matrices work by rows as well as by columns. 00:22:55.490 --> 00:22:55.990 Okay. 00:22:55.990 --> 00:22:57.730 How does it work by rows? 00:22:57.730 --> 00:23:05.310 What -- so that's a row vector. 00:23:05.310 --> 00:23:08.860 I could say that's a one by three matrix, a row 00:23:08.860 --> 00:23:11.335 vector multiplying a three by three matrix. 00:23:15.300 --> 00:23:17.300 What's the output? 00:23:17.300 --> 00:23:23.440 What's the product of a row times a matrix? 00:23:23.440 --> 00:23:26.260 And -- okay, it's a row. 00:23:26.260 --> 00:23:29.080 A row -- a column -- 00:23:29.080 --> 00:23:29.580 I'm sorry. 00:23:29.580 --> 00:23:31.690 A matrix times a column is a column. 00:23:31.690 --> 00:23:35.350 So matrix times a -- yeah. 00:23:35.350 --> 00:23:41.760 Matrix times a column is a column. 00:23:41.760 --> 00:23:44.080 And we know what column it is. 00:23:44.080 --> 00:23:47.400 Over here, I'm doing a row times a matrix. 00:23:47.400 --> 00:23:49.500 And what's the answer? 00:23:49.500 --> 00:23:53.260 It's one of that first row, so it's 1 times -- 00:23:53.260 --> 00:24:05.000 1 times row one, plus 2 times row two plus 7 times row three. 00:24:05.000 --> 00:24:07.810 When -- as we do matrix multiplication, 00:24:07.810 --> 00:24:13.650 keep your eye on what it's doing with whole vectors. 00:24:13.650 --> 00:24:18.140 And what it's doing -- what it's doing in this case is 00:24:18.140 --> 00:24:20.520 it's combining the rows. 00:24:20.520 --> 00:24:24.750 And we have a combination, a linear combination of the rows. 00:24:24.750 --> 00:24:26.130 Okay, I want to use that. 00:24:33.160 --> 00:24:38.190 Okay, so my question is what's the matrix that does this first 00:24:38.190 --> 00:24:44.180 step, that takes -- subtracts 3 of equation one from equation 00:24:44.180 --> 00:24:44.680 two? 00:24:44.680 --> 00:24:46.550 That's what I want to do. 00:24:46.550 --> 00:24:48.800 So this is going to be a matrix that's 00:24:48.800 --> 00:25:03.780 going to subtract 3 times row one from row two, 00:25:03.780 --> 00:25:05.160 and leaves the other rows the 00:25:05.160 --> 00:25:05.670 same. 00:25:05.670 --> 00:25:09.150 Just in -- I mean, the answer is going to be that. 00:25:09.150 --> 00:25:12.600 So whatever matrix this is -- 00:25:12.600 --> 00:25:15.450 and you're going to, like, tell me what matrix will do it, 00:25:15.450 --> 00:25:19.820 it's the matrix that leaves the first row unchanged, 00:25:19.820 --> 00:25:23.680 leaves the last row unchanged, but takes 3 of these 00:25:23.680 --> 00:25:27.740 away from this so it puts a 0 there, a 2 there and a minus 2. 00:25:27.740 --> 00:25:29.470 Good. 00:25:29.470 --> 00:25:31.420 What matrix will do it? 00:25:31.420 --> 00:25:33.010 It's these. 00:25:33.010 --> 00:25:35.880 It should be a pretty simple matrix, 00:25:35.880 --> 00:25:40.360 because we're doing a very simple step. 00:25:40.360 --> 00:25:43.840 We're just doing this step that changes row two. 00:25:43.840 --> 00:25:46.060 So actually, row one is not changing. 00:25:46.060 --> 00:25:48.490 So tell me how the matrix should begin. 00:25:51.250 --> 00:26:01.130 One -- the first row of the matrix will be 1 0 0, 00:26:01.130 --> 00:26:05.050 because that's just the right thing that takes one of that 00:26:05.050 --> 00:26:08.030 row and none of the other rows, and that's what we want. 00:26:08.030 --> 00:26:11.160 What's the last row of the matrix? 00:26:11.160 --> 00:26:17.090 0 0 1, because that takes one of the third row 00:26:17.090 --> 00:26:19.030 and none of the other rows, that's great. 00:26:19.030 --> 00:26:20.370 Okay. 00:26:20.370 --> 00:26:24.420 Now, suppose I didn't want to do anything at all. 00:26:24.420 --> 00:26:28.340 Suppose my row -- well, I guess maybe I had a case here when I 00:26:28.340 --> 00:26:33.080 already had a 0 and, didn't have to do anything. 00:26:33.080 --> 00:26:40.540 What matrix does nothing, like, just leaves you where you were? 00:26:40.540 --> 00:26:42.420 If I put in -- 00:26:42.420 --> 00:26:49.280 if I put in 0 1 0, that would be -- that would be -- 00:26:49.280 --> 00:26:52.390 that's the matrix -- what's the name of that matrix? 00:26:52.390 --> 00:26:55.140 The identity matrix, right. 00:26:55.140 --> 00:26:56.720 So it does absolutely nothing. 00:26:56.720 --> 00:26:59.240 It just multiplies everything and leaves it where it is. 00:26:59.240 --> 00:27:03.120 It's like a one, like the number one, for matrices. 00:27:03.120 --> 00:27:06.360 But that's not what we want, because we want to change this 00:27:06.360 --> 00:27:08.100 row to -- 00:27:08.100 --> 00:27:11.060 so what's the correct -- 00:27:11.060 --> 00:27:16.610 what should I put in here now to do it right? 00:27:16.610 --> 00:27:18.280 I want to get -- what do I want? 00:27:18.280 --> 00:27:20.660 What I -- I'm after -- 00:27:20.660 --> 00:27:24.360 I want 3 of row one to get subtracted 00:27:24.360 --> 00:27:25.210 off. 00:27:25.210 --> 00:27:32.620 So what's the right matrix, finish that matrix for me. 00:27:32.620 --> 00:27:36.390 Negative 3 goes here? 00:27:36.390 --> 00:27:37.560 And what goes here? 00:27:37.560 --> 00:27:38.720 That 1. 00:27:38.720 --> 00:27:39.610 And what goes here? 00:27:39.610 --> 00:27:40.740 The 0. 00:27:40.740 --> 00:27:43.230 That's the good matrix. 00:27:43.230 --> 00:27:46.700 That's the matrix that takes minus 3 00:27:46.700 --> 00:27:50.800 of row one plus the row two and gives the new row 2. 00:27:50.800 --> 00:27:57.350 Should we just, like, check some particular 00:27:57.350 --> 00:27:58.130 entry? 00:27:58.130 --> 00:28:00.380 How do I check a particular entry 00:28:00.380 --> 00:28:03.640 of a matrix in matrix multiplication? 00:28:03.640 --> 00:28:09.050 Like, suppose I wanted to check the entry here that's in row 00:28:09.050 --> 00:28:11.850 two, column three. 00:28:11.850 --> 00:28:16.400 So where does the entry in row two, column three come from? 00:28:16.400 --> 00:28:19.660 I would look at row two of this guy 00:28:19.660 --> 00:28:26.230 and column three of this one to get that number. 00:28:26.230 --> 00:28:29.950 That number comes from the second row and the third column 00:28:29.950 --> 00:28:34.120 and I just take this dot product minus 3 -- 00:28:34.120 --> 00:28:39.360 I'm multiplying -- minus 3 plus 1 and 0 gives the minus 2. 00:28:39.360 --> 00:28:40.010 Yeah. 00:28:40.010 --> 00:28:41.640 It works. 00:28:41.640 --> 00:28:46.840 So we got various ways to multiply matrices now. 00:28:46.840 --> 00:28:49.870 We're sort of, like -- informally. 00:28:49.870 --> 00:28:52.600 We've got by columns, we've got -- well, 00:28:52.600 --> 00:28:56.670 we will have by columns, by rows, by each entry at a 00:28:56.670 --> 00:28:57.370 time. 00:28:57.370 --> 00:29:01.360 But it's good to see that matrix multiplication when one 00:29:01.360 --> 00:29:04.030 of the matrices is so simple. 00:29:04.030 --> 00:29:08.180 So this guy is our elementary matrix. 00:29:08.180 --> 00:29:12.530 Let's call it E for elementary or elimination. 00:29:12.530 --> 00:29:18.130 And let me put the indexes 2 1, because it's the matrix that we 00:29:18.130 --> 00:29:22.380 needed to fix the 2 1 position. 00:29:22.380 --> 00:29:25.790 It's the matrix that we needed to get this 2 1 00:29:25.790 --> 00:29:29.340 position to be Okay. 00:29:29.340 --> 00:29:30.000 Good enough. 00:29:30.000 --> 00:29:32.140 So what do I do next? 00:29:32.140 --> 00:29:34.550 I need another matrix, right? 00:29:34.550 --> 00:29:36.610 I need to -- 00:29:36.610 --> 00:29:39.420 there's another step here. 00:29:39.420 --> 00:29:43.330 And I want to express the whole elimination 00:29:43.330 --> 00:29:46.780 process in matrix language. 00:29:46.780 --> 00:29:54.510 So tell me what -- so next step, step two, which was what? 00:29:54.510 --> 00:29:59.970 Subtract -- what was -- what was the actual step that we did? 00:29:59.970 --> 00:30:03.670 I think I subtracted -- do you remember? 00:30:03.670 --> 00:30:06.730 I had a 2 in the pivot and a 4 below it, 00:30:06.730 --> 00:30:11.700 so I subtracted two times -- 00:30:11.700 --> 00:30:16.690 times row two from row three. 00:30:16.690 --> 00:30:18.870 From row three. 00:30:18.870 --> 00:30:21.040 Tell me the matrix that will do that. 00:30:23.910 --> 00:30:26.600 And tell me its name. 00:30:26.600 --> 00:30:31.390 Okay, it's going to be E, for elementary or elimination 00:30:31.390 --> 00:30:36.080 matrix and what's the index number that I used to tell me 00:30:36.080 --> 00:30:38.590 what E -- 00:30:38.590 --> 00:30:39.750 3, 2, right? 00:30:39.750 --> 00:30:43.690 Because it's fixing this 3 2 position. 00:30:43.690 --> 00:30:47.550 And what's the matrix, now? 00:30:47.550 --> 00:30:52.200 Okay, you remember -- so E 3 2 is supposed to multiply my guy 00:30:52.200 --> 00:30:58.840 that I have and it's supposed to produce the right result, 00:30:58.840 --> 00:31:01.320 which was -- it leaves -- it's supposed to leave the first 00:31:01.320 --> 00:31:05.420 row, it's supposed to leave the second row and it's supposed 00:31:05.420 --> 00:31:10.490 to straighten out that third row to this. 00:31:10.490 --> 00:31:12.775 And what's the matrix that does that? 00:31:15.620 --> 00:31:16.820 1 0 0, right? 00:31:16.820 --> 00:31:21.350 Because we don't change the first row and the next row 00:31:21.350 --> 00:31:24.190 we don't change either, and the last row 00:31:24.190 --> 00:31:27.240 is the one we do change. 00:31:27.240 --> 00:31:29.730 And what do I do? 00:31:29.730 --> 00:31:33.540 Let's see, I subtract two times -- 00:31:33.540 --> 00:31:35.120 so what's this row? 00:31:35.120 --> 00:31:37.100 What's this here? 00:31:37.100 --> 00:31:41.530 0, right, because the first row's not involved. 00:31:41.530 --> 00:31:44.520 It's just in the 3 2 position, isn't it? 00:31:44.520 --> 00:31:49.980 This the key number is this minus the multiplier that goes 00:31:49.980 --> 00:31:52.430 -- sitting there in that 3 2 position. 00:31:52.430 --> 00:31:59.233 Is it a minus 2 to subtract 2 and then this is a 1 so that -- 00:32:02.550 --> 00:32:06.660 the overall effect is to take minus 2 of this row plus 1 of 00:32:06.660 --> 00:32:07.200 that. 00:32:07.200 --> 00:32:07.700 Okay. 00:32:10.430 --> 00:32:15.280 So, I've now given you the pieces, the elimination 00:32:15.280 --> 00:32:18.720 matrices, the elementary matrices that take each step. 00:32:21.510 --> 00:32:23.890 So now what? 00:32:23.890 --> 00:32:27.510 Now the next point in the lecture 00:32:27.510 --> 00:32:32.450 is to put those steps together into a matrix that does it all 00:32:32.450 --> 00:32:34.930 and see how it all happens. 00:32:34.930 --> 00:32:37.220 So now I'm going to express the whole -- 00:32:37.220 --> 00:32:46.530 everything we did today so far on A was to start with A, 00:32:46.530 --> 00:32:53.150 we multiplied it by E 2 1, that was the first step -- 00:32:53.150 --> 00:33:01.080 and then we multiplied that result by E 3 2 and that led us 00:33:01.080 --> 00:33:04.970 to this thing and what was that matrix? 00:33:04.970 --> 00:33:10.890 U. 00:33:10.890 --> 00:33:13.840 You see why I like matrix notation, 00:33:13.840 --> 00:33:18.750 because there in, like, little space -- 00:33:18.750 --> 00:33:23.460 a few bits when its compressed on the web -- is everything -- 00:33:23.460 --> 00:33:25.490 is this whole lecture. 00:33:25.490 --> 00:33:26.550 Okay. 00:33:26.550 --> 00:33:33.720 Now there -- there are important facts about matrix 00:33:33.720 --> 00:33:35.940 multiplication. 00:33:35.940 --> 00:33:40.440 And we're close to maybe the most important. 00:33:40.440 --> 00:33:42.320 And that is this. 00:33:42.320 --> 00:33:44.470 Suppose I ask you this question. 00:33:44.470 --> 00:33:49.140 Suppose I start with a matrix A and I 00:33:49.140 --> 00:33:52.410 want to end with a matrix U and I 00:33:52.410 --> 00:33:55.230 want to say what matrix does the whole job? 00:33:57.750 --> 00:34:05.240 What matrix takes me from A to U, using the letters I've got? 00:34:07.880 --> 00:34:09.969 And the answer is simple. 00:34:09.969 --> 00:34:13.275 I'm not asking this as -- but it's highly important. 00:34:17.199 --> 00:34:19.429 How would I create the matrix that 00:34:19.429 --> 00:34:21.639 does the whole job at once, that does 00:34:21.639 --> 00:34:24.840 all of elimination in one shot? 00:34:24.840 --> 00:34:25.810 It would be -- 00:34:25.810 --> 00:34:29.489 I would just put these together, right? 00:34:29.489 --> 00:34:34.310 In other words, this is the thing I'm struggling to say. 00:34:34.310 --> 00:34:35.810 I can move those parentheses. 00:34:38.699 --> 00:34:41.050 If I keep the matrices in order -- 00:34:41.050 --> 00:34:45.030 I can't mess around with the order of the matrices, 00:34:45.030 --> 00:34:49.460 but I can change the order that I do the multiplications. 00:34:49.460 --> 00:34:53.004 I can multiply these two first -- 00:34:55.670 --> 00:34:59.490 in other words, you see what those parentheses are doing? 00:34:59.490 --> 00:35:04.910 It's saying -- multiply the Es first and that gives you 00:35:04.910 --> 00:35:07.660 the matrix that does everything at once. 00:35:07.660 --> 00:35:09.370 Okay. 00:35:09.370 --> 00:35:13.920 So this fact, that this is automatically the same as this 00:35:13.920 --> 00:35:14.870 -- 00:35:14.870 --> 00:35:20.590 for every matrix multiplication, which I'm conscious of still 00:35:20.590 --> 00:35:24.600 not telling you in every detail, but, like, 00:35:24.600 --> 00:35:27.910 you're seeing how it works -- and this is highly important -- 00:35:27.910 --> 00:35:32.150 and maybe tell me the long word that describes this law 00:35:32.150 --> 00:35:37.620 for matrices, that you can move the parentheses? 00:35:37.620 --> 00:35:40.670 It's called the associative law. 00:35:40.670 --> 00:35:42.551 I think you can now forget that. 00:35:45.380 --> 00:35:47.320 But don't forget the law. 00:35:47.320 --> 00:35:49.450 I mean, like, forget the word associative. 00:35:49.450 --> 00:35:50.240 I don't know. 00:35:50.240 --> 00:35:51.950 But don't forget the law. 00:35:51.950 --> 00:35:59.540 Because actually, we'll see so many steps in linear algebra, 00:35:59.540 --> 00:36:03.220 so many proofs, even, of main fact 00:36:03.220 --> 00:36:05.495 come from just moving the parentheses. 00:36:08.610 --> 00:36:15.400 And it's not that easy to prove that this is correct, 00:36:15.400 --> 00:36:18.240 you have to go into the gory details of matrix 00:36:18.240 --> 00:36:20.700 multiplication, do it both ways and see 00:36:20.700 --> 00:36:23.360 that you come out the same. 00:36:23.360 --> 00:36:27.360 Maybe I'll leave the author to do that. 00:36:27.360 --> 00:36:28.160 Okay. 00:36:28.160 --> 00:36:29.650 So there we go. 00:36:34.550 --> 00:36:40.160 So there's a single matrix, I could call it E -- 00:36:40.160 --> 00:36:45.780 while we're talking about these matrices, tell me one other -- 00:36:45.780 --> 00:36:49.140 there's another type of elementary matrix, 00:36:49.140 --> 00:36:52.510 and we already said why we might need it. 00:36:52.510 --> 00:36:54.520 We didn't need it in this case. 00:36:54.520 --> 00:36:58.580 But it's the matrix that exchanges two rows. 00:36:58.580 --> 00:37:01.990 It's called a permutation matrix. 00:37:01.990 --> 00:37:06.500 Can you just, like, tell me what that would So I'm just -- like, 00:37:06.500 --> 00:37:10.260 this is a slight digression and be? we'll -- yes, 00:37:10.260 --> 00:37:12.600 so let me get some -- let me figure out where I'm going 00:37:12.600 --> 00:37:13.970 to put a permutation matrix. 00:37:16.870 --> 00:37:19.120 You'll see I'm always squeezing stuff in. 00:37:19.120 --> 00:37:19.840 So permutation. 00:37:23.270 --> 00:37:33.530 Or, in fact this one you'll, like, exchange rows -- 00:37:33.530 --> 00:37:37.920 shall I exchange rows one and two, just to make life easy? 00:37:37.920 --> 00:37:41.450 So if I had my matrix -- no, let -- let me just do two by two. 00:37:41.450 --> 00:37:44.330 |a b; c d|. 00:37:44.330 --> 00:37:49.610 Suppose I want to find the matrix that exchanges 00:37:49.610 --> 00:37:50.330 those rows. 00:37:55.060 --> 00:37:58.350 What is it? 00:37:58.350 --> 00:38:01.260 So the matrix that exchanges those rows -- 00:38:01.260 --> 00:38:03.640 the row I want is c d and it's there. 00:38:03.640 --> 00:38:07.100 So I better take one of it. 00:38:07.100 --> 00:38:11.010 And the row I want here is up top, so I'll take one of that. 00:38:11.010 --> 00:38:13.180 So actually, I'm just -- 00:38:13.180 --> 00:38:16.850 the easy way -- this is my matrix that I'll call P, 00:38:16.850 --> 00:38:17.750 for permutation. 00:38:21.560 --> 00:38:26.650 It's the matrix -- actually, the easy way to find it is just do 00:38:26.650 --> 00:38:30.500 the thing to the identity matrix. 00:38:30.500 --> 00:38:32.650 Exchange the rows of the identity matrix 00:38:32.650 --> 00:38:38.830 and then that's the matrix that will do row exchanges for you. 00:38:38.830 --> 00:38:43.000 Suppose I wanted to exchange columns instead. 00:38:43.000 --> 00:38:45.290 Columns have hardly got into today's lecture, 00:38:45.290 --> 00:38:48.430 but they certainly are going to be around. 00:38:48.430 --> 00:38:53.010 How could I -- if I started with this matrix |a b; c d| 00:38:53.010 --> 00:38:55.740 then I wouldn't -- 00:38:55.740 --> 00:38:57.480 I'm not even going to write this down, 00:38:57.480 --> 00:39:03.460 I'm just going to ask you, because in elimination, we're 00:39:03.460 --> 00:39:05.240 doing rows. 00:39:05.240 --> 00:39:08.090 But suppose we wanted to exchange 00:39:08.090 --> 00:39:10.225 the columns of a matrix. 00:39:13.180 --> 00:39:15.450 How would I do that? 00:39:15.450 --> 00:39:19.110 What matrix multiplication would do that job? 00:39:19.110 --> 00:39:19.980 Actually, why not? 00:39:19.980 --> 00:39:22.020 I'll write it down. 00:39:22.020 --> 00:39:23.840 So this is -- 00:39:23.840 --> 00:39:28.220 I'll write it under here and then hide it again. 00:39:28.220 --> 00:39:28.740 Okay. 00:39:28.740 --> 00:39:32.580 Suppose I had my matrix |a b; c d| 00:39:32.580 --> 00:39:38.160 and I want to get to a c over here and b d here. 00:39:44.790 --> 00:39:46.620 What matrix does that job? 00:39:52.300 --> 00:39:58.520 Can I multiply -- can I cook up some matrix that produces that 00:39:58.520 --> 00:40:00.800 answer? 00:40:00.800 --> 00:40:04.200 You can see from where I put my hand I was really 00:40:04.200 --> 00:40:09.660 asking can I put a matrix here on the left that 00:40:09.660 --> 00:40:11.850 will exchange columns? 00:40:11.850 --> 00:40:16.260 And the answer is no. 00:40:16.260 --> 00:40:18.000 I'm just bringing out again this point 00:40:18.000 --> 00:40:22.940 that when I multiply on the left, I'm doing row operations. 00:40:22.940 --> 00:40:24.790 So if I want to do a column operation, 00:40:24.790 --> 00:40:29.260 where do I put that permutation matrix? 00:40:29.260 --> 00:40:30.970 On the right. 00:40:30.970 --> 00:40:35.220 If I put it here, where I just barely left room for it -- 00:40:35.220 --> 00:40:40.230 so I'll exchange the two columns of the identity. 00:40:40.230 --> 00:40:43.410 Then it comes out right, because now I'm 00:40:43.410 --> 00:40:45.980 multiplying a column at a time. 00:40:45.980 --> 00:40:49.230 This is the first column and says take one -- 00:40:49.230 --> 00:40:52.500 take none of that column, one of this one and then you 00:40:52.500 --> 00:40:53.500 got it. 00:40:53.500 --> 00:40:56.240 Over here, take one of this one, none 00:40:56.240 --> 00:40:58.420 of this one and you've got a c. 00:40:58.420 --> 00:41:01.990 So, in short, to do column operations, 00:41:01.990 --> 00:41:04.310 the matrix multiplies on the right. 00:41:04.310 --> 00:41:07.660 To do row operations, it multiplies on the left. 00:41:07.660 --> 00:41:11.990 Okay, okay, and it's row operations that we're really 00:41:11.990 --> 00:41:12.535 doing. 00:41:12.535 --> 00:41:13.035 Okay. 00:41:18.790 --> 00:41:23.230 And of course, I mentioned in passing, 00:41:23.230 --> 00:41:30.840 but I better say it very clearly that you can't exchange 00:41:30.840 --> 00:41:32.120 the orders of matrices. 00:41:32.120 --> 00:41:35.340 And that's just the point I was making again here. 00:41:35.340 --> 00:41:39.520 A times B is not the same as B times A. 00:41:39.520 --> 00:41:46.030 You have to keep these matrices in their Gauss given order 00:41:46.030 --> 00:41:49.750 here, right? 00:41:49.750 --> 00:41:52.930 But you can move the parentheses, 00:41:52.930 --> 00:41:57.310 so that, in other words, the commutative law, which 00:41:57.310 --> 00:42:03.850 would allow you to take it in the other order is false. 00:42:03.850 --> 00:42:05.930 So we have to keep it in that order. 00:42:05.930 --> 00:42:06.690 Okay. 00:42:06.690 --> 00:42:10.915 So what next? 00:42:13.730 --> 00:42:16.670 I could do this multiplication. 00:42:16.670 --> 00:42:19.110 I could do E 32. 00:42:19.110 --> 00:42:21.525 So let me come back to see what that was. 00:42:24.750 --> 00:42:28.470 Here was E 2 1. 00:42:28.470 --> 00:42:33.020 And here is E 3 2. 00:42:33.020 --> 00:42:39.320 And if I multiply those two matrices together -- 00:42:39.320 --> 00:42:43.064 E 3 2 and then E 2 1, I'll get a single matrix 00:42:43.064 --> 00:42:43.980 that does elimination. 00:42:49.240 --> 00:42:52.110 I don't want to do it that -- 00:42:52.110 --> 00:42:55.350 if I do that multiplication -- 00:42:55.350 --> 00:43:00.790 there -- there's a better way to do this. 00:43:00.790 --> 00:43:03.520 And so in this last few minutes of today's lecture, 00:43:03.520 --> 00:43:05.540 can I anticipate that better way? 00:43:08.320 --> 00:43:15.260 The better way is to think not how do I get from A to U, 00:43:15.260 --> 00:43:20.310 but how do I get from U back to A? 00:43:20.310 --> 00:43:24.730 So reversing steps is going to come in. 00:43:24.730 --> 00:43:28.950 Inverse -- I'll use the word inverse here. 00:43:28.950 --> 00:43:29.630 Okay. 00:43:29.630 --> 00:43:37.250 So let me make the first step at what's the inverse matrix? 00:43:37.250 --> 00:43:40.560 All the matrices you've seen on this board have inverses. 00:43:43.720 --> 00:43:47.910 I didn't write any bad matrices down. 00:43:47.910 --> 00:43:50.770 We spoke about possible failure, and for a moment, 00:43:50.770 --> 00:43:53.400 we put in a matrix that would fail. 00:43:53.400 --> 00:43:56.260 But right now, all these matrices are good, 00:43:56.260 --> 00:43:57.720 they're all invertible. 00:43:57.720 --> 00:44:01.010 And let's take the inverse -- well, 00:44:01.010 --> 00:44:04.800 let me say first what does the inverse mean and find it? 00:44:04.800 --> 00:44:05.300 Okay. 00:44:05.300 --> 00:44:10.850 So we're getting a little leg up on inverses. 00:44:10.850 --> 00:44:16.172 Okay, so this is the final moments of today. 00:44:18.770 --> 00:44:22.870 Sorry, he's still there. 00:44:22.870 --> 00:44:23.382 Okay. 00:44:23.382 --> 00:44:23.882 Inverses. 00:44:31.290 --> 00:44:33.320 Okay, and I'm just going to take one example 00:44:33.320 --> 00:44:34.870 and then we're done. 00:44:34.870 --> 00:44:39.480 The example I'll take will be that E. So my matrix 00:44:39.480 --> 00:44:45.640 is 1 0 0 minus 3 1 0 0 0 1. 00:44:48.240 --> 00:44:55.870 And I want to find the matrix that undoes that step. 00:44:55.870 --> 00:44:57.720 So what was that step? 00:44:57.720 --> 00:45:03.600 The step was subtract 3 times row one from row two. 00:45:03.600 --> 00:45:10.670 So what matrix will get me back? 00:45:10.670 --> 00:45:14.540 What matrix will bring back -- 00:45:14.540 --> 00:45:19.530 you know, if I started with a 2 12 2 and I changed it to a 2 6 00:45:19.530 --> 00:45:25.580 2 because of this guy, I want to get back to the 2 12 I want 00:45:25.580 --> 00:45:30.150 to find the matrix which -- which undoes elimination, 00:45:30.150 --> 00:45:33.240 the matrix which multiplies this to give the identity. 00:45:38.390 --> 00:45:42.940 And you can tell me what I should do in words first, 00:45:42.940 --> 00:45:45.570 and then we'll write down the matrix that does it. 00:45:45.570 --> 00:45:50.130 If this step subtracted 3 times row 1 from row 2, 00:45:50.130 --> 00:45:51.500 what's the inverse step? 00:45:51.500 --> 00:46:02.120 I add 3 times row one to row two, right? 00:46:02.120 --> 00:46:03.080 I add it back. 00:46:03.080 --> 00:46:05.610 The -- what I subtracted away, I add back. 00:46:05.610 --> 00:46:08.770 So the inverse matrix in this case is -- 00:46:08.770 --> 00:46:12.660 I now want to add 3 times row one to row two, 00:46:12.660 --> 00:46:17.020 so I won't change row one, I won't change row three 00:46:17.020 --> 00:46:23.000 and I'll add 3 times row one to row two. 00:46:23.000 --> 00:46:27.740 That's a case where the inverse is clear. 00:46:27.740 --> 00:46:33.530 It's clear in words what to do, because what this did 00:46:33.530 --> 00:46:37.330 was simple to express. 00:46:37.330 --> 00:46:43.290 It just changed row two by subtracting 3 of row one. 00:46:43.290 --> 00:46:45.670 So to invert it, I go that way. 00:46:45.670 --> 00:46:47.940 And if you -- if we do that calculation, 00:46:47.940 --> 00:46:51.240 3 times this row plus 1 times this row, 00:46:51.240 --> 00:46:53.980 comes out the right row of the identity. 00:46:53.980 --> 00:46:57.050 Okay, so inverses are an -- 00:46:57.050 --> 00:47:05.020 so if this matrix was E and this matrix is I for identity, then 00:47:05.020 --> 00:47:08.370 what's the notation for this guy? 00:47:08.370 --> 00:47:11.010 E to the minus one. 00:47:11.010 --> 00:47:12.710 E inverse. 00:47:12.710 --> 00:47:13.950 Okay. 00:47:13.950 --> 00:47:16.040 Let's stop there for today. 00:47:16.040 --> 00:47:20.660 That's a little jump on what's coming on Monday. 00:47:20.660 --> 00:47:23.050 So, see you Monday.