0:00:00.000,0:00:00.680 0:00:00.680,0:00:04.510 I have this matrix A here that I[br]want to put into reduced row 0:00:04.510,0:00:05.450 echelon form. 0:00:05.450,0:00:07.160 And we've done this[br]multiple times. 0:00:07.160,0:00:09.670 You just perform a bunch[br]of row operations. 0:00:09.670,0:00:12.470 But what I want to show you in[br]this video is that those row 0:00:12.470,0:00:16.520 operations are equivalent to[br]linear transformations on the 0:00:16.520,0:00:19.450 column vectors of A. 0:00:19.450,0:00:21.490 So let me show you by example. 0:00:21.490,0:00:24.450 So if we just want to put A into[br]reduced row echelon form, 0:00:24.450,0:00:26.860 the first step that we might[br]want to do if we wanted to 0:00:26.860,0:00:31.580 zero out these entries right[br]here, is-- let me do it right 0:00:31.580,0:00:34.890 here-- is we'll keep our[br]first entry the same. 0:00:34.890,0:00:36.830 So for each of these column[br]vectors, we're going to keep 0:00:36.830,0:00:38.010 the first entry the same. 0:00:38.010,0:00:41.590 So they're going to be[br]1, minus 1, minus 1. 0:00:41.590,0:00:44.360 And actually, let me[br]simultaneously construct my 0:00:44.360,0:00:45.800 transformation. 0:00:45.800,0:00:48.340 So I'm saying that my row[br]operation I'm going to perform 0:00:48.340,0:00:51.690 is equivalent to a linear[br]transformation 0:00:51.690,0:00:52.630 on the column vector. 0:00:52.630,0:00:55.160 So it's going to be a[br]transformation that's going to 0:00:55.160,0:01:00.880 take some column vector,[br]a1, a2, and a3. 0:01:00.880,0:01:03.130 It's going to take each of these[br]and then do something to 0:01:03.130,0:01:05.239 them, do something to them[br]in a linear way. 0:01:05.239,0:01:07.330 They'll be linear[br]transformations. 0:01:07.330,0:01:09.470 So we're keeping the[br]first entry of our 0:01:09.470,0:01:11.090 column vector the same. 0:01:11.090,0:01:14.670 So this is just going[br]to be a1. 0:01:14.670,0:01:16.330 This is a line right here. 0:01:16.330,0:01:17.250 That's going to be a1. 0:01:17.250,0:01:19.050 Now, what can we do if[br]we want to get to 0:01:19.050,0:01:20.780 reduced row echelon form? 0:01:20.780,0:01:22.610 We'd want to make[br]this equal to 0. 0:01:22.610,0:01:26.360 So we would want to replace our[br]second row with the second 0:01:26.360,0:01:29.230 row plus the first row, because[br]then these guys would 0:01:29.230,0:01:30.500 turn out to be 0. 0:01:30.500,0:01:32.140 So let me write that on[br]my transformation. 0:01:32.140,0:01:35.490 I'm going to replace the second[br]row with the second row 0:01:35.490,0:01:39.090 plus the first row. 0:01:39.090,0:01:40.400 Let me write it out here. 0:01:40.400,0:01:43.410 Minus 1 plus 1 is 0. 0:01:43.410,0:01:45.810 2 plus minus 1 is 1. 0:01:45.810,0:01:48.950 3 plus minus 1 is 2. 0:01:48.950,0:01:51.070 Now, we also want[br]to get a 0 here. 0:01:51.070,0:01:54.360 So let me replace my third[br]row with my third row 0:01:54.360,0:01:55.900 minus my first row. 0:01:55.900,0:01:59.320 So I'm going to replace my third[br]row with my third row 0:01:59.320,0:02:01.690 minus my first row. 0:02:01.690,0:02:05.240 So 1 minus 1 is 0. 0:02:05.240,0:02:08.660 1 minus minus 1 is 2. 0:02:08.660,0:02:14.100 4 minus minus 1 is 5,[br]just like that. 0:02:14.100,0:02:16.790 So you see this was just a[br]linear transformation. 0:02:16.790,0:02:19.390 And any linear transformation[br]you could actually represent 0:02:19.390,0:02:22.280 as a matrix vector product. 0:02:22.280,0:02:24.140 So for example, this[br]transformation, I could 0:02:24.140,0:02:26.150 represent it. 0:02:26.150,0:02:28.230 To figure out its transformation[br]matrix, so if 0:02:28.230,0:02:32.600 we say that T of x is equal to,[br]I don't know, let's call 0:02:32.600,0:02:36.220 it some matrix S times x. 0:02:36.220,0:02:37.740 We already used the matrix A. 0:02:37.740,0:02:40.110 So I have to pick[br]another letter. 0:02:40.110,0:02:41.160 So how do we find S? 0:02:41.160,0:02:43.570 Well, we just apply the[br]transformation to all of the 0:02:43.570,0:02:46.370 column vectors, or the standard[br]basis vectors of the 0:02:46.370,0:02:47.240 identity matrix. 0:02:47.240,0:02:48.460 So let's do that. 0:02:48.460,0:02:50.760 So the identity matrix-- I'll[br]draw it really small like 0:02:50.760,0:02:55.080 this-- the identity matrix looks[br]like this, 1, 0, 0, 0, 0:02:55.080,0:02:57.900 1, 0, 0, 0, 1. 0:02:57.900,0:02:59.880 That's what that identity[br]matrix looks like. 0:02:59.880,0:03:02.580 To find the transformation[br]matrix, we just apply this guy 0:03:02.580,0:03:04.660 to each of the column[br]vectors of this. 0:03:04.660,0:03:06.286 So what do we get? 0:03:06.286,0:03:09.270 I'll do it a little[br]bit bigger. 0:03:09.270,0:03:11.140 We apply it to each of[br]these column vectors. 0:03:11.140,0:03:13.370 But we see the first row[br]always stays the same. 0:03:13.370,0:03:16.250 So the first row is always going[br]to be the same thing. 0:03:16.250,0:03:18.850 So 1, 0, 0. 0:03:18.850,0:03:21.180 I'm essentially applying it[br]simultaneously to each of 0:03:21.180,0:03:24.290 these column vectors, saying,[br]look, when you transform each 0:03:24.290,0:03:27.710 of these column vectors, their[br]first entry stays the same. 0:03:27.710,0:03:31.890 The second entry becomes[br]the second entry 0:03:31.890,0:03:32.910 plus the first entry. 0:03:32.910,0:03:35.730 So 0 plus 1 is 1. 0:03:35.730,0:03:38.510 1 plus 0 is 1. 0:03:38.510,0:03:41.350 0 plus 0 is 0. 0:03:41.350,0:03:45.440 Then the third entry gets[br]replaced with the third entry 0:03:45.440,0:03:46.690 minus the first entry. 0:03:46.690,0:03:49.660 So 0 minus 1 is minus 1. 0:03:49.660,0:03:52.500 0 minus 0 is 0. 0:03:52.500,0:03:54.930 1 minus 0 is 1. 0:03:54.930,0:03:58.510 Now notice, when I apply this[br]transformation to the column 0:03:58.510,0:04:02.010 vectors of our identity matrix,[br]I essentially just 0:04:02.010,0:04:03.760 performed those same[br]row operations 0:04:03.760,0:04:04.730 that I did up there. 0:04:04.730,0:04:07.160 I performed those exact same[br]row operations on this 0:04:07.160,0:04:08.330 identity matrix. 0:04:08.330,0:04:10.520 But we know that this is[br]actually the transformation 0:04:10.520,0:04:13.110 matrix, that if we multiply[br]it by each of these column 0:04:13.110,0:04:16.769 vectors, or by each of these[br]column vectors, we're going to 0:04:16.769,0:04:18.430 get these column vectors. 0:04:18.430,0:04:20.240 So you can view it this way. 0:04:20.240,0:04:22.990 This right here, this[br]is equal to S. 0:04:22.990,0:04:25.510 This is our transformation[br]matrix. 0:04:25.510,0:04:32.350 So we could say that if we[br]create a new matrix whose 0:04:32.350,0:04:37.320 columns are S times this column[br]vector, S times 1, 0:04:37.320,0:04:39.420 minus 1, 1. 0:04:39.420,0:04:47.540 And then the next column is S[br]times-- I wanted to do it in 0:04:47.540,0:04:54.670 that other color-- S times[br]this guy, minus 1, 2, 1. 0:04:54.670,0:05:02.930 And then the third column is[br]going to be S times this third 0:05:02.930,0:05:09.180 column vector, minus 1, 3, 4. 0:05:09.180,0:05:11.950 We now know we're applying this[br]transformation, this is 0:05:11.950,0:05:13.920 S, times each of these[br]column vectors. 0:05:13.920,0:05:15.890 That is the matrix[br]representation of this 0:05:15.890,0:05:17.630 transformation. 0:05:17.630,0:05:22.300 This guy right here will[br]be transformed 0:05:22.300,0:05:25.070 to this right here. 0:05:25.070,0:05:30.580 Let me do it down here. 0:05:30.580,0:05:33.670 I wanted to show that stuff that[br]I had above here as well. 0:05:33.670,0:05:35.310 Well, I'll just draw an arrow. 0:05:35.310,0:05:36.440 That's probably the[br]simplest thing. 0:05:36.440,0:05:39.900 This matrix right here[br]will become that 0:05:39.900,0:05:41.430 matrix right there. 0:05:41.430,0:05:43.610 So another way you could[br]write it, this is 0:05:43.610,0:05:44.570 equivalent to what? 0:05:44.570,0:05:45.520 What is this equivalent to? 0:05:45.520,0:05:47.900 When you take a matrix and you[br]multiply it times each of the 0:05:47.900,0:05:50.400 column vectors, when you[br]transform each of the column 0:05:50.400,0:05:53.970 vectors by this matrix, this[br]is the definition of a 0:05:53.970,0:05:55.270 matrix-matrix product. 0:05:55.270,0:05:59.440 This is equal to our matrix S--[br]I'll do it in pink-- this 0:05:59.440,0:06:05.920 is equal to our matrix S, which[br]is 1, 0, 0, 1, 1, 0, 0:06:05.920,0:06:15.790 minus 1, 0, 1, times our matrix[br]A, times 1, minus 1, 1, 0:06:15.790,0:06:22.090 minus 1, 2, 1, minus 1, 3, 4. 0:06:22.090,0:06:25.630 So let me make this[br]very clear. 0:06:25.630,0:06:28.430 This is our transformation[br]matrix S. 0:06:28.430,0:06:30.440 This is our matrix A. 0:06:30.440,0:06:34.130 And when you perform this[br]product you're going to get 0:06:34.130,0:06:37.720 this guy right over here. 0:06:37.720,0:06:40.450 I'll just copy and paste it. 0:06:40.450,0:06:45.230 Edit, copy, and let[br]me paste it. 0:06:45.230,0:06:48.210 You're going to get that[br]guy just like that. 0:06:48.210,0:06:50.480 Now the whole reason why I'm[br]doing that is just to remind 0:06:50.480,0:06:53.710 you that when we perform each of[br]these row operations, we're 0:06:53.710,0:06:54.600 just multiplying. 0:06:54.600,0:06:57.170 We're performing a linear[br]transformation on each of 0:06:57.170,0:06:58.090 these columns. 0:06:58.090,0:07:00.730 And it is completely equivalent[br]to just multiplying 0:07:00.730,0:07:02.790 this guy by some matrix S. 0:07:02.790,0:07:04.710 In this case, we took the[br]trouble of figuring out what 0:07:04.710,0:07:06.150 that matrix S is. 0:07:06.150,0:07:08.550 But any of these row operations[br]that we've been 0:07:08.550,0:07:12.300 doing, you can always represent[br]them by a matrix 0:07:12.300,0:07:13.550 multiplication. 0:07:13.550,0:07:17.060 0:07:17.060,0:07:19.045 So this leads to a very[br]interesting idea. 0:07:19.045,0:07:22.650 0:07:22.650,0:07:25.700 When you put something in[br]reduced row echelon form, let 0:07:25.700,0:07:26.950 me do it up here. 0:07:26.950,0:07:29.730 0:07:29.730,0:07:32.180 Actually, let's just finish what[br]we started with this guy. 0:07:32.180,0:07:34.230 Let's put this guy in reduced[br]row echelon form. 0:07:34.230,0:07:37.440 0:07:37.440,0:07:39.080 Let me call this first S. 0:07:39.080,0:07:40.400 Let's call that S1. 0:07:40.400,0:07:42.730 So this guy right here[br]is equal to that 0:07:42.730,0:07:46.050 first S1 times A. 0:07:46.050,0:07:47.580 We already showed that[br]that's true. 0:07:47.580,0:07:50.040 Now let's perform another[br]transformation. 0:07:50.040,0:07:53.170 Let's just do another set of[br]row operations to get us to 0:07:53.170,0:07:55.000 reduced row echelon form. 0:07:55.000,0:07:58.860 So let's keep our middle[br]row the same, 0, 1, 2. 0:07:58.860,0:08:02.190 And let's replace the first row[br]with the first row plus 0:08:02.190,0:08:04.630 the second row, because I[br]want to make this a 0. 0:08:04.630,0:08:06.980 So 1 plus 0 is 1. 0:08:06.980,0:08:10.310 Let me do it in another color. 0:08:10.310,0:08:12.810 Minus 1 plus 1 is 0. 0:08:12.810,0:08:15.620 Minus 1 plus 2 is 1. 0:08:15.620,0:08:21.640 Now, I want to replace the third[br]row with, let's say the 0:08:21.640,0:08:28.210 third row minus 2 times[br]the first row. 0:08:28.210,0:08:31.300 So that's 0 minus 2,[br]times 0, is 0. 0:08:31.300,0:08:33.820 2 minus 2, times 1, is 0. 0:08:33.820,0:08:37.350 5 minus 2, times 2, is 1. 0:08:37.350,0:08:40.130 5 minus 4 is 1. 0:08:40.130,0:08:41.549 We're almost there. 0:08:41.549,0:08:44.870 We just have to zero out[br]these guys right there. 0:08:44.870,0:08:47.220 Let's see if we can get this[br]into reduced row echelon form. 0:08:47.220,0:08:47.900 So what is this? 0:08:47.900,0:08:49.880 I just performed another[br]linear transformation. 0:08:49.880,0:08:50.800 Actually, let me write this. 0:08:50.800,0:08:53.710 Let's say if this was our first[br]linear transformation, 0:08:53.710,0:08:55.390 what I just did is I performed[br]another linear 0:08:55.390,0:08:56.660 transformation, T2. 0:08:56.660,0:08:59.940 I'll write it in a different[br]notation, where you give me 0:08:59.940,0:09:04.100 some vector, some column[br]vector, x1, x2, x3. 0:09:04.100,0:09:05.650 What did I just do? 0:09:05.650,0:09:08.390 What was the transformation[br]that I just performed? 0:09:08.390,0:09:12.380 My new vector, I made the top[br]row equal to the top row plus 0:09:12.380,0:09:13.325 the second row. 0:09:13.325,0:09:15.690 So it's x1 plus x2. 0:09:15.690,0:09:17.500 I kept the second[br]row the same. 0:09:17.500,0:09:20.580 And then the third row, I[br]replaced it with the third row 0:09:20.580,0:09:22.920 minus 2 times the second row. 0:09:22.920,0:09:25.290 That was a linear transformation[br]we just did. 0:09:25.290,0:09:27.450 And we could represent this[br]linear transformation as 0:09:27.450,0:09:31.300 being, we could say T2 applied[br]to some vector x is equal to 0:09:31.300,0:09:36.120 some transformation vector[br]S2, times our vector x. 0:09:36.120,0:09:42.140 0:09:42.140,0:09:45.340 Because if we applied this[br]transformation matrix to each 0:09:45.340,0:09:48.590 of these columns, it's[br]equivalent to multiplying this 0:09:48.590,0:09:50.940 guy by this transformation[br]matrix. 0:09:50.940,0:09:53.430 So you could say that this guy[br]right here-- we haven't 0:09:53.430,0:09:56.270 figured out what this is, but[br]I think you get the idea-- 0:09:56.270,0:09:59.200 this matrix right here is going[br]to be equal to this guy. 0:09:59.200,0:10:03.310 It's going to be equal[br]to S2 times this guy. 0:10:03.310,0:10:04.670 What is this guy right here? 0:10:04.670,0:10:07.805 Well, this guy is equal[br]to S1 times A. 0:10:07.805,0:10:12.510 It's going to be S2[br]times S1, times A. 0:10:12.510,0:10:13.760 Fair enough. 0:10:13.760,0:10:16.930 0:10:16.930,0:10:19.200 And you could have gotten[br]straight here if you just 0:10:19.200,0:10:20.940 multiplied S2 times S1. 0:10:20.940,0:10:22.250 This could be some[br]other matrix. 0:10:22.250,0:10:24.610 If you just multiplied it by[br]A, you'd go straight from 0:10:24.610,0:10:26.070 there to there. 0:10:26.070,0:10:26.670 Fair enough. 0:10:26.670,0:10:28.595 Now, we still haven't gotten[br]this guy into reduced row 0:10:28.595,0:10:30.010 echelon form. 0:10:30.010,0:10:31.220 So let's try to get there. 0:10:31.220,0:10:33.125 I've run out of space below[br]him, so I'm going 0:10:33.125,0:10:35.270 to have to go up. 0:10:35.270,0:10:36.520 So let's go upwards. 0:10:36.520,0:10:40.620 0:10:40.620,0:10:43.650 What I want to do is, I'm going[br]to keep the third row 0:10:43.650,0:10:48.790 the same, 0, 0, 1. 0:10:48.790,0:10:54.700 Let me replace the second row[br]with the second row minus 2 0:10:54.700,0:10:56.150 times the third row. 0:10:56.150,0:10:59.680 So we'll get a 0, we'll get a 1[br]minus 2, times 0, and we'll 0:10:59.680,0:11:02.250 get a 2 minus 2, times 1. 0:11:02.250,0:11:04.100 So that's a 0. 0:11:04.100,0:11:06.500 Let's replaced the first[br]row with the first row 0:11:06.500,0:11:08.280 minus the third row. 0:11:08.280,0:11:10.970 So 1 minus 0 is 1. 0:11:10.970,0:11:13.880 0 minus 0 is 0. 0:11:13.880,0:11:19.310 1 minus 1 is 0, just[br]like that. 0:11:19.310,0:11:21.470 Let's just actually write what[br]our transformation was. 0:11:21.470,0:11:22.686 Let's call it T3. 0:11:22.686,0:11:26.490 I'll do it in purple. 0:11:26.490,0:11:30.225 T3 is the transformation of some[br]vector x-- let me write 0:11:30.225,0:11:34.710 it like this-- of some[br]vector x1, x2, x3. 0:11:34.710,0:11:37.570 0:11:37.570,0:11:38.290 What did we do? 0:11:38.290,0:11:41.050 We replaced the first row with[br]the first row minus the third 0:11:41.050,0:11:44.300 row, x1 minus x3. 0:11:44.300,0:11:47.580 We replaced the second row with[br]the second row minus 2 0:11:47.580,0:11:48.970 times the third row. 0:11:48.970,0:11:51.870 So it's x2 minus 2 times x3. 0:11:51.870,0:11:53.960 Then the third row just[br]stayed the same. 0:11:53.960,0:11:57.510 So obviously, this could[br]also be represented. 0:11:57.510,0:12:01.840 T3 of x could be equal to some[br]other transformation matrix, 0:12:01.840,0:12:04.230 S3 times x. 0:12:04.230,0:12:07.040 So this transformation, when[br]you multiply it to each of 0:12:07.040,0:12:12.090 these columns, is equivalent to[br]multiplying this guy times 0:12:12.090,0:12:14.910 this transformation matrix,[br]which we haven't found yet. 0:12:14.910,0:12:15.560 We can write it. 0:12:15.560,0:12:20.430 So this is going to be equal to[br]S3 times this matrix right 0:12:20.430,0:12:27.150 here, which is S2, S1, A. 0:12:27.150,0:12:28.330 And what do we have here? 0:12:28.330,0:12:30.000 We got the identity matrix. 0:12:30.000,0:12:32.070 We put it in reduced[br]row echelon form. 0:12:32.070,0:12:33.580 We got the identity matrix. 0:12:33.580,0:12:36.530 We already know from previous[br]videos the reduced row echelon 0:12:36.530,0:12:38.750 form of something is the[br]identity matrix. 0:12:38.750,0:12:41.830 Then we are dealing with an[br]invertible transformation, or 0:12:41.830,0:12:44.140 an invertible matrix. 0:12:44.140,0:12:46.350 Because this obviously could be[br]the transformation for some 0:12:46.350,0:12:47.580 transformation. 0:12:47.580,0:12:51.670 Let's just call this[br]transformation, I don't know, 0:12:51.670,0:12:52.970 did I already use T? 0:12:52.970,0:12:57.420 Let's just call it Tnaught for[br]our transformation applied to 0:12:57.420,0:13:00.130 some vector x, that might[br]be equal to Ax. 0:13:00.130,0:13:04.390 So we know that this[br]is invertible. 0:13:04.390,0:13:06.170 We put it in reduced[br]row echelon form. 0:13:06.170,0:13:07.850 We put its transformation[br]matrix in 0:13:07.850,0:13:09.560 reduced row echelon form. 0:13:09.560,0:13:11.130 And we got the identity[br]matrix. 0:13:11.130,0:13:12.880 So that tells us that[br]this is invertible. 0:13:12.880,0:13:14.990 But something even more[br]interesting happened. 0:13:14.990,0:13:18.130 We got here by performing[br]some row operations. 0:13:18.130,0:13:21.620 And we said those row operations[br]were completely 0:13:21.620,0:13:26.080 equivalent to multiplying this[br]guy right here by multiplying 0:13:26.080,0:13:29.890 our original transformation[br]matrix by a series of 0:13:29.890,0:13:33.080 transformation matrices that[br]represent our row operations. 0:13:33.080,0:13:37.150 And when we multiplied all this,[br]this was equal to the 0:13:37.150,0:13:38.990 identity matrix. 0:13:38.990,0:13:43.930 Now, in the last video we said[br]that the inverse matrix, so if 0:13:43.930,0:13:48.450 this is Tnaught, Tnaught inverse[br]could be represented-- 0:13:48.450,0:13:50.850 it's also a linear[br]transformation-- It can be 0:13:50.850,0:13:54.450 represented by some inverse[br]matrix that we just called A 0:13:54.450,0:13:56.070 inverse times x. 0:13:56.070,0:14:02.610 And we saw that the inverse[br]transformation matrix times 0:14:02.610,0:14:06.580 our transformation matrix is[br]equal to the identity matrix. 0:14:06.580,0:14:09.540 We saw this last time. 0:14:09.540,0:14:11.060 We proved this to you. 0:14:11.060,0:14:12.710 Now, something very[br]interesting here. 0:14:12.710,0:14:16.750 We have a series of matrix[br]products times this guy, times 0:14:16.750,0:14:20.010 this guy, that also got me[br]the identity matrix. 0:14:20.010,0:14:23.640 So this guy right here, this[br]series of matrix products, 0:14:23.640,0:14:29.750 this must be the same thing as[br]my inverse matrix, as my 0:14:29.750,0:14:32.170 inverse transformation matrix. 0:14:32.170,0:14:35.720 And so we could actually[br]calculate it if we wanted to. 0:14:35.720,0:14:38.100 Just like we did, we actually[br]figured out what S1 was. 0:14:38.100,0:14:39.560 We did it down here. 0:14:39.560,0:14:41.520 We could do a similar operation[br]to figure out what 0:14:41.520,0:14:46.370 S2 was, S3 was, and then[br]multiply them all out. 0:14:46.370,0:14:50.810 We would have actually[br]constructed A inverse. 0:14:50.810,0:14:53.240 I guess, something more[br]interesting we could do 0:14:53.240,0:15:00.820 instead of doing that, what if[br]we applied these same matrix 0:15:00.820,0:15:05.020 products to the identity[br]matrix. 0:15:05.020,0:15:06.370 So the whole time we did[br]here, when we did 0:15:06.370,0:15:07.950 our first row operation. 0:15:07.950,0:15:10.500 So we have here, we[br]have the matrix A. 0:15:10.500,0:15:13.120 Let's say we have an identity[br]matrix on the right. 0:15:13.120,0:15:15.050 Let's call that I,[br]right there. 0:15:15.050,0:15:17.930 Now, our first linear[br]transformation we did-- we saw 0:15:17.930,0:15:20.240 that right here-- that[br]was equivalent to 0:15:20.240,0:15:23.910 multiplying S1 times A. 0:15:23.910,0:15:26.330 The first set of row operations[br]was this. 0:15:26.330,0:15:27.510 It got us here. 0:15:27.510,0:15:30.520 Now, if we perform that same set[br]of row operations on the 0:15:30.520,0:15:32.630 identity matrix, what[br]are we going to get? 0:15:32.630,0:15:35.050 We're going to get[br]the matrix S1. 0:15:35.050,0:15:37.580 S1 times the identity[br]matrix is just S1. 0:15:37.580,0:15:41.490 All of the columns of anything[br]times the identity times the 0:15:41.490,0:15:43.760 standard basis columns, it'll[br]just be equal to itself. 0:15:43.760,0:15:45.930 You'll just be left[br]with that S1. 0:15:45.930,0:15:47.820 This is S1 times I. 0:15:47.820,0:15:49.290 That's just S1. 0:15:49.290,0:15:50.090 Fair enough. 0:15:50.090,0:15:52.310 Now, you performed your next row[br]operation and you ended up 0:15:52.310,0:15:56.320 with S2 times S1, times A. 0:15:56.320,0:15:58.710 Now if you performed that same[br]row operation on this guy 0:15:58.710,0:16:00.820 right there, what[br]would you have? 0:16:00.820,0:16:05.430 You would have S2 times S1,[br]times the identity matrix. 0:16:05.430,0:16:08.300 Now, our last row operation we[br]represented with the matrix 0:16:08.300,0:16:09.800 product S3. 0:16:09.800,0:16:12.690 We're multiplying it by the[br]transformation matrix S3. 0:16:12.690,0:16:16.990 So if you did that, you[br]have S3, S2, S1 A. 0:16:16.990,0:16:19.550 But if you perform the same[br]exact row operations on this 0:16:19.550,0:16:24.940 guy right here, you have[br]S3, S2, S1, times 0:16:24.940,0:16:26.360 the identity matrix. 0:16:26.360,0:16:28.510 Now when you did this, when[br]you performed these row 0:16:28.510,0:16:32.690 operations here, this got you[br]to the identity matrix. 0:16:32.690,0:16:35.310 Well, what are these going[br]to get you to? 0:16:35.310,0:16:37.800 When you just performed the same[br]exact row operations you 0:16:37.800,0:16:40.270 performed on A to get to the[br]identity matrix, if you 0:16:40.270,0:16:43.110 performed those same exact row[br]operations on the identity 0:16:43.110,0:16:44.630 matrix, what do you get? 0:16:44.630,0:16:46.990 You get this guy right here. 0:16:46.990,0:16:48.790 Anything times that identity[br]matrix is going 0:16:48.790,0:16:50.930 to be equal to itself. 0:16:50.930,0:16:52.350 So what is that right there? 0:16:52.350,0:16:53.600 That is A inverse. 0:16:53.600,0:16:56.370 0:16:56.370,0:17:00.850 So we have a generalized way[br]of figuring out the inverse 0:17:00.850,0:17:02.630 for transformation matrix. 0:17:02.630,0:17:04.819 What I can do is, let's[br]say I have some 0:17:04.819,0:17:07.160 transformation matrix A. 0:17:07.160,0:17:09.420 I can set up an augmented[br]matrix where I put the 0:17:09.420,0:17:13.750 identity matrix right there,[br]just like that, and I perform 0:17:13.750,0:17:15.000 a bunch of row operations. 0:17:15.000,0:17:17.670 0:17:17.670,0:17:20.060 And you could represent them[br]as matrix products. 0:17:20.060,0:17:23.069 But you perform a bunch of row[br]operations on all of them. 0:17:23.069,0:17:25.180 You perform the same operations[br]you perform on A as 0:17:25.180,0:17:27.119 you would do on the[br]identity matrix. 0:17:27.119,0:17:31.340 By the time you have A as an[br]identity matrix, you have A in 0:17:31.340,0:17:33.250 reduced row echelon form. 0:17:33.250,0:17:38.950 By the time A is like that, your[br]identity matrix, having 0:17:38.950,0:17:42.290 performed the same exact[br]operations on it, it is going 0:17:42.290,0:17:46.300 to be transformed into[br]A's inverse. 0:17:46.300,0:17:50.340 This is a very useful tool for[br]solving actual inverses. 0:17:50.340,0:17:52.150 Now, I've explained[br]the theoretical 0:17:52.150,0:17:53.180 reason why this works. 0:17:53.180,0:17:54.740 In the next video we'll[br]actually solve this. 0:17:54.740,0:17:57.610 Maybe we'll do it for the[br]example that I started off 0:17:57.610,0:17:59.740 with in this video.