WEBVTT 00:00:08.711 --> 00:00:09.210 Hi. 00:00:09.210 --> 00:00:14.230 This is the first lecture in MIT's course 18.06, 00:00:14.230 --> 00:00:19.430 linear algebra, and I'm Gilbert Strang. 00:00:19.430 --> 00:00:23.440 The text for the course is this book, 00:00:23.440 --> 00:00:26.070 Introduction to Linear Algebra. 00:00:26.070 --> 00:00:32.729 And the course web page, which has got a lot of exercises from 00:00:32.729 --> 00:00:38.470 the past, MatLab codes, the syllabus for the course, 00:00:38.470 --> 00:00:44.060 is web.mit.edu/18.06. 00:00:44.060 --> 00:00:47.350 And this is the first lecture, lecture one. 00:00:49.920 --> 00:00:55.490 So, and later we'll give the web address for viewing these, 00:00:55.490 --> 00:00:57.270 videotapes. 00:00:57.270 --> 00:00:59.200 Okay, so what's in the first lecture? 00:01:02.680 --> 00:01:05.530 This is my plan. 00:01:05.530 --> 00:01:09.340 The fundamental problem of linear algebra, 00:01:09.340 --> 00:01:13.880 which is to solve a system of linear equations. 00:01:13.880 --> 00:01:16.210 So let's start with a case when we 00:01:16.210 --> 00:01:21.440 have some number of equations, say n equations and n unknowns. 00:01:21.440 --> 00:01:23.820 So an equal number of equations and unknowns. 00:01:23.820 --> 00:01:26.860 That's the normal, nice case. 00:01:26.860 --> 00:01:31.870 And what I want to do is -- with examples, of course -- 00:01:31.870 --> 00:01:37.230 to describe, first, what I call the Row picture. 00:01:37.230 --> 00:01:40.870 That's the picture of one equation at a time. 00:01:40.870 --> 00:01:43.980 It's the picture you've seen before in two 00:01:43.980 --> 00:01:46.350 by two equations where lines meet. 00:01:46.350 --> 00:01:49.460 So in a minute, you'll see lines meeting. 00:01:49.460 --> 00:01:52.230 The second picture, I'll put a star 00:01:52.230 --> 00:01:55.070 beside that, because that's such an important one. 00:01:55.070 --> 00:02:01.600 And maybe new to you is the picture -- a column at a time. 00:02:01.600 --> 00:02:05.440 And those are the rows and columns of a matrix. 00:02:05.440 --> 00:02:08.970 So the third -- the algebra way to look at the problem is 00:02:08.970 --> 00:02:13.790 the matrix form and using a matrix that I'll call A. 00:02:13.790 --> 00:02:17.490 Okay, so can I do an example? 00:02:17.490 --> 00:02:23.420 The whole semester will be examples and then see what's 00:02:23.420 --> 00:02:25.110 going on with the example. 00:02:25.110 --> 00:02:27.990 So, take an example. 00:02:27.990 --> 00:02:30.430 Two equations, two unknowns. 00:02:30.430 --> 00:02:38.290 So let me take 2x -y =0, let's say. 00:02:38.290 --> 00:02:45.350 And -x 2y=3. 00:02:45.350 --> 00:02:46.530 Okay. 00:02:46.530 --> 00:02:50.680 let me -- I can even say right away -- 00:02:50.680 --> 00:02:57.650 what's the matrix, that is, what's the coefficient matrix? 00:02:57.650 --> 00:03:00.590 The matrix that involves these numbers -- 00:03:00.590 --> 00:03:05.190 a matrix is just a rectangular array of numbers. 00:03:05.190 --> 00:03:09.610 Here it's two rows and two columns, so 2 and -- 00:03:09.610 --> 00:03:14.420 minus 1 in the first row minus 1 and 2 in the second row, 00:03:14.420 --> 00:03:16.700 that's the matrix. 00:03:16.700 --> 00:03:21.610 And the right-hand -- the, unknown -- 00:03:21.610 --> 00:03:23.090 well, we've got two unknowns. 00:03:23.090 --> 00:03:27.370 So we've got a vector, with two components, x and y, 00:03:27.370 --> 00:03:31.150 and we've got two right-hand sides that go into a vector 00:03:31.150 --> 00:03:32.940 0 3. 00:03:32.940 --> 00:03:38.260 I couldn't resist writing the matrix form, right -- 00:03:38.260 --> 00:03:41.050 even before the pictures. 00:03:41.050 --> 00:03:45.030 So I always will think of this as the matrix A, 00:03:45.030 --> 00:03:49.440 the matrix of coefficients, then there's a vector of 00:03:49.440 --> 00:03:50.030 unknowns. 00:03:50.030 --> 00:03:51.790 Here we've only got two unknowns. 00:03:51.790 --> 00:03:54.100 Later we'll have any number of unknowns. 00:03:54.100 --> 00:03:57.620 And that vector of unknowns, well I'll often -- 00:03:57.620 --> 00:03:59.820 I'll make that x -- 00:03:59.820 --> 00:04:01.150 extra bold. 00:04:01.150 --> 00:04:04.850 A and the right-hand side is also a vector 00:04:04.850 --> 00:04:08.230 that I'll always call b. 00:04:08.230 --> 00:04:13.790 So linear equations are A x equal b and the idea now 00:04:13.790 --> 00:04:16.820 is to solve this particular example 00:04:16.820 --> 00:04:20.940 and then step back to see the bigger picture. 00:04:20.940 --> 00:04:24.630 Okay, what's the picture for this example, the Row picture? 00:04:24.630 --> 00:04:26.740 Okay, so here comes the Row picture. 00:04:30.210 --> 00:04:32.890 So that means I take one row at a time 00:04:32.890 --> 00:04:38.520 and I'm drawing here the xy plane 00:04:38.520 --> 00:04:43.990 and I'm going to plot all the points that 00:04:43.990 --> 00:04:47.860 satisfy that first equation. 00:04:47.860 --> 00:04:53.120 So I'm looking at all the points that satisfy 2x-y =0. 00:04:53.120 --> 00:04:59.000 It's often good to start with which point on the horizontal 00:04:59.000 --> 00:05:02.480 line -- on this horizontal line, y is zero. 00:05:02.480 --> 00:05:06.510 The x axis has y as zero and that -- in this case, actually, 00:05:06.510 --> 00:05:07.720 then x is zero. 00:05:07.720 --> 00:05:11.230 So the point, the origin -- 00:05:11.230 --> 00:05:16.560 the point with coordinates (0,0) is on the line. 00:05:16.560 --> 00:05:18.290 It solves that equation. 00:05:18.290 --> 00:05:22.340 Okay, tell me in -- well, I guess I have to tell you 00:05:22.340 --> 00:05:26.980 another point that solves this same equation. 00:05:26.980 --> 00:05:32.090 Let me suppose x is one, so I'll take x to be one. 00:05:32.090 --> 00:05:34.930 Then y should be two, right? 00:05:34.930 --> 00:05:40.250 So there's the point one two that also solves this equation. 00:05:40.250 --> 00:05:42.840 And I could put in more points. 00:05:42.840 --> 00:05:45.720 But, but let me put in all the points 00:05:45.720 --> 00:05:50.810 at once, because they all lie on a straight line. 00:05:50.810 --> 00:05:53.540 This is a linear equation and that word linear 00:05:53.540 --> 00:05:54.310 got the letters 00:05:54.310 --> 00:05:56.760 Okay, thanks. for line in it. 00:05:56.760 --> 00:05:59.760 That's the equation -- this is the line that ... 00:05:59.760 --> 00:06:08.690 of solutions to 2x-y=0 my first row, first equation. 00:06:08.690 --> 00:06:14.060 So typically, maybe, x equal a half, y equal one will work. 00:06:14.060 --> 00:06:15.660 And sure enough it does. 00:06:15.660 --> 00:06:17.600 Okay, that's the first one. 00:06:17.600 --> 00:06:22.750 Now the second one is not going to go through the origin. 00:06:22.750 --> 00:06:24.390 It's always important. 00:06:24.390 --> 00:06:26.900 Do we go through the origin or not? 00:06:26.900 --> 00:06:30.020 In this case, yes, because there's a zero over there. 00:06:30.020 --> 00:06:32.270 In this case we don't go through the origin, 00:06:32.270 --> 00:06:36.220 because if x and y are zero, we don't get three. 00:06:36.220 --> 00:06:40.350 So, let me again say suppose y is zero, 00:06:40.350 --> 00:06:42.560 what x do we actually get? 00:06:42.560 --> 00:06:48.230 If y is zero, then I get x is minus three. 00:06:48.230 --> 00:06:52.610 So if y is zero, I go along minus three. 00:06:52.610 --> 00:06:56.360 So there's one point on this second line. 00:06:56.360 --> 00:07:03.450 Now let me say, well, suppose x is minus one -- 00:07:03.450 --> 00:07:05.370 just to take another x. 00:07:05.370 --> 00:07:09.280 If x is minus one, then this is a one 00:07:09.280 --> 00:07:15.870 and I think y should be a one, because if x is minus one, 00:07:15.870 --> 00:07:19.790 then I think y should be a one and we'll get that point. 00:07:19.790 --> 00:07:20.900 Is that right? 00:07:20.900 --> 00:07:23.330 If x is minus one, that's a one. 00:07:23.330 --> 00:07:25.640 If y is a one, that's a two and the one 00:07:25.640 --> 00:07:28.550 and the two make three and that point's on the equation. 00:07:28.550 --> 00:07:29.330 Okay. 00:07:29.330 --> 00:07:31.740 Now, I should just draw the line, right, 00:07:31.740 --> 00:07:34.470 connecting those two points at -- 00:07:34.470 --> 00:07:36.410 that will give me the whole line. 00:07:36.410 --> 00:07:40.480 And if I've done this reasonably well, 00:07:40.480 --> 00:07:42.640 I think it's going to happen to go through -- well, 00:07:42.640 --> 00:07:46.520 not happen -- it was arranged to go through that point. 00:07:46.520 --> 00:07:50.600 So I think that the second line is this one, 00:07:50.600 --> 00:07:56.100 and this is the all-important point that lies on both lines. 00:07:56.100 --> 00:07:58.830 Shall we just check that that point which 00:07:58.830 --> 00:08:03.690 is the point x equal one and y was two, right? 00:08:03.690 --> 00:08:07.790 That's the point there and that, I believe, 00:08:07.790 --> 00:08:10.530 solves both equations. 00:08:10.530 --> 00:08:11.500 Let's just check this. 00:08:11.500 --> 00:08:18.590 If x is one, I have a minus one plus four equals three, okay. 00:08:18.590 --> 00:08:21.680 Apologies for drawing this picture 00:08:21.680 --> 00:08:25.150 that you've seen before. 00:08:25.150 --> 00:08:29.690 But this -- seeing the row picture -- 00:08:29.690 --> 00:08:33.140 first of all, for n equal 2, two equations and two unknowns, 00:08:33.140 --> 00:08:35.460 it's the right place to start. 00:08:35.460 --> 00:08:36.270 Okay. 00:08:36.270 --> 00:08:37.690 So we've got the solution. 00:08:37.690 --> 00:08:39.679 The point that lies on both lines. 00:08:39.679 --> 00:08:43.600 Now can I come to the column picture? 00:08:43.600 --> 00:08:47.880 Pay attention, this is the key point. 00:08:47.880 --> 00:08:49.930 So the column picture. 00:08:49.930 --> 00:08:53.520 I'm now going to look at the columns of the matrix. 00:08:53.520 --> 00:08:56.790 I'm going to look at this part and this part. 00:08:56.790 --> 00:09:05.010 I'm going to say that the x part is really x times -- 00:09:05.010 --> 00:09:06.740 you see, I'm putting the two -- 00:09:06.740 --> 00:09:10.160 I'm kind of getting the two equations at once -- 00:09:10.160 --> 00:09:15.290 that part and then I have a y and in the first equation 00:09:15.290 --> 00:09:19.260 it's multiplying a minus one and in the second equation a two, 00:09:19.260 --> 00:09:22.030 and on the right-hand side, zero and three. 00:09:26.500 --> 00:09:29.860 You see, the columns of the matrix, the columns of A 00:09:29.860 --> 00:09:33.050 are here and the right-hand side b is there. 00:09:33.050 --> 00:09:36.940 And now what is the equation asking for? 00:09:36.940 --> 00:09:42.750 It's asking us to find -- somehow to combine that vector 00:09:42.750 --> 00:09:47.470 and this one in the right amounts to get that one. 00:09:47.470 --> 00:09:51.830 It's asking us to find the right linear combination -- 00:09:51.830 --> 00:09:55.420 this is called a linear combination. 00:09:55.420 --> 00:09:58.840 And it's the most fundamental operation in the whole course. 00:10:01.710 --> 00:10:06.330 It's a linear combination of the columns. 00:10:06.330 --> 00:10:09.550 That's what we're seeing on the left side. 00:10:09.550 --> 00:10:13.190 Again, I don't want to write down a big definition. 00:10:13.190 --> 00:10:15.010 You can see what it is. 00:10:15.010 --> 00:10:17.470 There's column one, there's column two. 00:10:17.470 --> 00:10:21.690 I multiply by some numbers and I add. 00:10:21.690 --> 00:10:25.940 That's a combination -- a linear combination and I want to make 00:10:25.940 --> 00:10:30.870 those numbers the right numbers to produce zero three. 00:10:30.870 --> 00:10:32.070 Okay. 00:10:32.070 --> 00:10:37.110 Now I want to draw a picture that, represents what this -- 00:10:37.110 --> 00:10:38.230 this is algebra. 00:10:38.230 --> 00:10:41.740 What's the geometry, what's the picture that goes with it? 00:10:41.740 --> 00:10:42.360 Okay. 00:10:42.360 --> 00:10:45.260 So again, these vectors have two components, 00:10:45.260 --> 00:10:50.730 so I better draw a picture like that. 00:10:50.730 --> 00:10:52.680 So can I put down these columns? 00:10:52.680 --> 00:10:55.470 I'll draw these columns as they are, 00:10:55.470 --> 00:10:58.700 and then I'll do a combination of them. 00:10:58.700 --> 00:11:04.000 So the first column is over two and down one, right? 00:11:04.000 --> 00:11:06.210 So there's the first column. 00:11:06.210 --> 00:11:07.860 The first column. 00:11:07.860 --> 00:11:10.960 Column one. 00:11:10.960 --> 00:11:13.830 It's the vector two minus one. 00:11:13.830 --> 00:11:18.580 The second column is -- 00:11:18.580 --> 00:11:22.000 minus one is the first component and up two. 00:11:22.000 --> 00:11:24.960 It's here. 00:11:24.960 --> 00:11:27.710 There's column two. 00:11:27.710 --> 00:11:32.280 So this, again, you see what its components are. 00:11:32.280 --> 00:11:35.530 Its components are minus one, two. 00:11:35.530 --> 00:11:36.030 Good. 00:11:36.030 --> 00:11:36.840 That's this guy. 00:11:39.471 --> 00:11:40.845 Now I have to take a combination. 00:11:43.740 --> 00:11:47.310 What combination shall I take? 00:11:47.310 --> 00:11:50.510 Why not the right combination, what the hell? 00:11:50.510 --> 00:11:51.010 Okay. 00:11:51.010 --> 00:11:53.840 So the combination I'm going to take 00:11:53.840 --> 00:11:57.250 is the right one to produce zero three 00:11:57.250 --> 00:11:59.620 and then we'll see it happen in the picture. 00:11:59.620 --> 00:12:04.530 So the right combination is to take x as one of those 00:12:04.530 --> 00:12:06.140 and two of these. 00:12:09.230 --> 00:12:12.760 It's because we already know that that's the right x and y, 00:12:12.760 --> 00:12:15.720 so why not take the correct combination here and see it 00:12:15.720 --> 00:12:16.280 happen? 00:12:16.280 --> 00:12:21.840 Okay, so how do I picture this linear combination? 00:12:21.840 --> 00:12:25.180 So I start with this vector that's already here -- 00:12:25.180 --> 00:12:28.220 so that's one of column one, that's one times column 00:12:28.220 --> 00:12:30.400 one, right there. 00:12:30.400 --> 00:12:33.720 And now I want to add on -- so I'm going to hook the next 00:12:33.720 --> 00:12:38.660 vector onto the front of the arrow will start the next 00:12:38.660 --> 00:12:40.590 vector and it will go this way. 00:12:40.590 --> 00:12:44.170 So let's see, can I do it right? 00:12:44.170 --> 00:12:46.240 If I added on one of these vectors, 00:12:46.240 --> 00:12:50.589 it would go left one and up two, so we'd go left one and up two, 00:12:50.589 --> 00:12:52.130 so it would probably get us to there. 00:12:52.130 --> 00:12:54.650 Maybe I'll do dotted line for that. 00:12:54.650 --> 00:12:55.610 Okay? 00:12:55.610 --> 00:12:59.710 That's one of column two tucked onto the end, 00:12:59.710 --> 00:13:03.390 but I wanted to tuck on two of column two. 00:13:03.390 --> 00:13:07.000 So that -- the second one -- we'll go up left one and up two 00:13:07.000 --> 00:13:07.660 also. 00:13:07.660 --> 00:13:09.440 It'll probably end there. 00:13:09.440 --> 00:13:10.520 And there's another one. 00:13:10.520 --> 00:13:14.350 So what I've put in here is two of column two. 00:13:17.310 --> 00:13:19.340 Added on. 00:13:19.340 --> 00:13:20.615 And where did I end up? 00:13:23.280 --> 00:13:28.230 What are the coordinates of this result? 00:13:28.230 --> 00:13:32.570 What do I get when I take one of this plus two of that? 00:13:32.570 --> 00:13:34.810 I do get that, of course. 00:13:34.810 --> 00:13:38.870 There it is, x is zero, y is three, that's b. 00:13:38.870 --> 00:13:41.530 That's the answer we wanted. 00:13:41.530 --> 00:13:43.530 And how do I do it? 00:13:43.530 --> 00:13:46.860 You see I do it just like the first component. 00:13:46.860 --> 00:13:50.700 I have a two and a minus two that produces a zero, 00:13:50.700 --> 00:13:54.680 and in the second component I have a minus one and a four, 00:13:54.680 --> 00:13:57.750 they combine to give the three. 00:13:57.750 --> 00:14:01.190 But look at this picture. 00:14:01.190 --> 00:14:03.230 So here's our key picture. 00:14:03.230 --> 00:14:10.280 I combine this column and this column to get this guy. 00:14:10.280 --> 00:14:11.360 That was the b. 00:14:11.360 --> 00:14:13.680 That's the zero three. 00:14:13.680 --> 00:14:14.540 Okay. 00:14:14.540 --> 00:14:18.790 So that idea of linear combination is crucial, 00:14:18.790 --> 00:14:21.130 and also -- 00:14:21.130 --> 00:14:22.915 do we want to think about this question? 00:14:25.420 --> 00:14:26.470 Sure, why not. 00:14:26.470 --> 00:14:29.830 What are all the combinations? 00:14:29.830 --> 00:14:32.655 If I took -- can I go back to xs and ys? 00:14:35.240 --> 00:14:38.610 This is a question for really -- 00:14:38.610 --> 00:14:41.450 it's going to come up over and over, 00:14:41.450 --> 00:14:45.350 but why don't we see it once now? 00:14:45.350 --> 00:14:50.610 If I took all the xs and all the ys, all the combinations, 00:14:50.610 --> 00:14:53.830 what would be all the results? 00:14:53.830 --> 00:14:56.200 And, actually, the result would be 00:14:56.200 --> 00:14:59.400 that I could get any right-hand side at all. 00:14:59.400 --> 00:15:02.680 The combinations of this and this 00:15:02.680 --> 00:15:06.250 would fill the whole plane. 00:15:06.250 --> 00:15:08.510 You can tuck that away. 00:15:08.510 --> 00:15:14.080 We'll, explore it further. 00:15:14.080 --> 00:15:19.610 But this idea of what linear combination gives b and what do 00:15:19.610 --> 00:15:21.650 all the linear combinations give, 00:15:21.650 --> 00:15:25.070 what are all the possible, achievable right-hand sides be 00:15:25.070 --> 00:15:26.610 -- that's going to be basic. 00:15:26.610 --> 00:15:27.220 Okay. 00:15:27.220 --> 00:15:31.980 Can I move to three equations and three unknowns? 00:15:31.980 --> 00:15:38.610 Because it's easy to picture the two by two case. 00:15:38.610 --> 00:15:40.980 Let me do a three by three example. 00:15:40.980 --> 00:15:43.330 Okay, I'll sort of start it the same way, 00:15:43.330 --> 00:15:50.210 say maybe 2x-y and maybe I'll take no zs as a zero and maybe 00:15:50.210 --> 00:15:55.940 a -x 2y and maybe a -z as a -- oh, 00:15:55.940 --> 00:16:01.130 let me make that a minus one and, just for variety let me 00:16:01.130 --> 00:16:09.930 take, -3z, -3ys, I should keep the ys in that line, 00:16:09.930 --> 00:16:14.400 and 4zs is, say, 4. 00:16:14.400 --> 00:16:16.480 Okay. 00:16:16.480 --> 00:16:19.100 That's three equations. 00:16:19.100 --> 00:16:22.380 I'm in three dimensions, x, y, z. 00:16:22.380 --> 00:16:27.210 And, I don't have a solution yet. 00:16:27.210 --> 00:16:32.230 So I want to understand the equations and then solve them. 00:16:32.230 --> 00:16:32.780 Okay. 00:16:32.780 --> 00:16:35.450 So how do I you understand them? 00:16:35.450 --> 00:16:37.350 The row picture one way. 00:16:37.350 --> 00:16:41.120 The column picture is another very important way. 00:16:41.120 --> 00:16:43.720 Just let's remember the matrix form, here, 00:16:43.720 --> 00:16:45.020 because that's easy. 00:16:45.020 --> 00:16:49.140 The matrix form -- what's our matrix A? 00:16:49.140 --> 00:16:54.280 Our matrix A is this right-hand side, the two and the minus one 00:16:54.280 --> 00:16:58.620 and the zero from the first row, the minus one and the two 00:16:58.620 --> 00:17:00.780 and the minus one from the second row, 00:17:00.780 --> 00:17:08.530 the zero, the minus three and the four from the third row. 00:17:08.530 --> 00:17:11.050 So it's a three by three matrix. 00:17:11.050 --> 00:17:12.970 Three equations, three unknowns. 00:17:12.970 --> 00:17:14.680 And what's our right-hand side? 00:17:14.680 --> 00:17:20.190 Of course, it's the vector, zero minus one, four. 00:17:20.190 --> 00:17:21.180 Okay. 00:17:21.180 --> 00:17:26.950 So that's the way, well, that's the short-hand to write out 00:17:26.950 --> 00:17:28.690 the three equations. 00:17:28.690 --> 00:17:31.429 But it's the picture that I'm looking for today. 00:17:31.429 --> 00:17:32.470 Okay, so the row picture. 00:17:32.470 --> 00:17:38.370 All right, so I'm in three dimensions, x, 00:17:38.370 --> 00:17:42.680 find out when there isn't a solution. 00:17:42.680 --> 00:17:45.950 y and z. 00:17:45.950 --> 00:17:51.150 And I want to take those equations one at a time and ask 00:17:51.150 --> 00:17:51.820 -- 00:17:51.820 --> 00:17:55.280 and make a picture of all the points that satisfy -- 00:17:55.280 --> 00:17:57.960 let's take equation number two. 00:17:57.960 --> 00:18:00.820 If I make a picture of all the points that satisfy -- 00:18:00.820 --> 00:18:05.900 all the x, y, z points that solve this equation -- 00:18:05.900 --> 00:18:09.550 well, first of all, the origin is not one of them. 00:18:09.550 --> 00:18:14.620 x, y, z -- it being 0, 0, 0 would not solve that equation. 00:18:14.620 --> 00:18:17.790 So what are some points that do solve the equation? 00:18:17.790 --> 00:18:23.750 Let's see, maybe if x is one, y and z could be zero. 00:18:23.750 --> 00:18:24.750 That would work, right? 00:18:24.750 --> 00:18:26.870 So there's one point. 00:18:26.870 --> 00:18:29.520 I'm looking at this second equation, 00:18:29.520 --> 00:18:33.120 here, just, to start with. 00:18:33.120 --> 00:18:33.670 Let's see. 00:18:33.670 --> 00:18:36.430 Also, I guess, if z could be one, 00:18:36.430 --> 00:18:38.710 x and y could be zero, so that would just 00:18:38.710 --> 00:18:41.930 go straight up that axis. 00:18:41.930 --> 00:18:46.880 And, probably I'd want a third point here. 00:18:46.880 --> 00:18:55.390 Let me take x to be zero, z to be zero, 00:18:55.390 --> 00:19:00.400 then y would be minus a half, right? 00:19:00.400 --> 00:19:06.120 So there's a third point, somewhere -- oh my -- okay. 00:19:06.120 --> 00:19:07.900 Let's see. 00:19:07.900 --> 00:19:13.080 I want to put in all the points that satisfy that equation. 00:19:13.080 --> 00:19:17.930 Do you know what that bunch of points will be? 00:19:17.930 --> 00:19:19.350 It's a plane. 00:19:19.350 --> 00:19:22.430 If we have a linear equation, then, fortunately, 00:19:22.430 --> 00:19:27.070 the graph of the thing, the plot of all the points that solve it 00:19:27.070 --> 00:19:30.390 are a plane. 00:19:30.390 --> 00:19:32.220 These three points determine a plane, 00:19:32.220 --> 00:19:37.570 but your lecturer is not Rembrandt 00:19:37.570 --> 00:19:43.380 and the art is going to be the weak point here. 00:19:43.380 --> 00:19:46.460 So I'm just going to draw a plane, right? 00:19:46.460 --> 00:19:48.310 There's a plane somewhere. 00:19:48.310 --> 00:19:50.800 That's my plane. 00:19:50.800 --> 00:19:54.960 That plane is all the points that solves this guy. 00:19:54.960 --> 00:19:59.880 Then, what about this one? 00:19:59.880 --> 00:20:02.760 Two x minus y plus zero z. 00:20:02.760 --> 00:20:05.410 So z actually can be anything. 00:20:05.410 --> 00:20:08.390 Again, it's going to be another plane. 00:20:08.390 --> 00:20:11.290 Each row in a three by three problem 00:20:11.290 --> 00:20:14.480 gives us a plane in three dimensions. 00:20:14.480 --> 00:20:17.730 So this one is going to be some other plane -- 00:20:17.730 --> 00:20:20.250 maybe I'll try to draw it like this. 00:20:20.250 --> 00:20:25.240 And those two planes meet in a line. 00:20:25.240 --> 00:20:29.280 So if I have two equations, just the first two 00:20:29.280 --> 00:20:33.380 equations in three dimensions, those give me a line. 00:20:33.380 --> 00:20:35.290 The line where those two planes meet. 00:20:35.290 --> 00:20:42.210 And now, the third guy is a third plane. 00:20:42.210 --> 00:20:49.290 And it goes somewhere. 00:20:49.290 --> 00:20:51.820 Okay, those three things meet in a point. 00:20:51.820 --> 00:20:55.090 Now I don't know where that point is, frankly. 00:20:55.090 --> 00:20:58.140 But -- linear algebra will find it. 00:20:58.140 --> 00:21:05.580 The main point is that the three planes, because they're not 00:21:05.580 --> 00:21:08.100 parallel, they're not special. 00:21:08.100 --> 00:21:11.780 They do meet in one point and that's the solution. 00:21:11.780 --> 00:21:16.910 But, maybe you can see that this row picture is 00:21:16.910 --> 00:21:19.120 getting a little hard to see. 00:21:19.120 --> 00:21:24.520 The row picture was a cinch when we looked at two lines 00:21:24.520 --> 00:21:25.220 meeting. 00:21:25.220 --> 00:21:27.740 When we look at three planes meeting, 00:21:27.740 --> 00:21:32.890 it's not so clear and in four dimensions probably 00:21:32.890 --> 00:21:34.780 a little less clear. 00:21:34.780 --> 00:21:37.530 So, can I quit on the row picture? 00:21:37.530 --> 00:21:41.870 Or quit on the row picture before I've successfully 00:21:41.870 --> 00:21:45.940 found the point where the three planes meet? 00:21:45.940 --> 00:21:51.510 All I really want to see is that the row picture consists 00:21:51.510 --> 00:21:55.780 of three planes and, if everything works right, 00:21:55.780 --> 00:21:59.340 three planes meet in one point and that's a solution. 00:21:59.340 --> 00:22:04.730 Now, you can tell I prefer the column picture. 00:22:04.730 --> 00:22:06.990 Okay, so let me take the column picture. 00:22:06.990 --> 00:22:09.820 That's x times -- 00:22:09.820 --> 00:22:14.400 so there were two xs in the first equation minus one x is, 00:22:14.400 --> 00:22:16.690 and no xs in the third. 00:22:16.690 --> 00:22:19.000 It's just the first column of that. 00:22:19.000 --> 00:22:21.300 And how many ys are there? 00:22:21.300 --> 00:22:24.730 There's minus one in the first equations, two in the second 00:22:24.730 --> 00:22:27.430 and maybe minus three in the third. 00:22:27.430 --> 00:22:29.830 Just the second column of my matrix. 00:22:29.830 --> 00:22:37.740 And z times no zs minus one zs and four zs. 00:22:37.740 --> 00:22:41.270 And it's those three columns, right, 00:22:41.270 --> 00:22:46.640 that I have to combine to produce the right-hand side, 00:22:46.640 --> 00:22:48.945 which is zero minus one four. 00:22:52.830 --> 00:22:54.910 Okay. 00:22:54.910 --> 00:22:57.940 So what have we got on this left-hand side? 00:22:57.940 --> 00:22:59.470 A linear combination. 00:22:59.470 --> 00:23:02.540 It's a linear combination now of three vectors, 00:23:02.540 --> 00:23:05.720 and they happen to be -- each one is a three dimensional 00:23:05.720 --> 00:23:10.000 vector, so we want to know what combination of those three 00:23:10.000 --> 00:23:12.090 vectors produces that one. 00:23:12.090 --> 00:23:15.820 Shall I try to draw the column picture, then? 00:23:15.820 --> 00:23:18.830 So, since these vectors have three components -- 00:23:18.830 --> 00:23:21.660 so it's some multiple -- let me draw in the first column 00:23:21.660 --> 00:23:23.340 as before -- 00:23:23.340 --> 00:23:27.287 x is two and y is minus one. 00:23:27.287 --> 00:23:28.620 Maybe there is the first column. 00:23:32.320 --> 00:23:38.570 y -- the second column has maybe a minus one and a two and the y 00:23:38.570 --> 00:23:44.180 is a minus three, somewhere, there possibly, column two. 00:23:44.180 --> 00:23:46.950 And the third column has -- 00:23:46.950 --> 00:23:52.760 no zero minus one four, so how shall I draw that? 00:23:52.760 --> 00:23:57.000 So this was the first component. 00:23:57.000 --> 00:23:59.230 The second component was a minus one. 00:23:59.230 --> 00:24:01.270 Maybe up here. 00:24:01.270 --> 00:24:08.921 That's column three, that's the column zero minus one and four. 00:24:08.921 --> 00:24:09.420 This guy. 00:24:12.050 --> 00:24:15.430 So, again, what's my problem? 00:24:15.430 --> 00:24:18.260 What this equation is asking me to do 00:24:18.260 --> 00:24:21.340 is to combine these three vectors 00:24:21.340 --> 00:24:27.900 with a right combination to produce this one. 00:24:27.900 --> 00:24:33.620 Well, you can see what the right combination is, because 00:24:33.620 --> 00:24:38.250 in this special problem, specially chosen 00:24:38.250 --> 00:24:42.780 by the lecturer, that right-hand side that I'm trying to get 00:24:42.780 --> 00:24:45.060 is actually one of these columns. 00:24:45.060 --> 00:24:46.660 So I know how to get that one. 00:24:46.660 --> 00:24:48.420 So what's the solution? 00:24:48.420 --> 00:24:50.910 What combination will work? 00:24:50.910 --> 00:24:53.940 I just want one of these and none of these. 00:24:53.940 --> 00:24:59.070 So x should be zero, y should be zero and z should 00:24:59.070 --> 00:25:00.040 be one. 00:25:02.860 --> 00:25:04.520 That's the combination. 00:25:04.520 --> 00:25:07.600 One of those is obviously the right one. 00:25:07.600 --> 00:25:09.870 Column three is actually the same 00:25:09.870 --> 00:25:12.090 as b in this particular problem. 00:25:15.640 --> 00:25:17.860 I made it work that way just so we 00:25:17.860 --> 00:25:21.770 would get an answer, (0,0,1), so somehow that's 00:25:21.770 --> 00:25:25.890 the point where those three planes met 00:25:25.890 --> 00:25:28.710 and I couldn't see it before. 00:25:28.710 --> 00:25:31.830 Of course, I won't always be able to see it from the column 00:25:31.830 --> 00:25:33.170 picture, either. 00:25:33.170 --> 00:25:39.390 It's the next lecture, actually, which is about elimination, 00:25:39.390 --> 00:25:46.670 which is the systematic way that everybody -- 00:25:46.670 --> 00:25:51.950 every bit of software, too -- 00:25:51.950 --> 00:25:56.830 production, large-scale software would solve the equations. 00:25:56.830 --> 00:25:59.020 So the lecture that's coming up. 00:25:59.020 --> 00:26:02.450 If I was to add that to the syllabus, 00:26:02.450 --> 00:26:09.050 will be about how to find x, y, z in all cases. 00:26:09.050 --> 00:26:15.110 Can I just think again, though, about the big picture? 00:26:15.110 --> 00:26:18.910 By the big picture I mean let's keep this same matrix 00:26:18.910 --> 00:26:21.990 on the left but imagine that we have 00:26:21.990 --> 00:26:24.250 a different right-hand side. 00:26:24.250 --> 00:26:27.430 Oh, let me take a different right-hand side. 00:26:27.430 --> 00:26:29.100 So I'll change that right-hand side 00:26:29.100 --> 00:26:34.610 to something that actually is also pretty special. 00:26:34.610 --> 00:26:36.810 Let me change it to -- 00:26:36.810 --> 00:26:38.940 if I add those first two columns, 00:26:38.940 --> 00:26:43.220 that would give me a one and a one and a minus three. 00:26:43.220 --> 00:26:46.210 There's a very special right-hand side. 00:26:46.210 --> 00:26:51.420 I just cooked it up by adding this one to this one. 00:26:51.420 --> 00:26:54.640 Now, what's the solution with this new right-hand side? 00:26:54.640 --> 00:26:58.960 The solution with this new right-hand side is clear. 00:26:58.960 --> 00:27:04.880 took one of these and none of those. 00:27:04.880 --> 00:27:08.030 So actually, it just changed around to this 00:27:08.030 --> 00:27:11.150 when I took this new right-hand side. 00:27:11.150 --> 00:27:12.010 Okay. 00:27:12.010 --> 00:27:19.070 So in the row picture, I have three different planes, 00:27:19.070 --> 00:27:23.800 three new planes meeting now at this point. 00:27:23.800 --> 00:27:27.000 In the column picture, I have the same three columns, 00:27:27.000 --> 00:27:30.640 but now I'm combining them to produce this guy, 00:27:30.640 --> 00:27:34.100 and it turned out that column one plus column two which would 00:27:34.100 --> 00:27:38.500 be somewhere -- there is the right column -- 00:27:38.500 --> 00:27:42.260 one of this and one of this would give me the new b. 00:27:45.190 --> 00:27:45.690 Okay. 00:27:45.690 --> 00:27:48.730 So we squeezed in an extra example. 00:27:48.730 --> 00:27:55.970 But now think about all bs, all right-hand sides. 00:27:55.970 --> 00:27:58.920 Can I solve these equations for every right-hand side? 00:28:02.030 --> 00:28:04.980 Can I ask that question? 00:28:04.980 --> 00:28:07.020 So that's the algebra question. 00:28:07.020 --> 00:28:11.870 Can I solve A x=b for every b? 00:28:11.870 --> 00:28:13.490 Let me write that down. 00:28:13.490 --> 00:28:24.320 Can I solve A x =b for every right-hand side b? 00:28:24.320 --> 00:28:26.420 I mean, is there a solution? 00:28:26.420 --> 00:28:29.670 And then, if there is, elimination 00:28:29.670 --> 00:28:31.940 will give me a way to find it. 00:28:31.940 --> 00:28:34.520 I really wanted to ask, is there a solution 00:28:34.520 --> 00:28:36.850 for every right-hand side? 00:28:36.850 --> 00:28:40.390 So now, can I put that in different words -- 00:28:40.390 --> 00:28:43.140 in this linear combination words? 00:28:43.140 --> 00:28:52.510 So in linear combination words, do the linear combinations 00:28:52.510 --> 00:29:03.060 of the columns fill three dimensional space? 00:29:05.740 --> 00:29:12.500 Every b means all the bs in three dimensional space. 00:29:12.500 --> 00:29:16.050 Do you see that I'm just asking the same question 00:29:16.050 --> 00:29:19.600 in different words? 00:29:19.600 --> 00:29:21.040 Solving A x -- 00:29:21.040 --> 00:29:25.020 A x -- that's very important. 00:29:25.020 --> 00:29:31.790 A times x -- when I multiply a matrix by a vector, 00:29:31.790 --> 00:29:34.940 I get a combination of the columns. 00:29:34.940 --> 00:29:38.650 I'll write that down in a moment. 00:29:38.650 --> 00:29:43.550 But in my column picture, that's really what I'm doing. 00:29:43.550 --> 00:29:47.080 I'm taking linear combinations of these three columns 00:29:47.080 --> 00:29:50.620 and I'm trying to find b. 00:29:50.620 --> 00:29:57.750 And, actually, the answer for this matrix will be yes. 00:29:57.750 --> 00:30:04.330 For this matrix A -- for these columns, the answer is yes. 00:30:04.330 --> 00:30:19.590 This matrix -- that I chose for an example is a good matrix. 00:30:19.590 --> 00:30:21.780 A non-singular matrix. 00:30:21.780 --> 00:30:23.180 An invertible matrix. 00:30:23.180 --> 00:30:26.700 Those will be the matrices that we like best. 00:30:26.700 --> 00:30:29.490 There could be other -- 00:30:29.490 --> 00:30:35.070 and we will see other matrices where the answer becomes, no -- 00:30:35.070 --> 00:30:38.320 oh, actually, you can see when it would become no. 00:30:38.320 --> 00:30:43.890 What could go wrong? find out -- because if elimination fails, 00:30:43.890 --> 00:30:46.820 How could it go wrong that out of these -- 00:30:46.820 --> 00:30:51.310 out of three columns and all their combinations -- 00:30:51.310 --> 00:30:58.390 when would I not be able to produce some b off here? 00:30:58.390 --> 00:31:00.610 When could it go wrong? 00:31:00.610 --> 00:31:04.350 Do you see that the combinations -- 00:31:04.350 --> 00:31:06.370 let me say when it goes wrong. 00:31:06.370 --> 00:31:12.310 If these three columns all lie in the same plane, 00:31:12.310 --> 00:31:18.110 then their combinations will lie in that same plane. 00:31:18.110 --> 00:31:19.640 So then we're in trouble. 00:31:19.640 --> 00:31:23.880 If the three columns of my matrix -- 00:31:23.880 --> 00:31:28.180 if those three vectors happen to lie in the same plane -- 00:31:28.180 --> 00:31:31.090 for example, if column three is just 00:31:31.090 --> 00:31:36.360 the sum of column one and column two, I would be in trouble. 00:31:36.360 --> 00:31:40.060 That would be a matrix A where the answer would be no, 00:31:40.060 --> 00:31:44.360 because the combinations -- 00:31:44.360 --> 00:31:48.260 if column three is in the same plane as column one and two, 00:31:48.260 --> 00:31:50.250 I don't get anything new from that. 00:31:50.250 --> 00:31:54.610 All the combinations are in the plane and only right-hand sides 00:31:54.610 --> 00:32:00.010 b that I could get would be the ones in that plane. 00:32:00.010 --> 00:32:03.680 So I could solve it for some right-hand sides, when 00:32:03.680 --> 00:32:08.320 b is in the plane, but most right-hand sides 00:32:08.320 --> 00:32:11.270 would be out of the plane and unreachable. 00:32:11.270 --> 00:32:14.190 So that would be a singular case. 00:32:14.190 --> 00:32:16.830 The matrix would be not invertible. 00:32:16.830 --> 00:32:19.810 There would not be a solution for every b. 00:32:19.810 --> 00:32:22.511 The answer would become no for that. 00:32:22.511 --> 00:32:23.010 Okay. 00:32:25.740 --> 00:32:27.040 I don't know -- 00:32:27.040 --> 00:32:29.190 shall we take just a little shot at thinking 00:32:29.190 --> 00:32:34.510 about nine dimensions? 00:32:34.510 --> 00:32:38.645 Imagine that we have vectors with nine components. 00:32:41.170 --> 00:32:44.920 Well, it's going to be hard to visualize those. 00:32:44.920 --> 00:32:46.800 I don't pretend to do it. 00:32:46.800 --> 00:32:51.950 But somehow, pretend you do. 00:32:51.950 --> 00:32:55.610 Pretend we have -- if this was nine equations and nine 00:32:55.610 --> 00:32:59.260 unknowns, then we would have nine columns, 00:32:59.260 --> 00:33:02.720 and each one would be a vector in nine-dimensional space 00:33:02.720 --> 00:33:06.930 and we would be looking at their linear combinations. 00:33:06.930 --> 00:33:09.340 So we would be having the linear combinations 00:33:09.340 --> 00:33:12.320 of nine vectors in nine-dimensional space, 00:33:12.320 --> 00:33:15.380 and we would be trying to find the combination that hit 00:33:15.380 --> 00:33:18.000 the correct right-hand side b. 00:33:18.000 --> 00:33:22.990 And we might also ask the question can we always do it? 00:33:22.990 --> 00:33:25.380 Can we get every right-hand side b? 00:33:25.380 --> 00:33:29.840 And certainly it will depend on those nine columns. 00:33:29.840 --> 00:33:32.480 Sometimes the answer will be yes -- 00:33:32.480 --> 00:33:35.500 if I picked a random matrix, it would be yes, actually. 00:33:35.500 --> 00:33:39.630 If I used MatLab and just used the random command, picked 00:33:39.630 --> 00:33:44.820 out a nine by nine matrix, I guarantee it would be 00:33:44.820 --> 00:33:45.320 good. 00:33:45.320 --> 00:33:47.140 It would be non-singular, it would 00:33:47.140 --> 00:33:49.270 be invertible, all beautiful. 00:33:49.270 --> 00:34:00.750 But if I choose those columns so that they're not independent, 00:34:00.750 --> 00:34:05.790 so that the ninth column is the same as the eighth column, 00:34:05.790 --> 00:34:09.020 then it contributes nothing new and there 00:34:09.020 --> 00:34:13.080 would be right-hand sides b that I couldn't get. 00:34:13.080 --> 00:34:18.210 Can you sort of think about nine vectors 00:34:18.210 --> 00:34:22.050 in nine-dimensional space an take their combinations? 00:34:22.050 --> 00:34:26.460 That's really the central thought -- 00:34:26.460 --> 00:34:30.920 that you get kind of used to in linear algebra. 00:34:30.920 --> 00:34:33.340 Even though you can't really visualize it, 00:34:33.340 --> 00:34:36.139 you sort of think you can after a while. 00:34:36.139 --> 00:34:40.830 Those nine columns and all their combinations 00:34:40.830 --> 00:34:45.030 may very well fill out the whole nine-dimensional space. 00:34:45.030 --> 00:34:48.239 But if the ninth column happened to be the same as the eighth 00:34:48.239 --> 00:34:51.090 column and gave nothing new, then probably what it would 00:34:51.090 --> 00:34:53.102 fill out would be -- 00:34:55.820 --> 00:35:03.070 I hesitate even to say this -- it would be a sort of a plane 00:35:03.070 --> 00:35:04.320 -- 00:35:04.320 --> 00:35:10.030 an eight dimensional plane inside nine-dimensional space. 00:35:10.030 --> 00:35:12.890 And it's those eight dimensional planes 00:35:12.890 --> 00:35:16.040 inside nine-dimensional space that we 00:35:16.040 --> 00:35:18.830 have to work with eventually. 00:35:18.830 --> 00:35:25.970 For now, let's stay with a nice case where the matrices work, 00:35:25.970 --> 00:35:29.980 we can get every right-hand side b and here 00:35:29.980 --> 00:35:32.350 we see how to do it with columns. 00:35:32.350 --> 00:35:33.160 Okay. 00:35:33.160 --> 00:35:36.520 There was one step which I realized 00:35:36.520 --> 00:35:41.210 I was saying in words that I now want to write in letters. 00:35:41.210 --> 00:35:45.750 Because I'm coming back to the matrix form of the equation, 00:35:45.750 --> 00:35:49.210 so let me write it here. 00:35:49.210 --> 00:35:54.270 The matrix form of my equation, of my system 00:35:54.270 --> 00:35:57.970 is some matrix A times some vector x 00:35:57.970 --> 00:36:00.920 equals some right-hand side b. 00:36:00.920 --> 00:36:01.950 Okay. 00:36:01.950 --> 00:36:03.450 So this is a multiplication. 00:36:03.450 --> 00:36:04.520 A times x. 00:36:04.520 --> 00:36:07.430 Matrix times vector, and I just want to say 00:36:07.430 --> 00:36:11.620 how do you multiply a matrix by a vector? 00:36:11.620 --> 00:36:16.550 Okay, so I'm just going to create a matrix -- 00:36:16.550 --> 00:36:21.640 let me take two five one three -- 00:36:21.640 --> 00:36:28.140 and let me take a vector x to be, say, 1and 2. 00:36:28.140 --> 00:36:31.500 How do I multiply a matrix by a vector? 00:36:34.150 --> 00:36:40.530 But just think a little bit about matrix notation 00:36:40.530 --> 00:36:42.120 and how to do that in multiplication. 00:36:42.120 --> 00:36:45.390 So let me say how I multiply a matrix by a vector. 00:36:45.390 --> 00:36:47.760 Actually, there are two ways to do it. 00:36:47.760 --> 00:36:50.430 Let me tell you my favorite way. 00:36:50.430 --> 00:36:52.770 It's columns again. 00:36:52.770 --> 00:36:54.720 It's a column at a time. 00:36:54.720 --> 00:36:57.980 For me, this matrix multiplication 00:36:57.980 --> 00:37:03.290 says I take one of that column and two of that column and add. 00:37:03.290 --> 00:37:06.770 So this is the way I would think of it 00:37:06.770 --> 00:37:12.820 is one of the first column and two of the second column 00:37:12.820 --> 00:37:17.300 and let's just see what we get. 00:37:17.300 --> 00:37:21.510 So in the first component I'm getting a two and a ten. 00:37:21.510 --> 00:37:23.470 I'm getting a twelve there. 00:37:23.470 --> 00:37:26.220 In the second component I'm getting a one and a six, 00:37:26.220 --> 00:37:27.860 I'm getting a seven. 00:37:27.860 --> 00:37:35.460 So that matrix times that vector is twelve seven. 00:37:35.460 --> 00:37:39.207 Now, you could do that another way. 00:37:39.207 --> 00:37:40.540 You could do it a row at a time. 00:37:40.540 --> 00:37:43.760 And you would get this twelve -- and actually I pretty much did 00:37:43.760 --> 00:37:44.970 it here -- 00:37:44.970 --> 00:37:45.710 this way. 00:37:45.710 --> 00:37:48.470 Two -- I could take that row times my vector. 00:37:48.470 --> 00:37:53.180 This is the idea of a dot product. 00:37:53.180 --> 00:37:58.140 This vector times this vector, two times one plus five times 00:37:58.140 --> 00:38:00.720 two is the twelve. 00:38:00.720 --> 00:38:04.610 This vector times this vector -- one times one plus three times 00:38:04.610 --> 00:38:06.000 two is the seven. 00:38:06.000 --> 00:38:11.810 So I can do it by rows, and in each row times 00:38:11.810 --> 00:38:16.360 my x is what I'll later call a dot product. 00:38:16.360 --> 00:38:19.430 But I also like to see it by columns. 00:38:19.430 --> 00:38:22.490 I see this as a linear combination of a column. 00:38:22.490 --> 00:38:24.130 So here's my point. 00:38:24.130 --> 00:38:36.180 A times x is a combination of the columns of A. 00:38:36.180 --> 00:38:43.650 That's how I hope you will think of A times x when we need 00:38:43.650 --> 00:38:44.150 it. 00:38:44.150 --> 00:38:47.230 Right now we've got -- with small ones, 00:38:47.230 --> 00:38:51.390 we can always do it in different ways, but later, 00:38:51.390 --> 00:38:53.930 think of it that way. 00:38:53.930 --> 00:38:54.480 Okay. 00:38:54.480 --> 00:39:02.020 So that's the picture for a two by two system. 00:39:02.020 --> 00:39:05.970 And if the right-hand side B happened to be twelve seven, 00:39:05.970 --> 00:39:12.110 then of course the correct solution would be one two. 00:39:12.110 --> 00:39:12.610 Okay. 00:39:12.610 --> 00:39:17.950 So let me come back next time to a systematic way, 00:39:17.950 --> 00:39:23.470 using elimination, to find the solution, 00:39:23.470 --> 00:39:29.810 if there is one, to a system of any size and