1 00:00:08,711 --> 00:00:09,210 Hi. 2 00:00:09,210 --> 00:00:14,230 This is the first lecture in MIT's course 18.06, 3 00:00:14,230 --> 00:00:19,430 linear algebra, and I'm Gilbert Strang. 4 00:00:19,430 --> 00:00:23,440 The text for the course is this book, 5 00:00:23,440 --> 00:00:26,070 Introduction to Linear Algebra. 6 00:00:26,070 --> 00:00:32,729 And the course web page, which has got a lot of exercises from 7 00:00:32,729 --> 00:00:38,470 the past, MatLab codes, the syllabus for the course, 8 00:00:38,470 --> 00:00:44,060 is web.mit.edu/18.06. 9 00:00:44,060 --> 00:00:47,350 And this is the first lecture, lecture one. 10 00:00:49,920 --> 00:00:55,490 So, and later we'll give the web address for viewing these, 11 00:00:55,490 --> 00:00:57,270 videotapes. 12 00:00:57,270 --> 00:00:59,200 Okay, so what's in the first lecture? 13 00:01:02,680 --> 00:01:05,530 This is my plan. 14 00:01:05,530 --> 00:01:09,340 The fundamental problem of linear algebra, 15 00:01:09,340 --> 00:01:13,880 which is to solve a system of linear equations. 16 00:01:13,880 --> 00:01:16,210 So let's start with a case when we 17 00:01:16,210 --> 00:01:21,440 have some number of equations, say n equations and n unknowns. 18 00:01:21,440 --> 00:01:23,820 So an equal number of equations and unknowns. 19 00:01:23,820 --> 00:01:26,860 That's the normal, nice case. 20 00:01:26,860 --> 00:01:31,870 And what I want to do is -- with examples, of course -- 21 00:01:31,870 --> 00:01:37,230 to describe, first, what I call the Row picture. 22 00:01:37,230 --> 00:01:40,870 That's the picture of one equation at a time. 23 00:01:40,870 --> 00:01:43,980 It's the picture you've seen before in two 24 00:01:43,980 --> 00:01:46,350 by two equations where lines meet. 25 00:01:46,350 --> 00:01:49,460 So in a minute, you'll see lines meeting. 26 00:01:49,460 --> 00:01:52,230 The second picture, I'll put a star 27 00:01:52,230 --> 00:01:55,070 beside that, because that's such an important one. 28 00:01:55,070 --> 00:02:01,600 And maybe new to you is the picture -- a column at a time. 29 00:02:01,600 --> 00:02:05,440 And those are the rows and columns of a matrix. 30 00:02:05,440 --> 00:02:08,970 So the third -- the algebra way to look at the problem is 31 00:02:08,970 --> 00:02:13,790 the matrix form and using a matrix that I'll call A. 32 00:02:13,790 --> 00:02:17,490 Okay, so can I do an example? 33 00:02:17,490 --> 00:02:23,420 The whole semester will be examples and then see what's 34 00:02:23,420 --> 00:02:25,110 going on with the example. 35 00:02:25,110 --> 00:02:27,990 So, take an example. 36 00:02:27,990 --> 00:02:30,430 Two equations, two unknowns. 37 00:02:30,430 --> 00:02:38,290 So let me take 2x -y =0, let's say. 38 00:02:38,290 --> 00:02:45,350 And -x 2y=3. 39 00:02:45,350 --> 00:02:46,530 Okay. 40 00:02:46,530 --> 00:02:50,680 let me -- I can even say right away -- 41 00:02:50,680 --> 00:02:57,650 what's the matrix, that is, what's the coefficient matrix? 42 00:02:57,650 --> 00:03:00,590 The matrix that involves these numbers -- 43 00:03:00,590 --> 00:03:05,190 a matrix is just a rectangular array of numbers. 44 00:03:05,190 --> 00:03:09,610 Here it's two rows and two columns, so 2 and -- 45 00:03:09,610 --> 00:03:14,420 minus 1 in the first row minus 1 and 2 in the second row, 46 00:03:14,420 --> 00:03:16,700 that's the matrix. 47 00:03:16,700 --> 00:03:21,610 And the right-hand -- the, unknown -- 48 00:03:21,610 --> 00:03:23,090 well, we've got two unknowns. 49 00:03:23,090 --> 00:03:27,370 So we've got a vector, with two components, x and y, 50 00:03:27,370 --> 00:03:31,150 and we've got two right-hand sides that go into a vector 51 00:03:31,150 --> 00:03:32,940 0 3. 52 00:03:32,940 --> 00:03:38,260 I couldn't resist writing the matrix form, right -- 53 00:03:38,260 --> 00:03:41,050 even before the pictures. 54 00:03:41,050 --> 00:03:45,030 So I always will think of this as the matrix A, 55 00:03:45,030 --> 00:03:49,440 the matrix of coefficients, then there's a vector of 56 00:03:49,440 --> 00:03:50,030 unknowns. 57 00:03:50,030 --> 00:03:51,790 Here we've only got two unknowns. 58 00:03:51,790 --> 00:03:54,100 Later we'll have any number of unknowns. 59 00:03:54,100 --> 00:03:57,620 And that vector of unknowns, well I'll often -- 60 00:03:57,620 --> 00:03:59,820 I'll make that x -- 61 00:03:59,820 --> 00:04:01,150 extra bold. 62 00:04:01,150 --> 00:04:04,850 A and the right-hand side is also a vector 63 00:04:04,850 --> 00:04:08,230 that I'll always call b. 64 00:04:08,230 --> 00:04:13,790 So linear equations are A x equal b and the idea now 65 00:04:13,790 --> 00:04:16,820 is to solve this particular example 66 00:04:16,820 --> 00:04:20,940 and then step back to see the bigger picture. 67 00:04:20,940 --> 00:04:24,630 Okay, what's the picture for this example, the Row picture? 68 00:04:24,630 --> 00:04:26,740 Okay, so here comes the Row picture. 69 00:04:30,210 --> 00:04:32,890 So that means I take one row at a time 70 00:04:32,890 --> 00:04:38,520 and I'm drawing here the xy plane 71 00:04:38,520 --> 00:04:43,990 and I'm going to plot all the points that 72 00:04:43,990 --> 00:04:47,860 satisfy that first equation. 73 00:04:47,860 --> 00:04:53,120 So I'm looking at all the points that satisfy 2x-y =0. 74 00:04:53,120 --> 00:04:59,000 It's often good to start with which point on the horizontal 75 00:04:59,000 --> 00:05:02,480 line -- on this horizontal line, y is zero. 76 00:05:02,480 --> 00:05:06,510 The x axis has y as zero and that -- in this case, actually, 77 00:05:06,510 --> 00:05:07,720 then x is zero. 78 00:05:07,720 --> 00:05:11,230 So the point, the origin -- 79 00:05:11,230 --> 00:05:16,560 the point with coordinates (0,0) is on the line. 80 00:05:16,560 --> 00:05:18,290 It solves that equation. 81 00:05:18,290 --> 00:05:22,340 Okay, tell me in -- well, I guess I have to tell you 82 00:05:22,340 --> 00:05:26,980 another point that solves this same equation. 83 00:05:26,980 --> 00:05:32,090 Let me suppose x is one, so I'll take x to be one. 84 00:05:32,090 --> 00:05:34,930 Then y should be two, right? 85 00:05:34,930 --> 00:05:40,250 So there's the point one two that also solves this equation. 86 00:05:40,250 --> 00:05:42,840 And I could put in more points. 87 00:05:42,840 --> 00:05:45,720 But, but let me put in all the points 88 00:05:45,720 --> 00:05:50,810 at once, because they all lie on a straight line. 89 00:05:50,810 --> 00:05:53,540 This is a linear equation and that word linear 90 00:05:53,540 --> 00:05:54,310 got the letters 91 00:05:54,310 --> 00:05:56,760 Okay, thanks. for line in it. 92 00:05:56,760 --> 00:05:59,760 That's the equation -- this is the line that ... 93 00:05:59,760 --> 00:06:08,690 of solutions to 2x-y=0 my first row, first equation. 94 00:06:08,690 --> 00:06:14,060 So typically, maybe, x equal a half, y equal one will work. 95 00:06:14,060 --> 00:06:15,660 And sure enough it does. 96 00:06:15,660 --> 00:06:17,600 Okay, that's the first one. 97 00:06:17,600 --> 00:06:22,750 Now the second one is not going to go through the origin. 98 00:06:22,750 --> 00:06:24,390 It's always important. 99 00:06:24,390 --> 00:06:26,900 Do we go through the origin or not? 100 00:06:26,900 --> 00:06:30,020 In this case, yes, because there's a zero over there. 101 00:06:30,020 --> 00:06:32,270 In this case we don't go through the origin, 102 00:06:32,270 --> 00:06:36,220 because if x and y are zero, we don't get three. 103 00:06:36,220 --> 00:06:40,350 So, let me again say suppose y is zero, 104 00:06:40,350 --> 00:06:42,560 what x do we actually get? 105 00:06:42,560 --> 00:06:48,230 If y is zero, then I get x is minus three. 106 00:06:48,230 --> 00:06:52,610 So if y is zero, I go along minus three. 107 00:06:52,610 --> 00:06:56,360 So there's one point on this second line. 108 00:06:56,360 --> 00:07:03,450 Now let me say, well, suppose x is minus one -- 109 00:07:03,450 --> 00:07:05,370 just to take another x. 110 00:07:05,370 --> 00:07:09,280 If x is minus one, then this is a one 111 00:07:09,280 --> 00:07:15,870 and I think y should be a one, because if x is minus one, 112 00:07:15,870 --> 00:07:19,790 then I think y should be a one and we'll get that point. 113 00:07:19,790 --> 00:07:20,900 Is that right? 114 00:07:20,900 --> 00:07:23,330 If x is minus one, that's a one. 115 00:07:23,330 --> 00:07:25,640 If y is a one, that's a two and the one 116 00:07:25,640 --> 00:07:28,550 and the two make three and that point's on the equation. 117 00:07:28,550 --> 00:07:29,330 Okay. 118 00:07:29,330 --> 00:07:31,740 Now, I should just draw the line, right, 119 00:07:31,740 --> 00:07:34,470 connecting those two points at -- 120 00:07:34,470 --> 00:07:36,410 that will give me the whole line. 121 00:07:36,410 --> 00:07:40,480 And if I've done this reasonably well, 122 00:07:40,480 --> 00:07:42,640 I think it's going to happen to go through -- well, 123 00:07:42,640 --> 00:07:46,520 not happen -- it was arranged to go through that point. 124 00:07:46,520 --> 00:07:50,600 So I think that the second line is this one, 125 00:07:50,600 --> 00:07:56,100 and this is the all-important point that lies on both lines. 126 00:07:56,100 --> 00:07:58,830 Shall we just check that that point which 127 00:07:58,830 --> 00:08:03,690 is the point x equal one and y was two, right? 128 00:08:03,690 --> 00:08:07,790 That's the point there and that, I believe, 129 00:08:07,790 --> 00:08:10,530 solves both equations. 130 00:08:10,530 --> 00:08:11,500 Let's just check this. 131 00:08:11,500 --> 00:08:18,590 If x is one, I have a minus one plus four equals three, okay. 132 00:08:18,590 --> 00:08:21,680 Apologies for drawing this picture 133 00:08:21,680 --> 00:08:25,150 that you've seen before. 134 00:08:25,150 --> 00:08:29,690 But this -- seeing the row picture -- 135 00:08:29,690 --> 00:08:33,140 first of all, for n equal 2, two equations and two unknowns, 136 00:08:33,140 --> 00:08:35,460 it's the right place to start. 137 00:08:35,460 --> 00:08:36,270 Okay. 138 00:08:36,270 --> 00:08:37,690 So we've got the solution. 139 00:08:37,690 --> 00:08:39,679 The point that lies on both lines. 140 00:08:39,679 --> 00:08:43,600 Now can I come to the column picture? 141 00:08:43,600 --> 00:08:47,880 Pay attention, this is the key point. 142 00:08:47,880 --> 00:08:49,930 So the column picture. 143 00:08:49,930 --> 00:08:53,520 I'm now going to look at the columns of the matrix. 144 00:08:53,520 --> 00:08:56,790 I'm going to look at this part and this part. 145 00:08:56,790 --> 00:09:05,010 I'm going to say that the x part is really x times -- 146 00:09:05,010 --> 00:09:06,740 you see, I'm putting the two -- 147 00:09:06,740 --> 00:09:10,160 I'm kind of getting the two equations at once -- 148 00:09:10,160 --> 00:09:15,290 that part and then I have a y and in the first equation 149 00:09:15,290 --> 00:09:19,260 it's multiplying a minus one and in the second equation a two, 150 00:09:19,260 --> 00:09:22,030 and on the right-hand side, zero and three. 151 00:09:26,500 --> 00:09:29,860 You see, the columns of the matrix, the columns of A 152 00:09:29,860 --> 00:09:33,050 are here and the right-hand side b is there. 153 00:09:33,050 --> 00:09:36,940 And now what is the equation asking for? 154 00:09:36,940 --> 00:09:42,750 It's asking us to find -- somehow to combine that vector 155 00:09:42,750 --> 00:09:47,470 and this one in the right amounts to get that one. 156 00:09:47,470 --> 00:09:51,830 It's asking us to find the right linear combination -- 157 00:09:51,830 --> 00:09:55,420 this is called a linear combination. 158 00:09:55,420 --> 00:09:58,840 And it's the most fundamental operation in the whole course. 159 00:10:01,710 --> 00:10:06,330 It's a linear combination of the columns. 160 00:10:06,330 --> 00:10:09,550 That's what we're seeing on the left side. 161 00:10:09,550 --> 00:10:13,190 Again, I don't want to write down a big definition. 162 00:10:13,190 --> 00:10:15,010 You can see what it is. 163 00:10:15,010 --> 00:10:17,470 There's column one, there's column two. 164 00:10:17,470 --> 00:10:21,690 I multiply by some numbers and I add. 165 00:10:21,690 --> 00:10:25,940 That's a combination -- a linear combination and I want to make 166 00:10:25,940 --> 00:10:30,870 those numbers the right numbers to produce zero three. 167 00:10:30,870 --> 00:10:32,070 Okay. 168 00:10:32,070 --> 00:10:37,110 Now I want to draw a picture that, represents what this -- 169 00:10:37,110 --> 00:10:38,230 this is algebra. 170 00:10:38,230 --> 00:10:41,740 What's the geometry, what's the picture that goes with it? 171 00:10:41,740 --> 00:10:42,360 Okay. 172 00:10:42,360 --> 00:10:45,260 So again, these vectors have two components, 173 00:10:45,260 --> 00:10:50,730 so I better draw a picture like that. 174 00:10:50,730 --> 00:10:52,680 So can I put down these columns? 175 00:10:52,680 --> 00:10:55,470 I'll draw these columns as they are, 176 00:10:55,470 --> 00:10:58,700 and then I'll do a combination of them. 177 00:10:58,700 --> 00:11:04,000 So the first column is over two and down one, right? 178 00:11:04,000 --> 00:11:06,210 So there's the first column. 179 00:11:06,210 --> 00:11:07,860 The first column. 180 00:11:07,860 --> 00:11:10,960 Column one. 181 00:11:10,960 --> 00:11:13,830 It's the vector two minus one. 182 00:11:13,830 --> 00:11:18,580 The second column is -- 183 00:11:18,580 --> 00:11:22,000 minus one is the first component and up two. 184 00:11:22,000 --> 00:11:24,960 It's here. 185 00:11:24,960 --> 00:11:27,710 There's column two. 186 00:11:27,710 --> 00:11:32,280 So this, again, you see what its components are. 187 00:11:32,280 --> 00:11:35,530 Its components are minus one, two. 188 00:11:35,530 --> 00:11:36,030 Good. 189 00:11:36,030 --> 00:11:36,840 That's this guy. 190 00:11:39,471 --> 00:11:40,845 Now I have to take a combination. 191 00:11:43,740 --> 00:11:47,310 What combination shall I take? 192 00:11:47,310 --> 00:11:50,510 Why not the right combination, what the hell? 193 00:11:50,510 --> 00:11:51,010 Okay. 194 00:11:51,010 --> 00:11:53,840 So the combination I'm going to take 195 00:11:53,840 --> 00:11:57,250 is the right one to produce zero three 196 00:11:57,250 --> 00:11:59,620 and then we'll see it happen in the picture. 197 00:11:59,620 --> 00:12:04,530 So the right combination is to take x as one of those 198 00:12:04,530 --> 00:12:06,140 and two of these. 199 00:12:09,230 --> 00:12:12,760 It's because we already know that that's the right x and y, 200 00:12:12,760 --> 00:12:15,720 so why not take the correct combination here and see it 201 00:12:15,720 --> 00:12:16,280 happen? 202 00:12:16,280 --> 00:12:21,840 Okay, so how do I picture this linear combination? 203 00:12:21,840 --> 00:12:25,180 So I start with this vector that's already here -- 204 00:12:25,180 --> 00:12:28,220 so that's one of column one, that's one times column 205 00:12:28,220 --> 00:12:30,400 one, right there. 206 00:12:30,400 --> 00:12:33,720 And now I want to add on -- so I'm going to hook the next 207 00:12:33,720 --> 00:12:38,660 vector onto the front of the arrow will start the next 208 00:12:38,660 --> 00:12:40,590 vector and it will go this way. 209 00:12:40,590 --> 00:12:44,170 So let's see, can I do it right? 210 00:12:44,170 --> 00:12:46,240 If I added on one of these vectors, 211 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, 212 00:12:50,589 --> 00:12:52,130 so it would probably get us to there. 213 00:12:52,130 --> 00:12:54,650 Maybe I'll do dotted line for that. 214 00:12:54,650 --> 00:12:55,610 Okay? 215 00:12:55,610 --> 00:12:59,710 That's one of column two tucked onto the end, 216 00:12:59,710 --> 00:13:03,390 but I wanted to tuck on two of column two. 217 00:13:03,390 --> 00:13:07,000 So that -- the second one -- we'll go up left one and up two 218 00:13:07,000 --> 00:13:07,660 also. 219 00:13:07,660 --> 00:13:09,440 It'll probably end there. 220 00:13:09,440 --> 00:13:10,520 And there's another one. 221 00:13:10,520 --> 00:13:14,350 So what I've put in here is two of column two. 222 00:13:17,310 --> 00:13:19,340 Added on. 223 00:13:19,340 --> 00:13:20,615 And where did I end up? 224 00:13:23,280 --> 00:13:28,230 What are the coordinates of this result? 225 00:13:28,230 --> 00:13:32,570 What do I get when I take one of this plus two of that? 226 00:13:32,570 --> 00:13:34,810 I do get that, of course. 227 00:13:34,810 --> 00:13:38,870 There it is, x is zero, y is three, that's b. 228 00:13:38,870 --> 00:13:41,530 That's the answer we wanted. 229 00:13:41,530 --> 00:13:43,530 And how do I do it? 230 00:13:43,530 --> 00:13:46,860 You see I do it just like the first component. 231 00:13:46,860 --> 00:13:50,700 I have a two and a minus two that produces a zero, 232 00:13:50,700 --> 00:13:54,680 and in the second component I have a minus one and a four, 233 00:13:54,680 --> 00:13:57,750 they combine to give the three. 234 00:13:57,750 --> 00:14:01,190 But look at this picture. 235 00:14:01,190 --> 00:14:03,230 So here's our key picture. 236 00:14:03,230 --> 00:14:10,280 I combine this column and this column to get this guy. 237 00:14:10,280 --> 00:14:11,360 That was the b. 238 00:14:11,360 --> 00:14:13,680 That's the zero three. 239 00:14:13,680 --> 00:14:14,540 Okay. 240 00:14:14,540 --> 00:14:18,790 So that idea of linear combination is crucial, 241 00:14:18,790 --> 00:14:21,130 and also -- 242 00:14:21,130 --> 00:14:22,915 do we want to think about this question? 243 00:14:25,420 --> 00:14:26,470 Sure, why not. 244 00:14:26,470 --> 00:14:29,830 What are all the combinations? 245 00:14:29,830 --> 00:14:32,655 If I took -- can I go back to xs and ys? 246 00:14:35,240 --> 00:14:38,610 This is a question for really -- 247 00:14:38,610 --> 00:14:41,450 it's going to come up over and over, 248 00:14:41,450 --> 00:14:45,350 but why don't we see it once now? 249 00:14:45,350 --> 00:14:50,610 If I took all the xs and all the ys, all the combinations, 250 00:14:50,610 --> 00:14:53,830 what would be all the results? 251 00:14:53,830 --> 00:14:56,200 And, actually, the result would be 252 00:14:56,200 --> 00:14:59,400 that I could get any right-hand side at all. 253 00:14:59,400 --> 00:15:02,680 The combinations of this and this 254 00:15:02,680 --> 00:15:06,250 would fill the whole plane. 255 00:15:06,250 --> 00:15:08,510 You can tuck that away. 256 00:15:08,510 --> 00:15:14,080 We'll, explore it further. 257 00:15:14,080 --> 00:15:19,610 But this idea of what linear combination gives b and what do 258 00:15:19,610 --> 00:15:21,650 all the linear combinations give, 259 00:15:21,650 --> 00:15:25,070 what are all the possible, achievable right-hand sides be 260 00:15:25,070 --> 00:15:26,610 -- that's going to be basic. 261 00:15:26,610 --> 00:15:27,220 Okay. 262 00:15:27,220 --> 00:15:31,980 Can I move to three equations and three unknowns? 263 00:15:31,980 --> 00:15:38,610 Because it's easy to picture the two by two case. 264 00:15:38,610 --> 00:15:40,980 Let me do a three by three example. 265 00:15:40,980 --> 00:15:43,330 Okay, I'll sort of start it the same way, 266 00:15:43,330 --> 00:15:50,210 say maybe 2x-y and maybe I'll take no zs as a zero and maybe 267 00:15:50,210 --> 00:15:55,940 a -x 2y and maybe a -z as a -- oh, 268 00:15:55,940 --> 00:16:01,130 let me make that a minus one and, just for variety let me 269 00:16:01,130 --> 00:16:09,930 take, -3z, -3ys, I should keep the ys in that line, 270 00:16:09,930 --> 00:16:14,400 and 4zs is, say, 4. 271 00:16:14,400 --> 00:16:16,480 Okay. 272 00:16:16,480 --> 00:16:19,100 That's three equations. 273 00:16:19,100 --> 00:16:22,380 I'm in three dimensions, x, y, z. 274 00:16:22,380 --> 00:16:27,210 And, I don't have a solution yet. 275 00:16:27,210 --> 00:16:32,230 So I want to understand the equations and then solve them. 276 00:16:32,230 --> 00:16:32,780 Okay. 277 00:16:32,780 --> 00:16:35,450 So how do I you understand them? 278 00:16:35,450 --> 00:16:37,350 The row picture one way. 279 00:16:37,350 --> 00:16:41,120 The column picture is another very important way. 280 00:16:41,120 --> 00:16:43,720 Just let's remember the matrix form, here, 281 00:16:43,720 --> 00:16:45,020 because that's easy. 282 00:16:45,020 --> 00:16:49,140 The matrix form -- what's our matrix A? 283 00:16:49,140 --> 00:16:54,280 Our matrix A is this right-hand side, the two and the minus one 284 00:16:54,280 --> 00:16:58,620 and the zero from the first row, the minus one and the two 285 00:16:58,620 --> 00:17:00,780 and the minus one from the second row, 286 00:17:00,780 --> 00:17:08,530 the zero, the minus three and the four from the third row. 287 00:17:08,530 --> 00:17:11,050 So it's a three by three matrix. 288 00:17:11,050 --> 00:17:12,970 Three equations, three unknowns. 289 00:17:12,970 --> 00:17:14,680 And what's our right-hand side? 290 00:17:14,680 --> 00:17:20,190 Of course, it's the vector, zero minus one, four. 291 00:17:20,190 --> 00:17:21,180 Okay. 292 00:17:21,180 --> 00:17:26,950 So that's the way, well, that's the short-hand to write out 293 00:17:26,950 --> 00:17:28,690 the three equations. 294 00:17:28,690 --> 00:17:31,429 But it's the picture that I'm looking for today. 295 00:17:31,429 --> 00:17:32,470 Okay, so the row picture. 296 00:17:32,470 --> 00:17:38,370 All right, so I'm in three dimensions, x, 297 00:17:38,370 --> 00:17:42,680 find out when there isn't a solution. 298 00:17:42,680 --> 00:17:45,950 y and z. 299 00:17:45,950 --> 00:17:51,150 And I want to take those equations one at a time and ask 300 00:17:51,150 --> 00:17:51,820 -- 301 00:17:51,820 --> 00:17:55,280 and make a picture of all the points that satisfy -- 302 00:17:55,280 --> 00:17:57,960 let's take equation number two. 303 00:17:57,960 --> 00:18:00,820 If I make a picture of all the points that satisfy -- 304 00:18:00,820 --> 00:18:05,900 all the x, y, z points that solve this equation -- 305 00:18:05,900 --> 00:18:09,550 well, first of all, the origin is not one of them. 306 00:18:09,550 --> 00:18:14,620 x, y, z -- it being 0, 0, 0 would not solve that equation. 307 00:18:14,620 --> 00:18:17,790 So what are some points that do solve the equation? 308 00:18:17,790 --> 00:18:23,750 Let's see, maybe if x is one, y and z could be zero. 309 00:18:23,750 --> 00:18:24,750 That would work, right? 310 00:18:24,750 --> 00:18:26,870 So there's one point. 311 00:18:26,870 --> 00:18:29,520 I'm looking at this second equation, 312 00:18:29,520 --> 00:18:33,120 here, just, to start with. 313 00:18:33,120 --> 00:18:33,670 Let's see. 314 00:18:33,670 --> 00:18:36,430 Also, I guess, if z could be one, 315 00:18:36,430 --> 00:18:38,710 x and y could be zero, so that would just 316 00:18:38,710 --> 00:18:41,930 go straight up that axis. 317 00:18:41,930 --> 00:18:46,880 And, probably I'd want a third point here. 318 00:18:46,880 --> 00:18:55,390 Let me take x to be zero, z to be zero, 319 00:18:55,390 --> 00:19:00,400 then y would be minus a half, right? 320 00:19:00,400 --> 00:19:06,120 So there's a third point, somewhere -- oh my -- okay. 321 00:19:06,120 --> 00:19:07,900 Let's see. 322 00:19:07,900 --> 00:19:13,080 I want to put in all the points that satisfy that equation. 323 00:19:13,080 --> 00:19:17,930 Do you know what that bunch of points will be? 324 00:19:17,930 --> 00:19:19,350 It's a plane. 325 00:19:19,350 --> 00:19:22,430 If we have a linear equation, then, fortunately, 326 00:19:22,430 --> 00:19:27,070 the graph of the thing, the plot of all the points that solve it 327 00:19:27,070 --> 00:19:30,390 are a plane. 328 00:19:30,390 --> 00:19:32,220 These three points determine a plane, 329 00:19:32,220 --> 00:19:37,570 but your lecturer is not Rembrandt 330 00:19:37,570 --> 00:19:43,380 and the art is going to be the weak point here. 331 00:19:43,380 --> 00:19:46,460 So I'm just going to draw a plane, right? 332 00:19:46,460 --> 00:19:48,310 There's a plane somewhere. 333 00:19:48,310 --> 00:19:50,800 That's my plane. 334 00:19:50,800 --> 00:19:54,960 That plane is all the points that solves this guy. 335 00:19:54,960 --> 00:19:59,880 Then, what about this one? 336 00:19:59,880 --> 00:20:02,760 Two x minus y plus zero z. 337 00:20:02,760 --> 00:20:05,410 So z actually can be anything. 338 00:20:05,410 --> 00:20:08,390 Again, it's going to be another plane. 339 00:20:08,390 --> 00:20:11,290 Each row in a three by three problem 340 00:20:11,290 --> 00:20:14,480 gives us a plane in three dimensions. 341 00:20:14,480 --> 00:20:17,730 So this one is going to be some other plane -- 342 00:20:17,730 --> 00:20:20,250 maybe I'll try to draw it like this. 343 00:20:20,250 --> 00:20:25,240 And those two planes meet in a line. 344 00:20:25,240 --> 00:20:29,280 So if I have two equations, just the first two 345 00:20:29,280 --> 00:20:33,380 equations in three dimensions, those give me a line. 346 00:20:33,380 --> 00:20:35,290 The line where those two planes meet. 347 00:20:35,290 --> 00:20:42,210 And now, the third guy is a third plane. 348 00:20:42,210 --> 00:20:49,290 And it goes somewhere. 349 00:20:49,290 --> 00:20:51,820 Okay, those three things meet in a point. 350 00:20:51,820 --> 00:20:55,090 Now I don't know where that point is, frankly. 351 00:20:55,090 --> 00:20:58,140 But -- linear algebra will find it. 352 00:20:58,140 --> 00:21:05,580 The main point is that the three planes, because they're not 353 00:21:05,580 --> 00:21:08,100 parallel, they're not special. 354 00:21:08,100 --> 00:21:11,780 They do meet in one point and that's the solution. 355 00:21:11,780 --> 00:21:16,910 But, maybe you can see that this row picture is 356 00:21:16,910 --> 00:21:19,120 getting a little hard to see. 357 00:21:19,120 --> 00:21:24,520 The row picture was a cinch when we looked at two lines 358 00:21:24,520 --> 00:21:25,220 meeting. 359 00:21:25,220 --> 00:21:27,740 When we look at three planes meeting, 360 00:21:27,740 --> 00:21:32,890 it's not so clear and in four dimensions probably 361 00:21:32,890 --> 00:21:34,780 a little less clear. 362 00:21:34,780 --> 00:21:37,530 So, can I quit on the row picture? 363 00:21:37,530 --> 00:21:41,870 Or quit on the row picture before I've successfully 364 00:21:41,870 --> 00:21:45,940 found the point where the three planes meet? 365 00:21:45,940 --> 00:21:51,510 All I really want to see is that the row picture consists 366 00:21:51,510 --> 00:21:55,780 of three planes and, if everything works right, 367 00:21:55,780 --> 00:21:59,340 three planes meet in one point and that's a solution. 368 00:21:59,340 --> 00:22:04,730 Now, you can tell I prefer the column picture. 369 00:22:04,730 --> 00:22:06,990 Okay, so let me take the column picture. 370 00:22:06,990 --> 00:22:09,820 That's x times -- 371 00:22:09,820 --> 00:22:14,400 so there were two xs in the first equation minus one x is, 372 00:22:14,400 --> 00:22:16,690 and no xs in the third. 373 00:22:16,690 --> 00:22:19,000 It's just the first column of that. 374 00:22:19,000 --> 00:22:21,300 And how many ys are there? 375 00:22:21,300 --> 00:22:24,730 There's minus one in the first equations, two in the second 376 00:22:24,730 --> 00:22:27,430 and maybe minus three in the third. 377 00:22:27,430 --> 00:22:29,830 Just the second column of my matrix. 378 00:22:29,830 --> 00:22:37,740 And z times no zs minus one zs and four zs. 379 00:22:37,740 --> 00:22:41,270 And it's those three columns, right, 380 00:22:41,270 --> 00:22:46,640 that I have to combine to produce the right-hand side, 381 00:22:46,640 --> 00:22:48,945 which is zero minus one four. 382 00:22:52,830 --> 00:22:54,910 Okay. 383 00:22:54,910 --> 00:22:57,940 So what have we got on this left-hand side? 384 00:22:57,940 --> 00:22:59,470 A linear combination. 385 00:22:59,470 --> 00:23:02,540 It's a linear combination now of three vectors, 386 00:23:02,540 --> 00:23:05,720 and they happen to be -- each one is a three dimensional 387 00:23:05,720 --> 00:23:10,000 vector, so we want to know what combination of those three 388 00:23:10,000 --> 00:23:12,090 vectors produces that one. 389 00:23:12,090 --> 00:23:15,820 Shall I try to draw the column picture, then? 390 00:23:15,820 --> 00:23:18,830 So, since these vectors have three components -- 391 00:23:18,830 --> 00:23:21,660 so it's some multiple -- let me draw in the first column 392 00:23:21,660 --> 00:23:23,340 as before -- 393 00:23:23,340 --> 00:23:27,287 x is two and y is minus one. 394 00:23:27,287 --> 00:23:28,620 Maybe there is the first column. 395 00:23:32,320 --> 00:23:38,570 y -- the second column has maybe a minus one and a two and the y 396 00:23:38,570 --> 00:23:44,180 is a minus three, somewhere, there possibly, column two. 397 00:23:44,180 --> 00:23:46,950 And the third column has -- 398 00:23:46,950 --> 00:23:52,760 no zero minus one four, so how shall I draw that? 399 00:23:52,760 --> 00:23:57,000 So this was the first component. 400 00:23:57,000 --> 00:23:59,230 The second component was a minus one. 401 00:23:59,230 --> 00:24:01,270 Maybe up here. 402 00:24:01,270 --> 00:24:08,921 That's column three, that's the column zero minus one and four. 403 00:24:08,921 --> 00:24:09,420 This guy. 404 00:24:12,050 --> 00:24:15,430 So, again, what's my problem? 405 00:24:15,430 --> 00:24:18,260 What this equation is asking me to do 406 00:24:18,260 --> 00:24:21,340 is to combine these three vectors 407 00:24:21,340 --> 00:24:27,900 with a right combination to produce this one. 408 00:24:27,900 --> 00:24:33,620 Well, you can see what the right combination is, because 409 00:24:33,620 --> 00:24:38,250 in this special problem, specially chosen 410 00:24:38,250 --> 00:24:42,780 by the lecturer, that right-hand side that I'm trying to get 411 00:24:42,780 --> 00:24:45,060 is actually one of these columns. 412 00:24:45,060 --> 00:24:46,660 So I know how to get that one. 413 00:24:46,660 --> 00:24:48,420 So what's the solution? 414 00:24:48,420 --> 00:24:50,910 What combination will work? 415 00:24:50,910 --> 00:24:53,940 I just want one of these and none of these. 416 00:24:53,940 --> 00:24:59,070 So x should be zero, y should be zero and z should 417 00:24:59,070 --> 00:25:00,040 be one. 418 00:25:02,860 --> 00:25:04,520 That's the combination. 419 00:25:04,520 --> 00:25:07,600 One of those is obviously the right one. 420 00:25:07,600 --> 00:25:09,870 Column three is actually the same 421 00:25:09,870 --> 00:25:12,090 as b in this particular problem. 422 00:25:15,640 --> 00:25:17,860 I made it work that way just so we 423 00:25:17,860 --> 00:25:21,770 would get an answer, (0,0,1), so somehow that's 424 00:25:21,770 --> 00:25:25,890 the point where those three planes met 425 00:25:25,890 --> 00:25:28,710 and I couldn't see it before. 426 00:25:28,710 --> 00:25:31,830 Of course, I won't always be able to see it from the column 427 00:25:31,830 --> 00:25:33,170 picture, either. 428 00:25:33,170 --> 00:25:39,390 It's the next lecture, actually, which is about elimination, 429 00:25:39,390 --> 00:25:46,670 which is the systematic way that everybody -- 430 00:25:46,670 --> 00:25:51,950 every bit of software, too -- 431 00:25:51,950 --> 00:25:56,830 production, large-scale software would solve the equations. 432 00:25:56,830 --> 00:25:59,020 So the lecture that's coming up. 433 00:25:59,020 --> 00:26:02,450 If I was to add that to the syllabus, 434 00:26:02,450 --> 00:26:09,050 will be about how to find x, y, z in all cases. 435 00:26:09,050 --> 00:26:15,110 Can I just think again, though, about the big picture? 436 00:26:15,110 --> 00:26:18,910 By the big picture I mean let's keep this same matrix 437 00:26:18,910 --> 00:26:21,990 on the left but imagine that we have 438 00:26:21,990 --> 00:26:24,250 a different right-hand side. 439 00:26:24,250 --> 00:26:27,430 Oh, let me take a different right-hand side. 440 00:26:27,430 --> 00:26:29,100 So I'll change that right-hand side 441 00:26:29,100 --> 00:26:34,610 to something that actually is also pretty special. 442 00:26:34,610 --> 00:26:36,810 Let me change it to -- 443 00:26:36,810 --> 00:26:38,940 if I add those first two columns, 444 00:26:38,940 --> 00:26:43,220 that would give me a one and a one and a minus three. 445 00:26:43,220 --> 00:26:46,210 There's a very special right-hand side. 446 00:26:46,210 --> 00:26:51,420 I just cooked it up by adding this one to this one. 447 00:26:51,420 --> 00:26:54,640 Now, what's the solution with this new right-hand side? 448 00:26:54,640 --> 00:26:58,960 The solution with this new right-hand side is clear. 449 00:26:58,960 --> 00:27:04,880 took one of these and none of those. 450 00:27:04,880 --> 00:27:08,030 So actually, it just changed around to this 451 00:27:08,030 --> 00:27:11,150 when I took this new right-hand side. 452 00:27:11,150 --> 00:27:12,010 Okay. 453 00:27:12,010 --> 00:27:19,070 So in the row picture, I have three different planes, 454 00:27:19,070 --> 00:27:23,800 three new planes meeting now at this point. 455 00:27:23,800 --> 00:27:27,000 In the column picture, I have the same three columns, 456 00:27:27,000 --> 00:27:30,640 but now I'm combining them to produce this guy, 457 00:27:30,640 --> 00:27:34,100 and it turned out that column one plus column two which would 458 00:27:34,100 --> 00:27:38,500 be somewhere -- there is the right column -- 459 00:27:38,500 --> 00:27:42,260 one of this and one of this would give me the new b. 460 00:27:45,190 --> 00:27:45,690 Okay. 461 00:27:45,690 --> 00:27:48,730 So we squeezed in an extra example. 462 00:27:48,730 --> 00:27:55,970 But now think about all bs, all right-hand sides. 463 00:27:55,970 --> 00:27:58,920 Can I solve these equations for every right-hand side? 464 00:28:02,030 --> 00:28:04,980 Can I ask that question? 465 00:28:04,980 --> 00:28:07,020 So that's the algebra question. 466 00:28:07,020 --> 00:28:11,870 Can I solve A x=b for every b? 467 00:28:11,870 --> 00:28:13,490 Let me write that down. 468 00:28:13,490 --> 00:28:24,320 Can I solve A x =b for every right-hand side b? 469 00:28:24,320 --> 00:28:26,420 I mean, is there a solution? 470 00:28:26,420 --> 00:28:29,670 And then, if there is, elimination 471 00:28:29,670 --> 00:28:31,940 will give me a way to find it. 472 00:28:31,940 --> 00:28:34,520 I really wanted to ask, is there a solution 473 00:28:34,520 --> 00:28:36,850 for every right-hand side? 474 00:28:36,850 --> 00:28:40,390 So now, can I put that in different words -- 475 00:28:40,390 --> 00:28:43,140 in this linear combination words? 476 00:28:43,140 --> 00:28:52,510 So in linear combination words, do the linear combinations 477 00:28:52,510 --> 00:29:03,060 of the columns fill three dimensional space? 478 00:29:05,740 --> 00:29:12,500 Every b means all the bs in three dimensional space. 479 00:29:12,500 --> 00:29:16,050 Do you see that I'm just asking the same question 480 00:29:16,050 --> 00:29:19,600 in different words? 481 00:29:19,600 --> 00:29:21,040 Solving A x -- 482 00:29:21,040 --> 00:29:25,020 A x -- that's very important. 483 00:29:25,020 --> 00:29:31,790 A times x -- when I multiply a matrix by a vector, 484 00:29:31,790 --> 00:29:34,940 I get a combination of the columns. 485 00:29:34,940 --> 00:29:38,650 I'll write that down in a moment. 486 00:29:38,650 --> 00:29:43,550 But in my column picture, that's really what I'm doing. 487 00:29:43,550 --> 00:29:47,080 I'm taking linear combinations of these three columns 488 00:29:47,080 --> 00:29:50,620 and I'm trying to find b. 489 00:29:50,620 --> 00:29:57,750 And, actually, the answer for this matrix will be yes. 490 00:29:57,750 --> 00:30:04,330 For this matrix A -- for these columns, the answer is yes. 491 00:30:04,330 --> 00:30:19,590 This matrix -- that I chose for an example is a good matrix. 492 00:30:19,590 --> 00:30:21,780 A non-singular matrix. 493 00:30:21,780 --> 00:30:23,180 An invertible matrix. 494 00:30:23,180 --> 00:30:26,700 Those will be the matrices that we like best. 495 00:30:26,700 --> 00:30:29,490 There could be other -- 496 00:30:29,490 --> 00:30:35,070 and we will see other matrices where the answer becomes, no -- 497 00:30:35,070 --> 00:30:38,320 oh, actually, you can see when it would become no. 498 00:30:38,320 --> 00:30:43,890 What could go wrong? find out -- because if elimination fails, 499 00:30:43,890 --> 00:30:46,820 How could it go wrong that out of these -- 500 00:30:46,820 --> 00:30:51,310 out of three columns and all their combinations -- 501 00:30:51,310 --> 00:30:58,390 when would I not be able to produce some b off here? 502 00:30:58,390 --> 00:31:00,610 When could it go wrong? 503 00:31:00,610 --> 00:31:04,350 Do you see that the combinations -- 504 00:31:04,350 --> 00:31:06,370 let me say when it goes wrong. 505 00:31:06,370 --> 00:31:12,310 If these three columns all lie in the same plane, 506 00:31:12,310 --> 00:31:18,110 then their combinations will lie in that same plane. 507 00:31:18,110 --> 00:31:19,640 So then we're in trouble. 508 00:31:19,640 --> 00:31:23,880 If the three columns of my matrix -- 509 00:31:23,880 --> 00:31:28,180 if those three vectors happen to lie in the same plane -- 510 00:31:28,180 --> 00:31:31,090 for example, if column three is just 511 00:31:31,090 --> 00:31:36,360 the sum of column one and column two, I would be in trouble. 512 00:31:36,360 --> 00:31:40,060 That would be a matrix A where the answer would be no, 513 00:31:40,060 --> 00:31:44,360 because the combinations -- 514 00:31:44,360 --> 00:31:48,260 if column three is in the same plane as column one and two, 515 00:31:48,260 --> 00:31:50,250 I don't get anything new from that. 516 00:31:50,250 --> 00:31:54,610 All the combinations are in the plane and only right-hand sides 517 00:31:54,610 --> 00:32:00,010 b that I could get would be the ones in that plane. 518 00:32:00,010 --> 00:32:03,680 So I could solve it for some right-hand sides, when 519 00:32:03,680 --> 00:32:08,320 b is in the plane, but most right-hand sides 520 00:32:08,320 --> 00:32:11,270 would be out of the plane and unreachable. 521 00:32:11,270 --> 00:32:14,190 So that would be a singular case. 522 00:32:14,190 --> 00:32:16,830 The matrix would be not invertible. 523 00:32:16,830 --> 00:32:19,810 There would not be a solution for every b. 524 00:32:19,810 --> 00:32:22,511 The answer would become no for that. 525 00:32:22,511 --> 00:32:23,010 Okay. 526 00:32:25,740 --> 00:32:27,040 I don't know -- 527 00:32:27,040 --> 00:32:29,190 shall we take just a little shot at thinking 528 00:32:29,190 --> 00:32:34,510 about nine dimensions? 529 00:32:34,510 --> 00:32:38,645 Imagine that we have vectors with nine components. 530 00:32:41,170 --> 00:32:44,920 Well, it's going to be hard to visualize those. 531 00:32:44,920 --> 00:32:46,800 I don't pretend to do it. 532 00:32:46,800 --> 00:32:51,950 But somehow, pretend you do. 533 00:32:51,950 --> 00:32:55,610 Pretend we have -- if this was nine equations and nine 534 00:32:55,610 --> 00:32:59,260 unknowns, then we would have nine columns, 535 00:32:59,260 --> 00:33:02,720 and each one would be a vector in nine-dimensional space 536 00:33:02,720 --> 00:33:06,930 and we would be looking at their linear combinations. 537 00:33:06,930 --> 00:33:09,340 So we would be having the linear combinations 538 00:33:09,340 --> 00:33:12,320 of nine vectors in nine-dimensional space, 539 00:33:12,320 --> 00:33:15,380 and we would be trying to find the combination that hit 540 00:33:15,380 --> 00:33:18,000 the correct right-hand side b. 541 00:33:18,000 --> 00:33:22,990 And we might also ask the question can we always do it? 542 00:33:22,990 --> 00:33:25,380 Can we get every right-hand side b? 543 00:33:25,380 --> 00:33:29,840 And certainly it will depend on those nine columns. 544 00:33:29,840 --> 00:33:32,480 Sometimes the answer will be yes -- 545 00:33:32,480 --> 00:33:35,500 if I picked a random matrix, it would be yes, actually. 546 00:33:35,500 --> 00:33:39,630 If I used MatLab and just used the random command, picked 547 00:33:39,630 --> 00:33:44,820 out a nine by nine matrix, I guarantee it would be 548 00:33:44,820 --> 00:33:45,320 good. 549 00:33:45,320 --> 00:33:47,140 It would be non-singular, it would 550 00:33:47,140 --> 00:33:49,270 be invertible, all beautiful. 551 00:33:49,270 --> 00:34:00,750 But if I choose those columns so that they're not independent, 552 00:34:00,750 --> 00:34:05,790 so that the ninth column is the same as the eighth column, 553 00:34:05,790 --> 00:34:09,020 then it contributes nothing new and there 554 00:34:09,020 --> 00:34:13,080 would be right-hand sides b that I couldn't get. 555 00:34:13,080 --> 00:34:18,210 Can you sort of think about nine vectors 556 00:34:18,210 --> 00:34:22,050 in nine-dimensional space an take their combinations? 557 00:34:22,050 --> 00:34:26,460 That's really the central thought -- 558 00:34:26,460 --> 00:34:30,920 that you get kind of used to in linear algebra. 559 00:34:30,920 --> 00:34:33,340 Even though you can't really visualize it, 560 00:34:33,340 --> 00:34:36,139 you sort of think you can after a while. 561 00:34:36,139 --> 00:34:40,830 Those nine columns and all their combinations 562 00:34:40,830 --> 00:34:45,030 may very well fill out the whole nine-dimensional space. 563 00:34:45,030 --> 00:34:48,239 But if the ninth column happened to be the same as the eighth 564 00:34:48,239 --> 00:34:51,090 column and gave nothing new, then probably what it would 565 00:34:51,090 --> 00:34:53,102 fill out would be -- 566 00:34:55,820 --> 00:35:03,070 I hesitate even to say this -- it would be a sort of a plane 567 00:35:03,070 --> 00:35:04,320 -- 568 00:35:04,320 --> 00:35:10,030 an eight dimensional plane inside nine-dimensional space. 569 00:35:10,030 --> 00:35:12,890 And it's those eight dimensional planes 570 00:35:12,890 --> 00:35:16,040 inside nine-dimensional space that we 571 00:35:16,040 --> 00:35:18,830 have to work with eventually. 572 00:35:18,830 --> 00:35:25,970 For now, let's stay with a nice case where the matrices work, 573 00:35:25,970 --> 00:35:29,980 we can get every right-hand side b and here 574 00:35:29,980 --> 00:35:32,350 we see how to do it with columns. 575 00:35:32,350 --> 00:35:33,160 Okay. 576 00:35:33,160 --> 00:35:36,520 There was one step which I realized 577 00:35:36,520 --> 00:35:41,210 I was saying in words that I now want to write in letters. 578 00:35:41,210 --> 00:35:45,750 Because I'm coming back to the matrix form of the equation, 579 00:35:45,750 --> 00:35:49,210 so let me write it here. 580 00:35:49,210 --> 00:35:54,270 The matrix form of my equation, of my system 581 00:35:54,270 --> 00:35:57,970 is some matrix A times some vector x 582 00:35:57,970 --> 00:36:00,920 equals some right-hand side b. 583 00:36:00,920 --> 00:36:01,950 Okay. 584 00:36:01,950 --> 00:36:03,450 So this is a multiplication. 585 00:36:03,450 --> 00:36:04,520 A times x. 586 00:36:04,520 --> 00:36:07,430 Matrix times vector, and I just want to say 587 00:36:07,430 --> 00:36:11,620 how do you multiply a matrix by a vector? 588 00:36:11,620 --> 00:36:16,550 Okay, so I'm just going to create a matrix -- 589 00:36:16,550 --> 00:36:21,640 let me take two five one three -- 590 00:36:21,640 --> 00:36:28,140 and let me take a vector x to be, say, 1and 2. 591 00:36:28,140 --> 00:36:31,500 How do I multiply a matrix by a vector? 592 00:36:34,150 --> 00:36:40,530 But just think a little bit about matrix notation 593 00:36:40,530 --> 00:36:42,120 and how to do that in multiplication. 594 00:36:42,120 --> 00:36:45,390 So let me say how I multiply a matrix by a vector. 595 00:36:45,390 --> 00:36:47,760 Actually, there are two ways to do it. 596 00:36:47,760 --> 00:36:50,430 Let me tell you my favorite way. 597 00:36:50,430 --> 00:36:52,770 It's columns again. 598 00:36:52,770 --> 00:36:54,720 It's a column at a time. 599 00:36:54,720 --> 00:36:57,980 For me, this matrix multiplication 600 00:36:57,980 --> 00:37:03,290 says I take one of that column and two of that column and add. 601 00:37:03,290 --> 00:37:06,770 So this is the way I would think of it 602 00:37:06,770 --> 00:37:12,820 is one of the first column and two of the second column 603 00:37:12,820 --> 00:37:17,300 and let's just see what we get. 604 00:37:17,300 --> 00:37:21,510 So in the first component I'm getting a two and a ten. 605 00:37:21,510 --> 00:37:23,470 I'm getting a twelve there. 606 00:37:23,470 --> 00:37:26,220 In the second component I'm getting a one and a six, 607 00:37:26,220 --> 00:37:27,860 I'm getting a seven. 608 00:37:27,860 --> 00:37:35,460 So that matrix times that vector is twelve seven. 609 00:37:35,460 --> 00:37:39,207 Now, you could do that another way. 610 00:37:39,207 --> 00:37:40,540 You could do it a row at a time. 611 00:37:40,540 --> 00:37:43,760 And you would get this twelve -- and actually I pretty much did 612 00:37:43,760 --> 00:37:44,970 it here -- 613 00:37:44,970 --> 00:37:45,710 this way. 614 00:37:45,710 --> 00:37:48,470 Two -- I could take that row times my vector. 615 00:37:48,470 --> 00:37:53,180 This is the idea of a dot product. 616 00:37:53,180 --> 00:37:58,140 This vector times this vector, two times one plus five times 617 00:37:58,140 --> 00:38:00,720 two is the twelve. 618 00:38:00,720 --> 00:38:04,610 This vector times this vector -- one times one plus three times 619 00:38:04,610 --> 00:38:06,000 two is the seven. 620 00:38:06,000 --> 00:38:11,810 So I can do it by rows, and in each row times 621 00:38:11,810 --> 00:38:16,360 my x is what I'll later call a dot product. 622 00:38:16,360 --> 00:38:19,430 But I also like to see it by columns. 623 00:38:19,430 --> 00:38:22,490 I see this as a linear combination of a column. 624 00:38:22,490 --> 00:38:24,130 So here's my point. 625 00:38:24,130 --> 00:38:36,180 A times x is a combination of the columns of A. 626 00:38:36,180 --> 00:38:43,650 That's how I hope you will think of A times x when we need 627 00:38:43,650 --> 00:38:44,150 it. 628 00:38:44,150 --> 00:38:47,230 Right now we've got -- with small ones, 629 00:38:47,230 --> 00:38:51,390 we can always do it in different ways, but later, 630 00:38:51,390 --> 00:38:53,930 think of it that way. 631 00:38:53,930 --> 00:38:54,480 Okay. 632 00:38:54,480 --> 00:39:02,020 So that's the picture for a two by two system. 633 00:39:02,020 --> 00:39:05,970 And if the right-hand side B happened to be twelve seven, 634 00:39:05,970 --> 00:39:12,110 then of course the correct solution would be one two. 635 00:39:12,110 --> 00:39:12,610 Okay. 636 00:39:12,610 --> 00:39:17,950 So let me come back next time to a systematic way, 637 00:39:17,950 --> 00:39:23,470 using elimination, to find the solution, 638 00:39:23,470 --> 00:39:29,810 if there is one, to a system of any size and