1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:19,655 at ocw.mit.edu. 8 00:00:34,230 --> 00:00:35,090 PROFESSOR: Hi. 9 00:00:35,090 --> 00:00:38,920 Today's lesson will complete our first excursion 10 00:00:38,920 --> 00:00:40,700 into the subject of linear algebra 11 00:00:40,700 --> 00:00:42,530 for the sake of linear algebra. 12 00:00:42,530 --> 00:00:45,960 And next time, we will turn our attention 13 00:00:45,960 --> 00:00:50,380 to applications of linear algebra and linear equations 14 00:00:50,380 --> 00:00:53,420 and the like, to applications towards functions 15 00:00:53,420 --> 00:00:56,930 of several variables in the cases where the functions are 16 00:00:56,930 --> 00:00:58,850 not necessarily linear. 17 00:00:58,850 --> 00:01:01,090 Now, you may recall that we concluded 18 00:01:01,090 --> 00:01:07,030 our last lesson on the note of talking about inverse matrices, 19 00:01:07,030 --> 00:01:09,880 pointing out that an inverse matrix existed, 20 00:01:09,880 --> 00:01:16,510 provided that the determinant of the entries making up 21 00:01:16,510 --> 00:01:20,860 the matrix was not 0, otherwise, the inverse matrix didn't 22 00:01:20,860 --> 00:01:23,890 exist, but that we then said, once we know this, 23 00:01:23,890 --> 00:01:27,850 how do we actually construct the inverse matrix? 24 00:01:27,850 --> 00:01:29,680 And what I hope to do in today's lesson 25 00:01:29,680 --> 00:01:33,000 is to show you a rather elegant manner for inverting a matrix, 26 00:01:33,000 --> 00:01:38,200 and at the same time, to point out several other consequences 27 00:01:38,200 --> 00:01:41,250 that will be helpful in our study of calculus 28 00:01:41,250 --> 00:01:44,110 of several variables in other contexts as we proceed. 29 00:01:44,110 --> 00:01:46,350 At a rate, today's lessons is called quite 30 00:01:46,350 --> 00:01:48,630 simply "Inverting a Matrix." 31 00:01:48,630 --> 00:01:52,450 And to start with a specific example in front of us, let's 32 00:01:52,450 --> 00:01:54,710 invert the matrix A inverse, where 33 00:01:54,710 --> 00:02:00,370 A is the matrix whose entries are 1, 1, 1; 2, 3, 4; 3, 4, 6. 34 00:02:00,370 --> 00:02:03,270 Keep in mind, among the various ways 35 00:02:03,270 --> 00:02:07,140 of motivating matrix algebra, we chose 36 00:02:07,140 --> 00:02:10,810 as our particular illustration the coding system 37 00:02:10,810 --> 00:02:14,430 of the matrix representing the coefficients in a system 38 00:02:14,430 --> 00:02:16,540 of m equations with n unknowns. 39 00:02:16,540 --> 00:02:19,140 In this particular case of a 3 by 3 matrix, 40 00:02:19,140 --> 00:02:22,300 we view the matrix A as coding the system 41 00:02:22,300 --> 00:02:24,970 of three linear equations and three unknowns, 42 00:02:24,970 --> 00:02:26,760 given by system 1. 43 00:02:26,760 --> 00:02:28,250 I call it equation 1. 44 00:02:28,250 --> 00:02:30,970 We have this system of three equations with three unknowns. 45 00:02:30,970 --> 00:02:36,500 By the way, observe that this looks very much like a problem 46 00:02:36,500 --> 00:02:39,250 that we tackled in high school, except for one 47 00:02:39,250 --> 00:02:40,990 very small change. 48 00:02:40,990 --> 00:02:44,630 In high school, you may remember that we tackled equations 49 00:02:44,630 --> 00:02:48,840 of the form like this, provided that the y's 50 00:02:48,840 --> 00:02:50,410 were replaced by constants. 51 00:02:50,410 --> 00:02:54,040 In other words, we weren't used to thinking of linear systems 52 00:02:54,040 --> 00:02:57,000 of equations in terms of functions this way, 53 00:02:57,000 --> 00:03:01,050 but we were used to thinking of things like x_1 plus x_2 plus 54 00:03:01,050 --> 00:03:06,480 x_3 equals 2, 2*y_1 plus 3*x_2 plus 4*x_3 equals 9. 55 00:03:06,480 --> 00:03:10,170 3*x_1 plus 4*x_2 plus 6*x_3 equals 13. 56 00:03:10,170 --> 00:03:12,830 And then the question was to solve specifically 57 00:03:12,830 --> 00:03:14,955 for x_1, x_2, and x_3. 58 00:03:14,955 --> 00:03:15,740 All right? 59 00:03:15,740 --> 00:03:17,974 Now, the only difference we're making now 60 00:03:17,974 --> 00:03:19,390 is that we're not going to specify 61 00:03:19,390 --> 00:03:21,350 what y_1, y_2, and y_3 are. 62 00:03:21,350 --> 00:03:24,210 We'll mention the relationship between our problem 63 00:03:24,210 --> 00:03:26,640 and the high school one later in the lecture. 64 00:03:26,640 --> 00:03:28,630 But what we are saying is this. 65 00:03:28,630 --> 00:03:32,160 We would like, in general, given these three equations and three 66 00:03:32,160 --> 00:03:34,780 unknowns, this system of linear equations, 67 00:03:34,780 --> 00:03:38,120 we would like to be able to solve for the x's in terms 68 00:03:38,120 --> 00:03:39,490 of the y's. 69 00:03:39,490 --> 00:03:42,340 And that is what I'm going to show you is identified 70 00:03:42,340 --> 00:03:43,770 with inverting the matrix. 71 00:03:43,770 --> 00:03:47,690 What I hope to show you as we go along is that if the matrix A 72 00:03:47,690 --> 00:03:51,040 represents these coefficients, assuming that we find 73 00:03:51,040 --> 00:03:53,860 the technique whereby we can express uniquely 74 00:03:53,860 --> 00:03:58,940 for the x's in terms of the y's, the coefficients that express 75 00:03:58,940 --> 00:04:02,270 the x's as linear combinations of the y's-- those coefficients 76 00:04:02,270 --> 00:04:05,850 involving the y's-- will turn out to be A inverse. 77 00:04:05,850 --> 00:04:07,570 We'll talk about that more later. 78 00:04:07,570 --> 00:04:09,450 I simply wanted to tell you that now 79 00:04:09,450 --> 00:04:12,700 so that you'd have a head start into what my plan of attack 80 00:04:12,700 --> 00:04:13,930 is going to be. 81 00:04:13,930 --> 00:04:16,640 First of all, let me mention a method which 82 00:04:16,640 --> 00:04:19,269 you may have seen previously in school, 83 00:04:19,269 --> 00:04:23,210 but a very effective method for systematizing or solving 84 00:04:23,210 --> 00:04:28,400 systems of linear equations in a way that works very, very 85 00:04:28,400 --> 00:04:29,810 systematically. 86 00:04:29,810 --> 00:04:31,740 The idea is something like this. 87 00:04:31,740 --> 00:04:37,814 We know that if we replace one equation-- well, let's 88 00:04:37,814 --> 00:04:38,480 put it this way. 89 00:04:38,480 --> 00:04:40,650 If we add equals to equals, the results are equal. 90 00:04:40,650 --> 00:04:42,690 If we multiply equals by equals, the results 91 00:04:42,690 --> 00:04:43,910 are equal, et cetera. 92 00:04:43,910 --> 00:04:47,000 Consequently, given a system of equations like this, 93 00:04:47,000 --> 00:04:50,150 we know that if we replace one of the equations 94 00:04:50,150 --> 00:04:54,620 by that equation plus a multiple of any other equation, 95 00:04:54,620 --> 00:04:57,580 you see-- if we add two equations together, things 96 00:04:57,580 --> 00:05:00,220 of this type, where, as we change the system, 97 00:05:00,220 --> 00:05:03,200 we don't change the solutions to the system. 98 00:05:03,200 --> 00:05:05,090 In other words, there are devices 99 00:05:05,090 --> 00:05:07,990 whereby we can systematically replace 100 00:05:07,990 --> 00:05:10,450 the given system of equations, hopefully, 101 00:05:10,450 --> 00:05:13,550 by simpler systems of equations where the simpler 102 00:05:13,550 --> 00:05:16,410 system of equations has the same set of solutions 103 00:05:16,410 --> 00:05:18,320 as the original system. 104 00:05:18,320 --> 00:05:21,510 And the method that I'm thinking of is called diagonalization. 105 00:05:21,510 --> 00:05:24,130 For example, what we often say is something like this. 106 00:05:24,130 --> 00:05:25,880 We say, lookit, wouldn't it be nice to get 107 00:05:25,880 --> 00:05:31,270 a system in which x_1 appears only in the first equation? 108 00:05:31,270 --> 00:05:33,190 And one way of doing that, notice, 109 00:05:33,190 --> 00:05:35,120 is we observe that in the second equation, 110 00:05:35,120 --> 00:05:38,710 we have 2*x_1 In the first equation, we have x_1. 111 00:05:38,710 --> 00:05:43,240 If we were to subtract twice the first equation from the second 112 00:05:43,240 --> 00:05:46,750 equation and use that for our new second equation, 113 00:05:46,750 --> 00:05:50,220 notice that the new second equation would have no x_1 term 114 00:05:50,220 --> 00:05:52,960 in it, because 2*x_1 minus 2*x_1 is 0. 115 00:05:52,960 --> 00:05:58,040 Similarly, since the leading term in the third equation is 116 00:05:58,040 --> 00:06:01,690 3*x_1 we observe that if we were to subtract three times 117 00:06:01,690 --> 00:06:03,510 the first equation from the third, 118 00:06:03,510 --> 00:06:07,060 the resulting third equation would have no x_1 term in it, 119 00:06:07,060 --> 00:06:10,580 so that the new system would have x_1 in the first equation 120 00:06:10,580 --> 00:06:12,020 and no place else. 121 00:06:12,020 --> 00:06:16,290 Similarly, we could move over now to our new second equation, 122 00:06:16,290 --> 00:06:21,840 and then eliminate x_2 from every equation but the second 123 00:06:21,840 --> 00:06:24,510 by subtracting off the appropriate multiple 124 00:06:24,510 --> 00:06:26,360 of the second equation. 125 00:06:26,360 --> 00:06:28,210 Now, what I'm going to do here is 126 00:06:28,210 --> 00:06:30,980 that, since it gets kind of messy working with the y_1's, 127 00:06:30,980 --> 00:06:32,450 y_2's, and y_3's and it stretches 128 00:06:32,450 --> 00:06:35,300 out all over the place, let me invent 129 00:06:35,300 --> 00:06:37,570 a new type of coding system. 130 00:06:37,570 --> 00:06:40,420 I will take these three equations and three unknowns 131 00:06:40,420 --> 00:06:45,250 and represent it by a 3 by 6 matrix as follows. 132 00:06:45,250 --> 00:06:49,650 I will start up with the following coding system. 133 00:06:49,650 --> 00:06:51,550 I will use place value. 134 00:06:51,550 --> 00:06:54,880 The first three columns will represent x_1, x_2, x_3, 135 00:06:54,880 --> 00:06:56,150 respectively. 136 00:06:56,150 --> 00:06:59,570 The next three columns will represent y_1, y_2, and y_3, 137 00:06:59,570 --> 00:07:00,330 respectively. 138 00:07:00,330 --> 00:07:03,850 I'll assume an equal sign divides this. 139 00:07:03,850 --> 00:07:07,560 And this will simply be a coding device for reading what? 140 00:07:07,560 --> 00:07:11,050 x_1 plus x_2 plus x_3 equals y_1. 141 00:07:11,050 --> 00:07:15,740 2*x_1 plus 3*x_2 plus 4*x_3 equals y_2. 142 00:07:15,740 --> 00:07:19,480 3*x_1 plus 4*x_2 plus 6*x_3 equals y_3. 143 00:07:19,480 --> 00:07:20,670 Notice why I say y_3. 144 00:07:20,670 --> 00:07:24,540 It's 0*y_1 plus 0*y_2 plus 1*y_3, which, of course, 145 00:07:24,540 --> 00:07:26,100 is y_3. 146 00:07:26,100 --> 00:07:31,880 Notice also that the left side of this 3 by 6 matrix 147 00:07:31,880 --> 00:07:38,650 is precisely the matrix A that we're investigating back here. 148 00:07:38,650 --> 00:07:40,990 I just want that for future reference. 149 00:07:40,990 --> 00:07:43,670 Now, let me go through this procedure. 150 00:07:43,670 --> 00:07:46,880 What I will now do is subtract twice the first equation 151 00:07:46,880 --> 00:07:49,620 from the second and make that my new second one. 152 00:07:49,620 --> 00:07:52,290 And I will subtract three times the first equation 153 00:07:52,290 --> 00:07:56,330 from the third to give me my new third equation, which 154 00:07:56,330 --> 00:07:59,030 means, in terms of the matrix notation, 155 00:07:59,030 --> 00:08:02,530 that I am replacing the second row of this matrix 156 00:08:02,530 --> 00:08:05,010 by the second row minus twice the first row. 157 00:08:05,010 --> 00:08:08,700 I'm replacing the third row by the third row minus three times 158 00:08:08,700 --> 00:08:09,700 the first row. 159 00:08:09,700 --> 00:08:14,360 I now wind up with this particular 3 by 6 matrix. 160 00:08:16,870 --> 00:08:20,280 Again, just by brief review, what this matrix system tells 161 00:08:20,280 --> 00:08:22,530 me, for example, is, among other things, 162 00:08:22,530 --> 00:08:27,620 that x_2 plus 2*x_3 is minus 2*y_1 plus y_2. 163 00:08:27,620 --> 00:08:30,420 You see, not only do I have a coding system here, 164 00:08:30,420 --> 00:08:33,610 but the right-hand side allows me 165 00:08:33,610 --> 00:08:37,419 to check how I got the resulting matrix on the left hand side. 166 00:08:37,419 --> 00:08:40,200 It tells me how the y's had to be combined to give me this 167 00:08:40,200 --> 00:08:42,090 combination of the x's. 168 00:08:42,090 --> 00:08:45,290 At any rate, I now want to eliminate 1 169 00:08:45,290 --> 00:08:49,180 every place in the second column except in the second row. 170 00:08:49,180 --> 00:08:52,230 So looking at this matrix now, what I will do 171 00:08:52,230 --> 00:08:56,080 is I will replace the first row by the first minus the second-- 172 00:08:56,080 --> 00:08:57,630 see, 1 minus 1 is 0. 173 00:08:57,630 --> 00:09:00,590 So my new first row will have a 0 here. 174 00:09:00,590 --> 00:09:02,450 I will then replace the third row 175 00:09:02,450 --> 00:09:05,790 by the third minus the second, again, meaning what? 176 00:09:05,790 --> 00:09:08,630 That my new third row will have a 0 here. 177 00:09:08,630 --> 00:09:11,770 Leaving the details for you to check for yourself, 178 00:09:11,770 --> 00:09:16,120 the resulting 3 by 6 matrix now looks like this. 179 00:09:16,120 --> 00:09:18,440 By the way, again notice what this tells me. 180 00:09:18,440 --> 00:09:20,200 Among other things, this tells me 181 00:09:20,200 --> 00:09:26,160 that x_3 is minus y_1 minus y_2 plus y_3, which 182 00:09:26,160 --> 00:09:29,260 somehow tells me that to find x_3, 183 00:09:29,260 --> 00:09:32,960 I essentially have to do what? 184 00:09:32,960 --> 00:09:39,890 I can subtract the sum of the first and second equation 185 00:09:39,890 --> 00:09:42,960 from the third equation, and that will give me x_3. 186 00:09:42,960 --> 00:09:45,840 By the way, just as a quick look over here, 187 00:09:45,840 --> 00:09:48,220 just to show you how nice this technique is, 188 00:09:48,220 --> 00:09:51,600 maybe somebody who had been very quick could have looked at this 189 00:09:51,600 --> 00:09:53,330 system of equations and said, you know, 190 00:09:53,330 --> 00:09:55,530 if I add the first two equations, 191 00:09:55,530 --> 00:10:00,590 this will give me a 3*x_1, this will give me a 4*x_2, 192 00:10:00,590 --> 00:10:03,640 and therefore, if I subtract that from the third equation, 193 00:10:03,640 --> 00:10:08,410 the x_1 term and the x_2 terms will drop out, et cetera. 194 00:10:08,410 --> 00:10:10,800 The thing I want to point out is that this matrix system 195 00:10:10,800 --> 00:10:13,300 makes geniuses out of all of us, that we do not 196 00:10:13,300 --> 00:10:16,880 have to be able to see these intricacies to be able to get 197 00:10:16,880 --> 00:10:19,050 down to a stage like this. 198 00:10:19,050 --> 00:10:21,390 By the way, again, what this tells me is 199 00:10:21,390 --> 00:10:25,890 I now have a simpler method for solving this original system 200 00:10:25,890 --> 00:10:26,810 of equations. 201 00:10:26,810 --> 00:10:28,710 Namely, I can now solve this system 202 00:10:28,710 --> 00:10:30,960 of equations-- the one that's coded by this-- 203 00:10:30,960 --> 00:10:32,680 and the solutions of this equation 204 00:10:32,680 --> 00:10:35,280 will be the same set as the solutions 205 00:10:35,280 --> 00:10:36,540 to my original equation. 206 00:10:36,540 --> 00:10:40,320 At any rate, continuing this diagonalization method still 207 00:10:40,320 --> 00:10:43,630 more, what I now do is I try to replace what? 208 00:10:43,630 --> 00:10:47,120 In the third column, I try to get 0's everywhere 209 00:10:47,120 --> 00:10:49,360 except in the third row. 210 00:10:49,360 --> 00:10:52,120 And to put that in still other words, 211 00:10:52,120 --> 00:10:54,040 I guess, in the language of matrices, 212 00:10:54,040 --> 00:10:56,990 what I'm going to try to do is get what? 213 00:10:56,990 --> 00:10:59,500 The three by three identity matrix 214 00:10:59,500 --> 00:11:05,130 to make up the first half over here-- 215 00:11:05,130 --> 00:11:07,020 that's in terms of the matrix language. 216 00:11:07,020 --> 00:11:09,630 Now again, to show you quickly what I'm trying to do here, 217 00:11:09,630 --> 00:11:11,510 I'm simply going to what over here? 218 00:11:11,510 --> 00:11:14,600 I'm going to replace the first row 219 00:11:14,600 --> 00:11:16,090 by the first plus the third. 220 00:11:16,090 --> 00:11:18,560 I'm going to replace the second row by the second row 221 00:11:18,560 --> 00:11:20,250 minus twice the third. 222 00:11:20,250 --> 00:11:24,430 I now wind up with this particular matrix. 223 00:11:24,430 --> 00:11:29,350 I claim that the right half-- see, 224 00:11:29,350 --> 00:11:31,680 the left half of this matrix is the identity matrix. 225 00:11:31,680 --> 00:11:34,180 I claim that the left half is the identity matrix. 226 00:11:34,180 --> 00:11:36,630 I calm the right half is A inverse. 227 00:11:39,490 --> 00:11:40,490 That's what my claim is. 228 00:11:40,490 --> 00:11:42,730 And by the way, let's see why I say that. 229 00:11:42,730 --> 00:11:46,150 What system of equations does A inverse code? 230 00:11:46,150 --> 00:11:47,740 Just read what this says. 231 00:11:47,740 --> 00:11:53,010 It says x_1 is equal to 2*y_1 minus 2*y_2 plus y_3. 232 00:11:53,010 --> 00:11:55,780 x_2 is equal to y_2 minus 2*y_3. 233 00:11:55,780 --> 00:11:59,870 And x_3 is minus y_1 minus y_2 plus y_3. 234 00:11:59,870 --> 00:12:02,820 In other words, A inverse codes this particular system. 235 00:12:02,820 --> 00:12:04,640 Let's call that system 2. 236 00:12:04,640 --> 00:12:08,690 How are system 2 and what we call system 1 related? 237 00:12:08,690 --> 00:12:13,000 The relationship was that system 1 expressed the y's in terms 238 00:12:13,000 --> 00:12:14,180 of the x's. 239 00:12:14,180 --> 00:12:17,910 System 2 shows what it would look like if the x's were 240 00:12:17,910 --> 00:12:19,670 expressed in terms of the y's. 241 00:12:19,670 --> 00:12:23,300 And so what we have done, in the language of-- whatever you want 242 00:12:23,300 --> 00:12:26,180 to call it-- but what we've done is we've inverted the role 243 00:12:26,180 --> 00:12:28,410 of the variables, that we started with the y's given 244 00:12:28,410 --> 00:12:29,640 in terms of the x's. 245 00:12:29,640 --> 00:12:33,200 We now have the x's expressed in terms of the y's. 246 00:12:33,200 --> 00:12:36,820 Now, since you may want to see this more in matrix language, 247 00:12:36,820 --> 00:12:41,170 let me show you another way of seeing the same result, that 248 00:12:41,170 --> 00:12:43,340 uses the word inverse in the way we're 249 00:12:43,340 --> 00:12:45,990 used to seeing inverse matrix defined 250 00:12:45,990 --> 00:12:49,190 in terms of our present course. 251 00:12:49,190 --> 00:12:52,250 The matrix algebra interpretation is simply this. 252 00:12:52,250 --> 00:12:55,600 Let's rewrite system 1 so that you can see it here. 253 00:12:55,600 --> 00:12:59,610 And the interesting point is the following. 254 00:12:59,610 --> 00:13:06,315 Think of these y's as forming a column vector or column 255 00:13:06,315 --> 00:13:08,440 matrix-- whichever way you want to read this thing. 256 00:13:08,440 --> 00:13:11,640 In other words, I am going to view this 257 00:13:11,640 --> 00:13:15,550 as being the matrix which has three rows and one column, 258 00:13:15,550 --> 00:13:20,030 namely the matrix y_1 y_2, y_3. 259 00:13:20,030 --> 00:13:23,030 I am now going to take the matrix of coefficients here, 260 00:13:23,030 --> 00:13:25,760 which, as we recall, is the matrix A. But that's what? 261 00:13:25,760 --> 00:13:30,170 [1, 1, 1; 2, 3, 4; 3, 4, 6]. 262 00:13:30,170 --> 00:13:35,740 And I am now going to write x_1, x_2, x_3 as a column vector 263 00:13:35,740 --> 00:13:37,580 rather than as a row vector. 264 00:13:37,580 --> 00:13:41,410 And in turn, I can think of that as being a column matrix-- 265 00:13:41,410 --> 00:13:46,240 again, 3 by 1 matrix-- three rows, one column. 266 00:13:46,240 --> 00:13:51,350 Now, what I claim is that this system of three equations 267 00:13:51,350 --> 00:13:53,640 and three unknowns, so to speak, is 268 00:13:53,640 --> 00:13:58,060 equivalent to the single matrix equation given by this. 269 00:13:58,060 --> 00:14:00,420 In fact, to make this easier to read, 270 00:14:00,420 --> 00:14:06,270 let's let capital Y denote the column matrix [y 1; y2; y3]. 271 00:14:06,270 --> 00:14:11,670 Let's let capital X denote the column matrix [x 1; x2; x3], 272 00:14:11,670 --> 00:14:14,640 and capital A, as before, the original matrix. 273 00:14:14,640 --> 00:14:17,790 And my claim is that this system of three equations and three 274 00:14:17,790 --> 00:14:21,220 unknowns is represented by the single matrix equation 275 00:14:21,220 --> 00:14:25,400 capital Y equals A times capital X. And just 276 00:14:25,400 --> 00:14:27,550 to show you by way of a very quick review why 277 00:14:27,550 --> 00:14:31,580 this is the case, remember how we multiply two matrices. 278 00:14:31,580 --> 00:14:34,810 To find the term in the first row, first column, 279 00:14:34,810 --> 00:14:37,900 we multiply the first row of the first 280 00:14:37,900 --> 00:14:40,560 by the first column of the second. 281 00:14:40,560 --> 00:14:43,230 And since that's supposed to equal this matrix, 282 00:14:43,230 --> 00:14:45,930 and since matrices are equal only if they're 283 00:14:45,930 --> 00:14:49,940 equal entry by entry, it means that this product 284 00:14:49,940 --> 00:14:51,670 must equal y_1. 285 00:14:51,670 --> 00:14:53,150 Notice what that says. 286 00:14:53,150 --> 00:14:57,890 It says x_1 plus x_2 plus x_3 equals y_1. 287 00:14:57,890 --> 00:15:05,400 Similarly, 2 times x_1 plus 3 times x_2 plus 4 times x_3 288 00:15:05,400 --> 00:15:08,600 is going to have to equal y_2. 289 00:15:08,600 --> 00:15:13,400 And similarly, 3 times x_1 plus 4 times x_2 plus 6 times x_3 290 00:15:13,400 --> 00:15:15,350 is going to have to equal y_3. 291 00:15:15,350 --> 00:15:18,030 And that's precisely the original system of equations 292 00:15:18,030 --> 00:15:20,420 that we began with. 293 00:15:20,420 --> 00:15:25,930 Now, notice that this expresses the matrix Y as something times 294 00:15:25,930 --> 00:15:29,430 the matrix X. If A inverse exists, 295 00:15:29,430 --> 00:15:33,010 multiply both sides of this matrix equation by A inverse. 296 00:15:33,010 --> 00:15:35,910 On the left hand side, we get A inverse Y. 297 00:15:35,910 --> 00:15:39,470 On the right-hand side, A inverse times A*X, 298 00:15:39,470 --> 00:15:42,880 by associativity, is A inverse A times X. 299 00:15:42,880 --> 00:15:45,370 A inverse times A is the identity matrix. 300 00:15:45,370 --> 00:15:48,140 The identity matrix doesn't change the given matrix 301 00:15:48,140 --> 00:15:49,300 that it's multiplying. 302 00:15:49,300 --> 00:15:53,350 So we have that A inverse Y is just X. 303 00:15:53,350 --> 00:15:56,400 And notice that the role of A inverse 304 00:15:56,400 --> 00:16:00,330 is, again, just as mentioned above, that what we've done 305 00:16:00,330 --> 00:16:03,310 is we have started with Y given as a matrix times 306 00:16:03,310 --> 00:16:06,860 X. We now have X expressed as a matrix times 307 00:16:06,860 --> 00:16:10,890 Y. In fact, if you now take the matrix that we called 308 00:16:10,890 --> 00:16:14,730 A inverse originally and carry out this operation, 309 00:16:14,730 --> 00:16:18,300 we will see that this is precisely what does happen. 310 00:16:18,300 --> 00:16:20,070 This is what the identification is. 311 00:16:20,070 --> 00:16:22,490 In other words, the system of equations 312 00:16:22,490 --> 00:16:26,180 is identifiable with a single matrix equation. 313 00:16:26,180 --> 00:16:29,140 And that matrix equation can be inverted, 314 00:16:29,140 --> 00:16:32,920 meaning we can solve for X in terms of Y if and only 315 00:16:32,920 --> 00:16:35,320 if A inverse exists. 316 00:16:35,320 --> 00:16:37,810 There is still a third interpretation, 317 00:16:37,810 --> 00:16:41,840 an interpretation that goes back to the idea of mappings. 318 00:16:41,840 --> 00:16:44,400 And that is, we can think of the original system 319 00:16:44,400 --> 00:16:48,660 1 as a mapping that carries three-dimensional space 320 00:16:48,660 --> 00:16:50,340 into three-dimensional space. 321 00:16:50,340 --> 00:16:52,150 How does it carry three-dimensional space 322 00:16:52,150 --> 00:16:53,380 into three-dimensional space? 323 00:16:53,380 --> 00:16:59,730 Well, we view it as carrying the 3-tuple (x_1, x_2, x_3) 324 00:16:59,730 --> 00:17:07,880 into the 3-tuple (y_1, y_2, y_3) by the system of equations 325 00:17:07,880 --> 00:17:09,744 defined by system 1. 326 00:17:09,744 --> 00:17:11,869 And to show you what I mean by that in more detail, 327 00:17:11,869 --> 00:17:17,310 suppose I pick x_1 to be 1, x_2 to be 1, x_3 to be 1. 328 00:17:17,310 --> 00:17:20,050 Just for the sake of argument, suppose I picked that. 329 00:17:20,050 --> 00:17:22,200 x_1, x_2, x_3 are all 1. 330 00:17:22,200 --> 00:17:23,650 What this means is-- lookit. 331 00:17:23,650 --> 00:17:25,400 Come back to the original system, 332 00:17:25,400 --> 00:17:28,130 replace x_1, x_2, and x_3 by 1. 333 00:17:28,130 --> 00:17:35,430 If we do that, we see that y_1 is 3, y_2 is 9, and y_3 is 13. 334 00:17:35,430 --> 00:17:37,060 Therefore, we would say what? 335 00:17:37,060 --> 00:17:43,327 That f bar maps (1, 1, 1) into (3, 9, 13). 336 00:17:43,327 --> 00:17:45,660 By the way, you're going to notice this sooner or later. 337 00:17:45,660 --> 00:17:47,284 This bothered me when I first did this. 338 00:17:47,284 --> 00:17:48,480 This is just a little aside. 339 00:17:48,480 --> 00:17:51,960 You may have noticed that when I was arbitrarily writing down 340 00:17:51,960 --> 00:17:56,184 numbers to pick for y_1, y_2, and y_3, when we first started 341 00:17:56,184 --> 00:17:57,600 at the beginning of the lecture, I 342 00:17:57,600 --> 00:18:02,650 said why don't we let y_1 be 2, y_2 be 9, and y_3 be 13. 343 00:18:02,650 --> 00:18:04,980 And it looks almost like I made a mistake 344 00:18:04,980 --> 00:18:07,590 and meant to get (2, 9, 13) back again. 345 00:18:07,590 --> 00:18:10,050 I got (3, 9, 13). 346 00:18:10,050 --> 00:18:16,610 Notice that to get (2, 9, 13), I couldn't pick x_1, x_2, and x_3 347 00:18:16,610 --> 00:18:21,060 all to be 1, but I could pick x_1 to be minus 1, 348 00:18:21,060 --> 00:18:24,130 x_2 to be 1, x_3 to be 2. 349 00:18:24,130 --> 00:18:29,300 And then what that says is if I let x_1 be minus 1, 350 00:18:29,300 --> 00:18:33,460 x_2 be plus 1, and x_3 be 3, it says 351 00:18:33,460 --> 00:18:36,610 that, under those conditions, y_1 would have been 2, 352 00:18:36,610 --> 00:18:41,120 y_2 would have been 9, and y_3 would have been 13. 353 00:18:41,120 --> 00:18:43,120 Of course, the question that you now may ask, 354 00:18:43,120 --> 00:18:46,560 which I hope gives you some sort of a cross reference to go by, 355 00:18:46,560 --> 00:18:48,220 is how did I know? 356 00:18:48,220 --> 00:18:50,640 See, starting with (1, 1, 1), it seemed 357 00:18:50,640 --> 00:18:54,280 very easy to find the image (3, 9, 13). 358 00:18:54,280 --> 00:18:57,220 Now, I say, starting with the image, 359 00:18:57,220 --> 00:19:01,550 how do I find that this is what mapped into (2, 9, 13)? 360 00:19:01,550 --> 00:19:02,594 How did I know? 361 00:19:02,594 --> 00:19:03,510 See, here it was easy. 362 00:19:03,510 --> 00:19:06,497 I started with the input x bar. 363 00:19:06,497 --> 00:19:08,080 But here, I'm starting with the output 364 00:19:08,080 --> 00:19:09,560 and I'm trying to find the input. 365 00:19:09,560 --> 00:19:12,740 How did I find (-1, 1, 2) that fast? 366 00:19:12,740 --> 00:19:15,400 And the answer is that's precisely where the role 367 00:19:15,400 --> 00:19:17,460 of the inverse matrix comes in. 368 00:19:17,460 --> 00:19:20,700 You see, going back to what we did just a little while 369 00:19:20,700 --> 00:19:25,070 ago, when we inverted system 1 to get system 2, 370 00:19:25,070 --> 00:19:26,440 what did we do? 371 00:19:26,440 --> 00:19:31,420 We showed how to find x_1, x_2, x_3 once y_1, y_2, 372 00:19:31,420 --> 00:19:32,700 and y_3 were given. 373 00:19:32,700 --> 00:19:35,690 So all I had to do here was, knowing that I wanted y_1 374 00:19:35,690 --> 00:19:39,950 to be 2, y_2 to be 9, and y_3 to be 13, 375 00:19:39,950 --> 00:19:42,770 I just shoved those values in here 376 00:19:42,770 --> 00:19:45,890 and saw what x_1, x_2, and x_3 were. 377 00:19:45,890 --> 00:19:49,250 In fact, this is why working with the y's is an improvement 378 00:19:49,250 --> 00:19:52,080 over what the high school system was of picking 379 00:19:52,080 --> 00:19:53,070 the y's to be constant. 380 00:19:53,070 --> 00:19:55,770 You see, once I've done this, I can 381 00:19:55,770 --> 00:20:00,440 find what x_1, x_2, x_3 look like in terms of the y's, 382 00:20:00,440 --> 00:20:04,560 all in one shot, regardless of what values of y somebody 383 00:20:04,560 --> 00:20:07,120 is going to give me to play around with later on. 384 00:20:07,120 --> 00:20:11,940 But returning now to our immediate problem, notice, 385 00:20:11,940 --> 00:20:16,260 all I'm saying is that when I'm given a system of n 386 00:20:16,260 --> 00:20:18,690 linear equations and n unknowns, or even 387 00:20:18,690 --> 00:20:20,400 n equations and unknowns-- in this case 388 00:20:20,400 --> 00:20:22,980 we had three linear equations and three unknowns-- 389 00:20:22,980 --> 00:20:26,090 I can view that in a very natural way as a mapping 390 00:20:26,090 --> 00:20:29,920 from three-dimensional space into three-dimensional space. 391 00:20:29,920 --> 00:20:33,720 Notice that the makers of coefficients A and the mapping 392 00:20:33,720 --> 00:20:35,585 f bar-- and I'll coin a word here, 393 00:20:35,585 --> 00:20:37,960 because I don't know exactly how I want you to see this-- 394 00:20:37,960 --> 00:20:39,510 but they're identifiable. 395 00:20:39,510 --> 00:20:41,660 They're essentially equivalent. 396 00:20:41,660 --> 00:20:45,630 Notice that, given x bar, I want to apply f bar 397 00:20:45,630 --> 00:20:47,650 to that to find y bar. 398 00:20:47,650 --> 00:20:49,950 Notice that, in terms of our matrix language, 399 00:20:49,950 --> 00:20:54,100 we show that, after all, the matrix capital 400 00:20:54,100 --> 00:20:58,230 Y, which had entries y_1, y_2, y_3, 401 00:20:58,230 --> 00:21:01,510 is just a different numeral for writing this vector. 402 00:21:01,510 --> 00:21:02,850 This is a 3-tuple. 403 00:21:02,850 --> 00:21:06,650 This is the same 3-tuple written as a column rather as a row. 404 00:21:06,650 --> 00:21:11,150 x bar is a 3-tuple, but capital X that same 3-tuple 405 00:21:11,150 --> 00:21:14,480 written as a column vector rather than as a row vector. 406 00:21:14,480 --> 00:21:16,920 But notice that the identification 407 00:21:16,920 --> 00:21:19,920 that if y bar is f bar of x bar, that 408 00:21:19,920 --> 00:21:23,340 says the same information as saying that capital Y 409 00:21:23,340 --> 00:21:26,880 is the matrix A times capital X. 410 00:21:26,880 --> 00:21:30,620 These convey equivalent pieces of information. 411 00:21:30,620 --> 00:21:33,840 In still other words, notice that, in finding 412 00:21:33,840 --> 00:21:37,630 the images of f bar here and inverting the images, 413 00:21:37,630 --> 00:21:42,160 I used precisely the same device that I used in the system 414 00:21:42,160 --> 00:21:44,660 without giving the mapping interpretation. 415 00:21:44,660 --> 00:21:48,160 In other words, these do convey the same information 416 00:21:48,160 --> 00:21:50,650 so that from a major point of view, 417 00:21:50,650 --> 00:21:55,520 we can identify the matrix A with the mapping f bar. 418 00:21:55,520 --> 00:21:59,590 And you see, what this means is that if we can identify 419 00:21:59,590 --> 00:22:03,860 A with f bar, we should be able to identify A inverse 420 00:22:03,860 --> 00:22:05,260 with the inverse function. 421 00:22:05,260 --> 00:22:07,635 See, that's the other way we've used the word "inverse"-- 422 00:22:07,635 --> 00:22:09,210 as an inverse function. 423 00:22:09,210 --> 00:22:13,400 The existence of A inverse is equivalent to the existence 424 00:22:13,400 --> 00:22:15,270 of f bar inverse. 425 00:22:15,270 --> 00:22:18,000 And by the way, we showed that. 426 00:22:18,000 --> 00:22:21,330 You see, using the system of equations that we started with, 427 00:22:21,330 --> 00:22:24,980 we showed that if x_1 is minus 1, x_2 is 1, 428 00:22:24,980 --> 00:22:29,920 and x_3 is 2, that y_1 would be 2, y_2 would be 9, 429 00:22:29,920 --> 00:22:32,030 and y_3 would be 13. 430 00:22:32,030 --> 00:22:36,200 We also showed that if we start with y_1 equals 2, 431 00:22:36,200 --> 00:22:39,320 y_2 equals 9, and y_3 equals 13, then 432 00:22:39,320 --> 00:22:43,610 the x's had to be given by minus 1, 1, and 2. 433 00:22:43,610 --> 00:22:45,510 In terms of the mapping, what we're saying 434 00:22:45,510 --> 00:22:53,290 is not only does (-1, 1, 2) map into (2, 9, 13), but (2, 9, 13) 435 00:22:53,290 --> 00:22:58,440 can only be obtained from the 3-tuple (-1, 1, 2). 436 00:22:58,440 --> 00:23:00,730 You see, in essence, in this particular case, 437 00:23:00,730 --> 00:23:02,680 the existence of A inverse showed 438 00:23:02,680 --> 00:23:06,080 that f bar was onto-- we could get any 3-tuple we wanted-- 439 00:23:06,080 --> 00:23:09,090 and it was also one-to-one. 440 00:23:09,090 --> 00:23:11,870 The important point, in terms of our theory of matrices, 441 00:23:11,870 --> 00:23:16,680 is that, since f bar inverse need not exist for a given f, 442 00:23:16,680 --> 00:23:21,130 A inverse need not exist for a given matrix A. In other words, 443 00:23:21,130 --> 00:23:24,430 if we're identifying the matrix with the function, 444 00:23:24,430 --> 00:23:26,550 if the inverse function doesn't exist, 445 00:23:26,550 --> 00:23:28,830 the inverse matrix doesn't exist. 446 00:23:28,830 --> 00:23:33,840 In other words, not all matrices are invertible. 447 00:23:33,840 --> 00:23:36,730 And let's emphasize that. 448 00:23:36,730 --> 00:23:40,090 We knew that from before, but let's emphasize this now 449 00:23:40,090 --> 00:23:42,320 in terms of our structure. 450 00:23:42,320 --> 00:23:44,390 Namely, let's take a new example. 451 00:23:44,390 --> 00:23:47,720 Let's try to invert a matrix which doesn't have an inverse. 452 00:23:47,720 --> 00:23:50,700 Obviously, if I try to do that, I should fail dismally. 453 00:23:50,700 --> 00:23:52,790 And let's see how that does work out. 454 00:23:52,790 --> 00:23:57,860 Let me take the matrix A to be [1, 1, 1; 2, 3, 4; 3, 4, 5]. 455 00:23:57,860 --> 00:23:59,590 I deliberately called this A again, 456 00:23:59,590 --> 00:24:01,500 even though we've used A before. 457 00:24:01,500 --> 00:24:03,790 I've deliberately made this A look very much 458 00:24:03,790 --> 00:24:05,710 like the A we worked with before. 459 00:24:05,710 --> 00:24:07,690 If you compare this with our original A, 460 00:24:07,690 --> 00:24:12,710 the only difference is I now have a 5 here rather than a 6. 461 00:24:12,710 --> 00:24:16,840 I simply wanted to show you how a subtle change can affect 462 00:24:16,840 --> 00:24:19,680 the existence of an inverse. 463 00:24:19,680 --> 00:24:21,880 Again, by way of very quick review, 464 00:24:21,880 --> 00:24:26,000 before we go any further, notice that this matrix 465 00:24:26,000 --> 00:24:28,400 codes what system of equations? 466 00:24:28,400 --> 00:24:30,870 This particular system of equations. 467 00:24:30,870 --> 00:24:34,030 And so that you don't see any hanky-panky 468 00:24:34,030 --> 00:24:37,400 going on here and wondering why I knew that something was going 469 00:24:37,400 --> 00:24:40,360 to go wrong here, let me show you in advance 470 00:24:40,360 --> 00:24:45,340 why I can't invert this, then show you how our matrix coding 471 00:24:45,340 --> 00:24:47,150 system gives us the same information, 472 00:24:47,150 --> 00:24:49,320 even if we weren't bright enough to see this. 473 00:24:49,320 --> 00:24:53,290 Just for the sake of argument, suppose I had a keen enough eye 474 00:24:53,290 --> 00:24:57,270 to observe right away that if I added the first two equations, 475 00:24:57,270 --> 00:25:01,160 I would get that y_1 plus y_2 is what? 476 00:25:01,160 --> 00:25:04,460 3*x_1 plus 4*x_2 plus 5*x_3. 477 00:25:07,710 --> 00:25:11,150 And now I look at this result, compare 478 00:25:11,150 --> 00:25:14,530 that with my third equation where the right-hand sides 479 00:25:14,530 --> 00:25:17,020 are identical, and conclude from this 480 00:25:17,020 --> 00:25:20,430 that this is not really three equations with three unknowns. 481 00:25:20,430 --> 00:25:23,810 This is really two equations with three unknowns, 482 00:25:23,810 --> 00:25:27,310 because this equation here is either redundant or 483 00:25:27,310 --> 00:25:31,610 incompatible, meaning, notice that these two facts tell me 484 00:25:31,610 --> 00:25:35,070 that y_1 plus y_2 has to equal y_3. 485 00:25:35,070 --> 00:25:36,720 To show you what I mean by that, let 486 00:25:36,720 --> 00:25:41,660 me pick specific values for y_1, y_2, and y_3, in which y_3 487 00:25:41,660 --> 00:25:44,520 is not equal to y_1 plus y_2. 488 00:25:44,520 --> 00:25:47,409 And to keep this as close to the problem 489 00:25:47,409 --> 00:25:48,950 that we were working with originally, 490 00:25:48,950 --> 00:25:53,900 let me pick y_1 to be 2, y_2 to be 9, and y_3 to be 13. 491 00:25:53,900 --> 00:25:59,010 Notice that this system of equations cannot possibly have 492 00:25:59,010 --> 00:26:00,750 a solution now. 493 00:26:00,750 --> 00:26:03,700 This system of equations here cannot possibly have 494 00:26:03,700 --> 00:26:06,805 a solution, because if it had a solution, it would imply what? 495 00:26:06,805 --> 00:26:08,520 That if I add these two equations, 496 00:26:08,520 --> 00:26:13,080 it would say that 3*x_1 plus 4*x_2 plus 5*x_3 equals 11-- 497 00:26:13,080 --> 00:26:14,700 if I add these two equations. 498 00:26:14,700 --> 00:26:17,730 That would say it also has to equal 13. 499 00:26:17,730 --> 00:26:19,807 And since 11 is equal to 13-- and I 500 00:26:19,807 --> 00:26:22,140 have to correct a mistake I made in an earlier lecture-- 501 00:26:22,140 --> 00:26:26,010 it's for large values of 11 that 11 is equal to 13. 502 00:26:26,010 --> 00:26:30,070 But other than that, 11 is unequal to 13. 503 00:26:30,070 --> 00:26:33,300 This means that it's impossible to find solutions 504 00:26:33,300 --> 00:26:34,400 to this equation. 505 00:26:34,400 --> 00:26:37,500 Well, we'll come back to that in a little while 506 00:26:37,500 --> 00:26:39,920 and we'll also emphasize this in the exercises. 507 00:26:39,920 --> 00:26:42,250 But the main point now is to show 508 00:26:42,250 --> 00:26:45,340 how we could have obtained this information by our matrix 509 00:26:45,340 --> 00:26:46,340 coding system. 510 00:26:46,340 --> 00:26:49,110 And we do exactly the same thing as we did before. 511 00:26:49,110 --> 00:26:52,740 We take this matrix and we augment it by the 3 512 00:26:52,740 --> 00:26:54,970 by 3 identity matrix. 513 00:26:54,970 --> 00:27:02,120 Remember again, what our coding system says is that the first 514 00:27:02,120 --> 00:27:06,330 three columns represent the x's, the next three the y's. 515 00:27:06,330 --> 00:27:08,790 For example, this says x_1 plus x_2 plus x_3 516 00:27:08,790 --> 00:27:10,360 equals y_1, et cetera. 517 00:27:10,360 --> 00:27:13,940 We now go through what I call the row reducing operations, 518 00:27:13,940 --> 00:27:14,780 which means what? 519 00:27:14,780 --> 00:27:19,780 We're going to replace rows by the rows plus or minus 520 00:27:19,780 --> 00:27:22,650 a suitable multiple of another row so that we do what? 521 00:27:22,650 --> 00:27:28,490 We wind up with a 1, eventually, only here and 0's elsewhere, 522 00:27:28,490 --> 00:27:32,574 a 1 here, 0's here, a 1 here and 0's here-- if we can do that. 523 00:27:32,574 --> 00:27:33,740 Let's see what happens here. 524 00:27:33,740 --> 00:27:35,390 What I'm going to do, of course, is 525 00:27:35,390 --> 00:27:37,056 I'm going to replace the second row here 526 00:27:37,056 --> 00:27:39,090 by the second minus twice the first. 527 00:27:39,090 --> 00:27:42,060 I'll replace the third row by the third minus 3 times 528 00:27:42,060 --> 00:27:42,740 the first. 529 00:27:42,740 --> 00:27:46,050 I now wind up with this equivalent matrix, 530 00:27:46,050 --> 00:27:48,370 that this matrix codes the same system of equations 531 00:27:48,370 --> 00:27:50,050 as the original matrix. 532 00:27:50,050 --> 00:27:52,930 Now, I'm going to replace the first row by the first row 533 00:27:52,930 --> 00:27:54,200 minus the second. 534 00:27:54,200 --> 00:27:57,250 I will replace the third row by the third row minus the second. 535 00:27:57,250 --> 00:27:59,040 And since I already have a 1 in here, 536 00:27:59,040 --> 00:28:01,340 I will leave the second row intact. 537 00:28:01,340 --> 00:28:06,050 And if I do that, I now wind up with this 3 by 6 matrix. 538 00:28:06,050 --> 00:28:08,140 And now a very interesting thing has 539 00:28:08,140 --> 00:28:13,890 happened-- maybe alarming, maybe unhappy, 540 00:28:13,890 --> 00:28:17,380 but nonetheless interesting. 541 00:28:17,380 --> 00:28:21,910 I observe that on the left-hand side of my 3 by 6 matrix here, 542 00:28:21,910 --> 00:28:26,500 I have a row consisting entirely of 0's, that in row reducing 543 00:28:26,500 --> 00:28:29,270 this, I have lost any chance of getting a 1 over here 544 00:28:29,270 --> 00:28:32,340 because this whole row has dropped out. 545 00:28:32,340 --> 00:28:35,449 See, one of two things had to happen when I row reduced. 546 00:28:35,449 --> 00:28:37,990 Either, when I finished, I would have had the identity matrix 547 00:28:37,990 --> 00:28:42,590 here, or else I would have had, in this left-hand half, 548 00:28:42,590 --> 00:28:44,120 at least one row of 0's. 549 00:28:44,120 --> 00:28:47,140 In this case, I got the one row of 0's. 550 00:28:47,140 --> 00:28:49,810 Let's translate what this thing says. 551 00:28:49,810 --> 00:28:54,890 It says x_1 minus x_3 is 3*y_1 minus y_2. 552 00:28:54,890 --> 00:28:59,350 x_2 plus 2*x_3 is minus 2*y_1 plus y_2. 553 00:28:59,350 --> 00:29:01,220 And it also says that what? 554 00:29:01,220 --> 00:29:04,780 0*x_1 plus 0*x_2 plus 0*x_3-- in other words, 0-- 555 00:29:04,780 --> 00:29:09,730 is equal to minus y_1 minus y_2 plus y_3. 556 00:29:09,730 --> 00:29:13,160 And let's write that to emphasize this. 557 00:29:13,160 --> 00:29:18,250 You see, what we really have from the first two equations 558 00:29:18,250 --> 00:29:21,820 is this system here-- two equations but in three 559 00:29:21,820 --> 00:29:23,810 unknowns-- x_1, x_2, x_3. 560 00:29:23,810 --> 00:29:26,830 We're assuming that the y's are knowns now. 561 00:29:26,830 --> 00:29:28,750 The third piece of information I've 562 00:29:28,750 --> 00:29:32,760 written in an accentuated form to emphasize the fact 563 00:29:32,760 --> 00:29:35,884 that the third equation tells us under what conditions 564 00:29:35,884 --> 00:29:38,050 we're in trouble as far as the choice of the y's are 565 00:29:38,050 --> 00:29:39,160 concerned. 566 00:29:39,160 --> 00:29:43,160 Namely, what this tells us is that minus y_1 567 00:29:43,160 --> 00:29:47,190 minus y_2 plus y_3 has to be 0 to be able to solve 568 00:29:47,190 --> 00:29:48,199 the system at all. 569 00:29:48,199 --> 00:29:49,740 That's the same as saying, of course, 570 00:29:49,740 --> 00:29:52,440 that y_3 is y_1 plus y_2. 571 00:29:52,440 --> 00:29:54,310 So the first thing we know is that, 572 00:29:54,310 --> 00:29:59,440 unless y_3 equals y_1 plus y_2, we cannot express 573 00:29:59,440 --> 00:30:02,750 the x's in terms of the y's. 574 00:30:02,750 --> 00:30:04,250 Why can't we express them? 575 00:30:04,250 --> 00:30:05,990 The system is incompatible. 576 00:30:05,990 --> 00:30:10,060 That's exactly, by the way, what we observed over here. 577 00:30:10,060 --> 00:30:13,320 Only now, we don't have to be that bright to notice it. 578 00:30:13,320 --> 00:30:15,810 See, here we have to astutely observe 579 00:30:15,810 --> 00:30:18,830 that the third equation was the sum of the first two. 580 00:30:18,830 --> 00:30:22,480 Our matrix coding system told us right away that what went wrong 581 00:30:22,480 --> 00:30:26,070 here was that y_3 equals y_1 plus y_2. 582 00:30:26,070 --> 00:30:28,760 So unless y_3 equals y_1 plus y_2, 583 00:30:28,760 --> 00:30:32,070 we cannot express the x's in terms of the y's. 584 00:30:32,070 --> 00:30:34,620 The second thing that goes wrong is 585 00:30:34,620 --> 00:30:39,750 that even if y_3 equals y_1 plus y_2-- then, for example, 586 00:30:39,750 --> 00:30:42,500 we still can't express the x's in terms of the y's, 587 00:30:42,500 --> 00:30:43,959 because-- I say "for example" here, 588 00:30:43,959 --> 00:30:45,708 because I could have picked any of the x's 589 00:30:45,708 --> 00:30:46,900 as far as that's concerned. 590 00:30:46,900 --> 00:30:51,630 But for example, x_3 is independent of y_1 and y_2. 591 00:30:51,630 --> 00:30:54,420 I add "and y3" in parentheses here, 592 00:30:54,420 --> 00:30:56,810 because notice, in the second case, 593 00:30:56,810 --> 00:30:59,460 y_3 is not independent of y_1 and y_2. 594 00:30:59,460 --> 00:31:03,610 y_3 is equal to y_1 plus y_2, so it's automatically specified 595 00:31:03,610 --> 00:31:05,569 as soon as I know y_1 and y_2. 596 00:31:05,569 --> 00:31:06,610 But here what I'm saying. 597 00:31:06,610 --> 00:31:11,320 If I now take this result here and come back 598 00:31:11,320 --> 00:31:14,210 to this system of equations, assuming 599 00:31:14,210 --> 00:31:18,960 that y_1 plus y_2 is y_3 so that I now have this constraint here 600 00:31:18,960 --> 00:31:23,670 met, notice that I can pick x_3 completely at random, 601 00:31:23,670 --> 00:31:29,500 independently of what the values of y_1 and y_2 are, 602 00:31:29,500 --> 00:31:32,170 and solve for x_1 and x_2. 603 00:31:32,170 --> 00:31:36,260 In other words, I have a degree of freedom here. 604 00:31:36,260 --> 00:31:38,990 The x's are not completely determined by the y's. 605 00:31:38,990 --> 00:31:41,660 One of the x's can be chosen at random. 606 00:31:41,660 --> 00:31:43,120 It floats around. 607 00:31:43,120 --> 00:31:45,770 OK, so you see what goes wrong here from the point of view 608 00:31:45,770 --> 00:31:47,360 of inverting the equations? 609 00:31:47,360 --> 00:31:52,560 If the constraint, meaning y_3 equals y_1 plus y_2 is not met, 610 00:31:52,560 --> 00:31:54,520 you can't invert at all. 611 00:31:54,520 --> 00:31:56,420 The system is incompatible. 612 00:31:56,420 --> 00:31:59,620 If the constraint is met, the x's aren't completely 613 00:31:59,620 --> 00:32:00,690 determined by the y's. 614 00:32:00,690 --> 00:32:02,490 You can pick one of the x's at random, 615 00:32:02,490 --> 00:32:04,790 and then solve for the remaining two. 616 00:32:07,600 --> 00:32:09,089 Now, what does this mean in terms 617 00:32:09,089 --> 00:32:10,380 of our function interpretation? 618 00:32:10,380 --> 00:32:12,400 And I think you may enjoy this, because I think 619 00:32:12,400 --> 00:32:14,700 it'll show things very nicely. 620 00:32:14,700 --> 00:32:17,410 What it means, for example, is this. 621 00:32:17,410 --> 00:32:20,200 Suppose I now look at the mapping that's 622 00:32:20,200 --> 00:32:22,990 defined by the system of equations 623 00:32:22,990 --> 00:32:24,860 that we were just talking about. 624 00:32:24,860 --> 00:32:29,450 Notice that what we're saying is that, unless y_3 equals 625 00:32:29,450 --> 00:32:32,880 y_1 plus y_2, there's no hope of finding 626 00:32:32,880 --> 00:32:36,060 x's that solve that particular equation. 627 00:32:36,060 --> 00:32:39,830 In other words, if it turns out, given this mapping, 628 00:32:39,830 --> 00:32:44,105 that y bar is (1, 1, 3), then there is no x bar such 629 00:32:44,105 --> 00:32:47,090 that f bar of x bar is equal to y bar, 630 00:32:47,090 --> 00:32:50,850 because 3 is unequal to 1 plus 1. 631 00:32:50,850 --> 00:32:52,540 You see, again, what we're saying, 632 00:32:52,540 --> 00:32:56,180 in terms of the mapping here, is to get a solution, 633 00:32:56,180 --> 00:32:59,810 the y's have to be no longer arbitrary, but chosen 634 00:32:59,810 --> 00:33:02,860 to satisfy a specific constraint. 635 00:33:02,860 --> 00:33:05,440 And in the system of equations 3 that we 636 00:33:05,440 --> 00:33:08,300 dealt with in this case, that constraint was y_3 637 00:33:08,300 --> 00:33:10,570 equals y_1 plus y_2. 638 00:33:10,570 --> 00:33:13,080 In other words, what this means in terms of the mapping 639 00:33:13,080 --> 00:33:15,800 is that f bar is not onto. 640 00:33:15,800 --> 00:33:18,120 See, notice, the equation wasn't invertible. 641 00:33:18,120 --> 00:33:20,100 A inverse didn't exist. 642 00:33:20,100 --> 00:33:23,170 Therefore, f bar inverse shouldn't exist, 643 00:33:23,170 --> 00:33:25,460 and for f bar inverse not to exist, 644 00:33:25,460 --> 00:33:28,940 it's sufficient that f bar be either not onto, or else 645 00:33:28,940 --> 00:33:30,070 not one-to-one. 646 00:33:30,070 --> 00:33:33,350 And we have now shown that f bar is not onto, because nothing 647 00:33:33,350 --> 00:33:35,590 maps into (1, 1, 3). 648 00:33:35,590 --> 00:33:37,230 We can go one step further and show 649 00:33:37,230 --> 00:33:39,000 that even if the constraint is met-- 650 00:33:39,000 --> 00:33:41,690 suppose we pick a y bar where y_3 651 00:33:41,690 --> 00:33:46,830 is equal to y_1 plus y_2, for example, (1, 1, 2). 652 00:33:46,830 --> 00:33:48,930 See, 2 is equal to 1 plus 1. 653 00:33:48,930 --> 00:33:52,410 Then the system of equations that we had before applies. 654 00:33:52,410 --> 00:33:55,470 What was that system of equations that we had before? 655 00:33:55,470 --> 00:34:01,360 It was x_1 minus x_3 equals 3*y_1 minus y_2. 656 00:34:01,360 --> 00:34:05,450 x_2 plus 2*x_3 equals minus 2*y_1 plus y_2. 657 00:34:05,450 --> 00:34:08,250 The third equation wasn't an equation. 658 00:34:08,250 --> 00:34:10,350 It was the constraint that required 659 00:34:10,350 --> 00:34:13,000 that y_3 equals y_1 plus y_2. 660 00:34:13,000 --> 00:34:14,170 So that's been met. 661 00:34:14,170 --> 00:34:17,010 So we now down to what? 662 00:34:17,010 --> 00:34:22,209 Two equations in three unknowns-- first of all, 663 00:34:22,209 --> 00:34:24,250 let's take a look and see what happens over here. 664 00:34:24,250 --> 00:34:26,560 In this case, y_1 is 1. 665 00:34:26,560 --> 00:34:27,880 y_2 is 1. 666 00:34:27,880 --> 00:34:30,780 So x_1 minus x_3 is simply 2. 667 00:34:30,780 --> 00:34:33,790 x_2 plus 2*x_3 is simply minus 1. 668 00:34:33,790 --> 00:34:40,739 I can now pick x_3 at random and solve for x_1 and x_2 669 00:34:40,739 --> 00:34:42,969 in terms of that, namely, what? 670 00:34:42,969 --> 00:34:50,610 x_1 is simply x_3 plus 2, and x_2 is minus 2*x_3 minus 1. 671 00:34:50,610 --> 00:34:52,540 In other words, what this tells me 672 00:34:52,540 --> 00:34:57,889 is that a whole bunch of vectors are mapped into (1, 1, 2). 673 00:34:57,889 --> 00:35:00,430 A whole bunch of 3-tuples-- if you want to use word "3-tuple" 674 00:35:00,430 --> 00:35:05,200 rather than vector-- are mapped into (1, 1, 2) by f bar. 675 00:35:05,200 --> 00:35:06,920 How are those vectors chosen? 676 00:35:06,920 --> 00:35:09,840 Well, you can pick the third component at random, 677 00:35:09,840 --> 00:35:12,370 in which case, the first component, x_1, 678 00:35:12,370 --> 00:35:16,680 is simply x_3 plus 2, and the second component, x_2, 679 00:35:16,680 --> 00:35:20,865 is simply minus 2*x_3 minus 1, where the x_3 is the value that 680 00:35:20,865 --> 00:35:22,170 you chose here. 681 00:35:22,170 --> 00:35:24,900 Since there are infinitely many ways of choosing x_3, 682 00:35:24,900 --> 00:35:27,010 there are infinitely many 3-tuples 683 00:35:27,010 --> 00:35:28,470 that map into (1, 1, 2). 684 00:35:28,470 --> 00:35:31,320 In particular, f bar is not one-to-one either, 685 00:35:31,320 --> 00:35:34,370 which means, in terms of a picture-- and in fact, 686 00:35:34,370 --> 00:35:36,640 let's see how we can work this very comfortably. 687 00:35:36,640 --> 00:35:39,550 Why not pick x_3 to be 0 for a start? 688 00:35:39,550 --> 00:35:42,630 If we pick x_3 to be 0, notice that this 3-tuple 689 00:35:42,630 --> 00:35:46,690 becomes (2, -1, 0). 690 00:35:46,690 --> 00:35:52,390 So (2, -1, 0)-- (2, -1, 0) maps into (1, 1, 2). 691 00:35:52,390 --> 00:35:58,910 If we pick x_3 to be say, 1, this becomes (3, -3, 1). 692 00:35:58,910 --> 00:36:03,600 So, for example, (3, -3, 1) also maps into (1, 1, 2). 693 00:36:03,600 --> 00:36:08,350 So that's the demonstration that f bar is not one-to-one. 694 00:36:08,350 --> 00:36:11,710 And the fact that nothing maps into (1, 1, 3) 695 00:36:11,710 --> 00:36:15,550 is the indication that f bar is not onto. 696 00:36:15,550 --> 00:36:19,240 By the way, by choosing a three-dimensional space, 697 00:36:19,240 --> 00:36:22,330 we can interpret this result geometrically. 698 00:36:22,330 --> 00:36:23,880 Now, what I'm going to do here is 699 00:36:23,880 --> 00:36:26,000 really optional from your point of view. 700 00:36:26,000 --> 00:36:28,570 If you don't like studies of three-dimensional space, 701 00:36:28,570 --> 00:36:31,585 ignore what I'm saying, otherwise-- I 702 00:36:31,585 --> 00:36:32,460 don't mean ignore it. 703 00:36:32,460 --> 00:36:34,390 Take it with a grain of salt. Look it over. 704 00:36:34,390 --> 00:36:34,890 Read it. 705 00:36:34,890 --> 00:36:37,740 It is not crucial to understanding abstractly what's 706 00:36:37,740 --> 00:36:38,500 going on. 707 00:36:38,500 --> 00:36:43,130 But I thought a visual example like this might be helpful. 708 00:36:43,130 --> 00:36:47,000 Notice that when you're mapping E^3 into E^3, 709 00:36:47,000 --> 00:36:50,150 you're mapping three-dimensional space into three-dimensional 710 00:36:50,150 --> 00:36:51,040 space. 711 00:36:51,040 --> 00:36:53,400 By the way, notice the n-tuple notation here. 712 00:36:53,400 --> 00:36:57,930 I prefer to write the x_1, x_2, x_3 space rather than 713 00:36:57,930 --> 00:37:00,650 the traditional x, y, z space. 714 00:37:00,650 --> 00:37:05,920 In other words, f bar is mapping x_1, x_2, x_3 space 715 00:37:05,920 --> 00:37:09,800 into y_1, y_2, y_3, space. 716 00:37:09,800 --> 00:37:12,380 And what we have shown so far is that, 717 00:37:12,380 --> 00:37:14,960 with our particular choice of function under consideration 718 00:37:14,960 --> 00:37:19,430 now, the only way f bar maps something into anything 719 00:37:19,430 --> 00:37:22,280 is if that anything has what property to it? 720 00:37:22,280 --> 00:37:26,020 What does the point (y_1, y_2, y_3) 721 00:37:26,020 --> 00:37:29,420 have to satisfy in order that we can be sure 722 00:37:29,420 --> 00:37:34,420 that something from the x_1, x_2, x_3 space maps into it? 723 00:37:34,420 --> 00:37:36,200 And the constraint was what? 724 00:37:36,200 --> 00:37:39,870 That y_3 equals y_1 plus y_2. 725 00:37:39,870 --> 00:37:42,790 Now, this is a good time to review our equation of planes 726 00:37:42,790 --> 00:37:44,430 in three-dimensional space. 727 00:37:44,430 --> 00:37:48,200 Sure, we're used to talking about z equals x plus y. 728 00:37:48,200 --> 00:37:50,510 But don't worry about that part notationally. 729 00:37:50,510 --> 00:37:55,380 y_3 equals y_1 plus y_2 is the equation of a plane. 730 00:37:55,380 --> 00:37:57,130 In fact, what plane is it? 731 00:37:57,130 --> 00:37:59,440 Instead of going into a long harangue about this, 732 00:37:59,440 --> 00:38:02,050 let me just find three points that are on this plane. 733 00:38:02,050 --> 00:38:06,630 In particular, (0, 0, 0) satisfies this plane. 734 00:38:06,630 --> 00:38:09,640 (2, 2, 4)-- see, 2 plus 2 is 4. 735 00:38:09,640 --> 00:38:11,050 1 plus 3 is 4. 736 00:38:11,050 --> 00:38:14,190 So three quick points I could pick off are the origin, (2, 2, 737 00:38:14,190 --> 00:38:16,060 4), (1, 3, 4=. 738 00:38:16,060 --> 00:38:18,550 These are three points not on the same straight line. 739 00:38:18,550 --> 00:38:20,060 They determine the plane. 740 00:38:20,060 --> 00:38:22,990 The image of f bar is this particular plane. 741 00:38:22,990 --> 00:38:25,740 In other words, f bar does not use up all the free space. 742 00:38:25,740 --> 00:38:29,860 It, in particular, uses up this one particular plane. 743 00:38:29,860 --> 00:38:33,540 In particular, let's look at our point (1, 1, 2). 744 00:38:33,540 --> 00:38:37,040 What did we see before mapped into (1, 1, 2)? 745 00:38:37,040 --> 00:38:40,910 We saw that any 3-tuple (x_1, x_2, x_3), 746 00:38:40,910 --> 00:38:45,170 where we picked x_3 at random, whereupon x_1 had to be x_3 747 00:38:45,170 --> 00:38:49,670 plus 2, x_2 had to be minus 2*x_3 minus 1. 748 00:38:49,670 --> 00:38:55,840 Notice that this system of equations defines a line. 749 00:38:55,840 --> 00:38:58,620 You see, think of x_3 as being a parameter here. 750 00:38:58,620 --> 00:39:02,410 If it bothers you, think of x_3 as being t. 751 00:39:02,410 --> 00:39:04,815 And what you have is that x_1 is t plus 2, 752 00:39:04,815 --> 00:39:09,420 x_2 is minus 2t minus 1, and x_3 is t. 753 00:39:09,420 --> 00:39:14,370 That is the parametric form of an equation 754 00:39:14,370 --> 00:39:15,470 for a straight line. 755 00:39:15,470 --> 00:39:18,190 You see, notice you have only one degree of freedom, t, 756 00:39:18,190 --> 00:39:20,830 and t appears only linearly. 757 00:39:20,830 --> 00:39:21,660 Now, lookit. 758 00:39:21,660 --> 00:39:25,810 To find the actual straight line geometrically, all I have to do 759 00:39:25,810 --> 00:39:28,030 is find two points on that line. 760 00:39:28,030 --> 00:39:32,760 To relate this with what we were doing just previous to this, 761 00:39:32,760 --> 00:39:34,580 notice that two points that we know 762 00:39:34,580 --> 00:39:39,650 are on this line are (2, -1, 0) and (3, -3, 1). 763 00:39:39,650 --> 00:39:41,230 Let's look at this graphically. 764 00:39:41,230 --> 00:39:45,880 Let's locate the point (2, -1, 0), (3, -3, 1) 765 00:39:45,880 --> 00:39:48,560 in the x_1, x_2, x_3 space. 766 00:39:48,560 --> 00:39:51,180 Draw the line that joins those two points. 767 00:39:51,180 --> 00:39:55,220 That line l has the property that every single one 768 00:39:55,220 --> 00:40:01,000 of these points on l map into the point (1, 1, 2) 769 00:40:01,000 --> 00:40:04,240 in the y_1, y_2, y_3 space. 770 00:40:04,240 --> 00:40:06,250 In other words, our mapping in this case 771 00:40:06,250 --> 00:40:08,500 is neither one-to-one nor onto. 772 00:40:08,500 --> 00:40:11,360 The entire image of this three-dimensional space 773 00:40:11,360 --> 00:40:13,140 is a single plane. 774 00:40:13,140 --> 00:40:15,030 And given a point not in that plane, 775 00:40:15,030 --> 00:40:17,740 therefore, nothing maps into that, 776 00:40:17,740 --> 00:40:22,910 but given a point in that plane, an entire line from the domain 777 00:40:22,910 --> 00:40:23,940 maps in there. 778 00:40:23,940 --> 00:40:27,030 And I simply point this out to you so that for those of you 779 00:40:27,030 --> 00:40:29,460 would like to have a picture of what's happening, 780 00:40:29,460 --> 00:40:31,420 I think this is a very nice way of seeing it. 781 00:40:31,420 --> 00:40:33,970 The reason I don't want to emphasize the picture more 782 00:40:33,970 --> 00:40:37,740 is simply because we will deal in the notes and the exercises 783 00:40:37,740 --> 00:40:41,510 with systems of more than three equations and three unknowns. 784 00:40:41,510 --> 00:40:43,920 We will deal with linear mapping, say, 785 00:40:43,920 --> 00:40:46,670 from E^5 into E^5, in which case, 786 00:40:46,670 --> 00:40:49,910 we can't draw the picture, so the picture is only a nice 787 00:40:49,910 --> 00:40:55,000 mental aid in setting things up in the 2 by 2 case or the 3 788 00:40:55,000 --> 00:40:56,070 by 3 case. 789 00:40:56,070 --> 00:40:58,700 But other than that, one has to rely 790 00:40:58,700 --> 00:41:00,890 on the equivalent of the matrix system 791 00:41:00,890 --> 00:41:02,330 for reducing these equations. 792 00:41:02,330 --> 00:41:05,340 So let me summarize, then, what our main aim was 793 00:41:05,340 --> 00:41:06,950 and what we have accomplished. 794 00:41:06,950 --> 00:41:11,570 Leaving out the motivations of everything for the time being, 795 00:41:11,570 --> 00:41:14,390 let me just summarize how we invert a matrix, 796 00:41:14,390 --> 00:41:17,190 provided that the matrix inverse exists. 797 00:41:17,190 --> 00:41:22,050 Given the n by n matrix A, we form the n by 2n matrix 798 00:41:22,050 --> 00:41:25,060 A augmented by the n by n identity matrix. 799 00:41:25,060 --> 00:41:27,220 In other words, what matrix is it? 800 00:41:27,220 --> 00:41:33,690 The left half is a_1 up to a_(1,n); a_(n,1), a_(n,n). 801 00:41:33,690 --> 00:41:35,590 In other words, it's the matrix A. 802 00:41:35,590 --> 00:41:39,390 And the right half is just the n by n identity matrix. 803 00:41:39,390 --> 00:41:41,170 We then row reduce this. 804 00:41:41,170 --> 00:41:44,380 Remember what mean by row reduce-- 805 00:41:44,380 --> 00:41:48,750 replacing rows by the rows plus or minus 806 00:41:48,750 --> 00:41:51,640 an appropriate multiple of another row, 807 00:41:51,640 --> 00:41:55,230 hoping to get the identity matrix to come over 808 00:41:55,230 --> 00:41:56,790 to the left-hand side over here. 809 00:41:56,790 --> 00:41:58,400 We row reduce this matrix. 810 00:41:58,400 --> 00:42:00,400 And then what we saw was what? 811 00:42:00,400 --> 00:42:02,020 We did it in the 3 by 3 case. 812 00:42:02,020 --> 00:42:04,380 We're now doing it in the n by n case. 813 00:42:04,380 --> 00:42:07,450 By the way, notice that if this throws you off abstractly, 814 00:42:07,450 --> 00:42:11,840 in the 3 by 3 case, the augmented matrix that we form 815 00:42:11,840 --> 00:42:13,410 was a 3 by 6. 816 00:42:13,410 --> 00:42:16,820 All we're saying is in the n by n case, you're tacking on, 817 00:42:16,820 --> 00:42:19,350 side by side, two n by n matrices, 818 00:42:19,350 --> 00:42:22,190 which makes the resulting matrix n by 2n. 819 00:42:22,190 --> 00:42:26,300 At any rate, what we do is we reduce this n by 2n matrix 820 00:42:26,300 --> 00:42:28,340 until one of two things happens. 821 00:42:28,340 --> 00:42:29,960 And one of the two must happen. 822 00:42:29,960 --> 00:42:34,440 Namely, either the left-hand half-- this half-- 823 00:42:34,440 --> 00:42:38,620 contains at least one row of 0's, in which case 824 00:42:38,620 --> 00:42:41,690 A inverse doesn't exist. 825 00:42:41,690 --> 00:42:43,370 The other alternative is that when 826 00:42:43,370 --> 00:42:47,920 we row reduce, the left-hand side-- the left half, 827 00:42:47,920 --> 00:42:51,580 in other words-- does become the identity matrix, in which case, 828 00:42:51,580 --> 00:42:53,670 the right half is A inverse. 829 00:42:53,670 --> 00:42:55,900 That's all there is to this mechanically. 830 00:42:55,900 --> 00:42:58,890 You row reduce the n by 2n matrix. 831 00:42:58,890 --> 00:42:59,390 All right. 832 00:42:59,390 --> 00:43:02,490 If the left-hand side contains a row of 0's, there 833 00:43:02,490 --> 00:43:04,760 is no inverse to the given matrix. 834 00:43:04,760 --> 00:43:07,550 If it reduces to the identity matrix, 835 00:43:07,550 --> 00:43:10,850 then the right half is the inverse matrix. 836 00:43:10,850 --> 00:43:13,020 Part 2 of our summary-- namely, what can we 837 00:43:13,020 --> 00:43:14,920 conclude if A inverse exists? 838 00:43:14,920 --> 00:43:19,730 In terms of matrix algebra, if A inverse exists, we can solve-- 839 00:43:19,730 --> 00:43:24,620 in other words, we can invert the equation Y equals A*X 840 00:43:24,620 --> 00:43:29,130 to conclude that X equals A inverse Y. 841 00:43:29,130 --> 00:43:32,590 And finally, in terms of mappings, 842 00:43:32,590 --> 00:43:37,740 if we define f bar from E sub n into E sub n-- 843 00:43:37,740 --> 00:43:40,120 f bar is a mapping from n-dimensional space 844 00:43:40,120 --> 00:43:45,570 into n-dimensional space-- defined by this, 845 00:43:45,570 --> 00:43:50,920 then f bar inverse exists if and only if A inverse exists, 846 00:43:50,920 --> 00:43:54,870 where A is the matrix of coefficients over here. 847 00:43:54,870 --> 00:43:58,135 And if A inverse doesn't exist, then 848 00:43:58,135 --> 00:44:02,170 f bar inverse not only doesn't exist, but f bar 849 00:44:02,170 --> 00:44:05,840 itself is neither one-to-one nor onto. 850 00:44:05,840 --> 00:44:10,230 Now, admittedly, today's lecture was a little bit 851 00:44:10,230 --> 00:44:11,490 on the long side. 852 00:44:11,490 --> 00:44:15,360 It was, though, in a sense a single concept, hopefully 853 00:44:15,360 --> 00:44:17,320 with many interpretations. 854 00:44:17,320 --> 00:44:22,290 It is a very, very crucial thing in many of our investigations 855 00:44:22,290 --> 00:44:25,110 to do at least one of two things. 856 00:44:25,110 --> 00:44:26,730 One thing is going to be, we are going 857 00:44:26,730 --> 00:44:30,490 to want to know whether the inverse of a matrix exists. 858 00:44:30,490 --> 00:44:32,660 In certain applications, we will not 859 00:44:32,660 --> 00:44:35,570 care to know what that inverse matrix is. 860 00:44:35,570 --> 00:44:38,980 All we're going to want to know is, does it exist? 861 00:44:38,980 --> 00:44:41,330 In that case, that's when it's sufficient to compute 862 00:44:41,330 --> 00:44:43,140 the determinant of a matrix. 863 00:44:43,140 --> 00:44:45,500 Namely, if the determinant is 0, it doesn't exist. 864 00:44:45,500 --> 00:44:49,560 If the determinant is not 0, the inverse matrix does exist. 865 00:44:49,560 --> 00:44:51,210 On other occasions, it will not be 866 00:44:51,210 --> 00:44:54,320 enough to know that the inverse exists. 867 00:44:54,320 --> 00:44:58,030 We will also require that we be able to construct 868 00:44:58,030 --> 00:45:00,200 A inverse from the given matrix A 869 00:45:00,200 --> 00:45:02,910 and that, in essence, was the lesson that we 870 00:45:02,910 --> 00:45:04,760 were trying to cover today. 871 00:45:04,760 --> 00:45:07,240 Starting with our next lesson, we 872 00:45:07,240 --> 00:45:11,270 will apply these results to systems of functions 873 00:45:11,270 --> 00:45:12,705 of several real variables. 874 00:45:12,705 --> 00:45:14,680 And until that time, good bye. 875 00:45:21,780 --> 00:45:24,150 Funding for the publication of this video 876 00:45:24,150 --> 00:45:29,020 was provided by the Gabriella and Paul Rosenbaum Foundation. 877 00:45:29,020 --> 00:45:33,200 Help OCW continue to provide free and open access to MIT 878 00:45:33,200 --> 00:45:40,900 courses by making a donation at ocw.mit.edu/donate.