1 00:00:05,870 --> 00:00:08,750 OK, when the camera says, we'll start. 2 00:00:08,750 --> 00:00:13,320 You want to give me a signal? 3 00:00:13,320 --> 00:00:17,290 OK, this is lecture eight in linear algebra, 4 00:00:17,290 --> 00:00:20,020 and this is the lecture where we completely 5 00:00:20,020 --> 00:00:22,820 solve linear equations. 6 00:00:22,820 --> 00:00:23,810 So Ax=b. 7 00:00:27,080 --> 00:00:29,040 That's our goal. 8 00:00:29,040 --> 00:00:32,159 If it has a solution. 9 00:00:32,159 --> 00:00:35,990 It certainly can happen that there is no solution. 10 00:00:35,990 --> 00:00:40,370 We have to identify that possibility by elimination. 11 00:00:40,370 --> 00:00:45,110 And then if there is a solution we want to find out is there 12 00:00:45,110 --> 00:00:48,650 only one solution or are -- is there a whole family 13 00:00:48,650 --> 00:00:51,190 of solutions, and then find them all. 14 00:00:51,190 --> 00:00:52,370 OK. 15 00:00:52,370 --> 00:00:57,420 Can I use as an example the same matrix 16 00:00:57,420 --> 00:01:01,010 that I had last time when we were 17 00:01:01,010 --> 00:01:02,890 looking for the null space. 18 00:01:02,890 --> 00:01:10,060 So the, the matrix has rows 1 2 2 2, 2 4 6 8, 19 00:01:10,060 --> 00:01:13,150 and the third row -- you remember the main point was 20 00:01:13,150 --> 00:01:21,890 the third row, 3 6 8 10, is the sum of row one plus row two. 21 00:01:21,890 --> 00:01:25,710 In other words, if I add those left-hand sides, 22 00:01:25,710 --> 00:01:28,710 I get the third left-hand side. 23 00:01:28,710 --> 00:01:31,530 So you can tell me right away what 24 00:01:31,530 --> 00:01:35,460 elimination is going to discover about the right-hand sides. 25 00:01:35,460 --> 00:01:41,570 What's -- there is a condition on b1, b2, 26 00:01:41,570 --> 00:01:44,690 and b3 for this system to have a solution. 27 00:01:44,690 --> 00:01:50,240 Most cases -- if I took these numbers to be one -- 28 00:01:50,240 --> 00:01:53,950 5, and 17, there would not be a solution. 29 00:01:53,950 --> 00:01:58,700 In fact, if I took those first numbers to be 1 and 5, 30 00:01:58,700 --> 00:02:03,600 what is the only b3 that would be OK? 31 00:02:03,600 --> 00:02:05,560 Six. 32 00:02:05,560 --> 00:02:09,860 If the left-hand -- if these left-hand sides add up to that, 33 00:02:09,860 --> 00:02:10,729 then B -- 34 00:02:10,729 --> 00:02:15,000 I need b1 plus b2 to equal b3. 35 00:02:15,000 --> 00:02:19,470 Let's just see how elimination discovers that. 36 00:02:19,470 --> 00:02:23,430 But we can see it coming, right? 37 00:02:23,430 --> 00:02:26,450 That if -- let me say it in other words. 38 00:02:26,450 --> 00:02:29,300 If some combination on the left-hand side 39 00:02:29,300 --> 00:02:33,090 gives all 0s then the same combination 40 00:02:33,090 --> 00:02:35,580 on the right-hand side must give 0. 41 00:02:35,580 --> 00:02:36,080 OK. 42 00:02:36,080 --> 00:02:43,670 So let me take that example and write down 43 00:02:43,670 --> 00:02:46,670 instead of copying out all the plus signs, 44 00:02:46,670 --> 00:02:49,830 let me write down the matrix. 45 00:02:49,830 --> 00:02:59,130 1 2 2 2, 2 4 6 8, and that 6 3 8 10, 46 00:02:59,130 --> 00:03:02,930 where the third row is the sum of the first two rows. 47 00:03:02,930 --> 00:03:06,700 Now how do we deal with the right-hand side? 48 00:03:06,700 --> 00:03:09,840 That's -- we want to do the same thing to the right-hand side 49 00:03:09,840 --> 00:03:12,700 that we're doing to these rows on the left side, 50 00:03:12,700 --> 00:03:17,920 so we just tack on the right-hand side as another 51 00:03:17,920 --> 00:03:20,660 vector, another column. 52 00:03:20,660 --> 00:03:26,220 So this is the augmented matrix. 53 00:03:29,640 --> 00:03:35,840 It's, it's the matrix A with the vector b tacked on. 54 00:03:35,840 --> 00:03:38,440 In Matlab, that's all you would need to type. 55 00:03:38,440 --> 00:03:39,050 OK. 56 00:03:39,050 --> 00:03:41,470 So we do elimination on that. 57 00:03:41,470 --> 00:03:43,580 Can we just do elimination quickly? 58 00:03:43,580 --> 00:03:46,790 The first pivot is fine, I subtract two of this 59 00:03:46,790 --> 00:03:49,470 away from this, three of this away from this, 60 00:03:49,470 --> 00:03:54,870 so I have 1 2 2 2 b1. 61 00:03:54,870 --> 00:04:00,840 Two of those away will give me 0 0 2 and 4, 62 00:04:00,840 --> 00:04:03,570 and that was b2 minus two b1. 63 00:04:03,570 --> 00:04:07,130 I, I have to do the same thing to that third, 64 00:04:07,130 --> 00:04:08,630 that last column. 65 00:04:08,630 --> 00:04:10,700 And then three of these away from this 66 00:04:10,700 --> 00:04:17,980 gave me 0 0 2 4 b3 minus three b1s. 67 00:04:17,980 --> 00:04:21,660 So that's the, that's elimination 68 00:04:21,660 --> 00:04:25,830 with the first column completed. 69 00:04:25,830 --> 00:04:26,750 We move on. 70 00:04:26,750 --> 00:04:29,570 There's the first pivot still. 71 00:04:29,570 --> 00:04:31,500 Here is the second pivot. 72 00:04:31,500 --> 00:04:34,660 We're always remembering, now, these are then 73 00:04:34,660 --> 00:04:36,650 going to be the pivot columns. 74 00:04:41,810 --> 00:04:47,640 And let me get the final result -- well, let me -- 75 00:04:47,640 --> 00:04:51,615 can I do it by eraser? 76 00:04:55,550 --> 00:05:00,680 We're capable of subtracting this row from this row, 77 00:05:00,680 --> 00:05:05,280 just by -- that'll knock this out completely and give me 78 00:05:05,280 --> 00:05:08,090 the row of 0s, and on the right-hand side, 79 00:05:08,090 --> 00:05:12,660 when I subtract this away from this, what do I have? 80 00:05:16,260 --> 00:05:22,550 I think I have b3 minus a b2, and I had minus three b1s. 81 00:05:22,550 --> 00:05:25,380 This is going to, it's going to be a minus a b1. 82 00:05:25,380 --> 00:05:28,000 Oh yeah that's exactly what I expect. 83 00:05:31,490 --> 00:05:34,430 So now the -- what's the last equation? 84 00:05:34,430 --> 00:05:38,930 The last equation, this represented by this zero row, 85 00:05:38,930 --> 00:05:45,750 that last equation is, says 0 equals b3 minus b2 minus b1. 86 00:05:45,750 --> 00:05:51,519 So that's the condition for solvability. 87 00:05:51,519 --> 00:05:53,310 That's the condition on the right-hand side 88 00:05:53,310 --> 00:05:54,450 that we expected. 89 00:05:54,450 --> 00:05:57,900 It says that b1+b2 has to match b3, 90 00:05:57,900 --> 00:06:02,350 and if our numbers happen to have been 1, 5, and 6 -- 91 00:06:02,350 --> 00:06:07,220 so let me take, suppose b is 1 5 6. 92 00:06:07,220 --> 00:06:09,920 That's an OK b. 93 00:06:09,920 --> 00:06:13,120 And when I do this elimination, what will I have? 94 00:06:13,120 --> 00:06:16,390 The b1 will still be a 1. 95 00:06:16,390 --> 00:06:18,950 b2 would be 5 minus 2, this would be a 3. 96 00:06:18,950 --> 00:06:24,860 5 -- my 6 minus 5 minus 1, this will be -- 97 00:06:24,860 --> 00:06:29,800 this is the main point -- this will be a 0, thanks. 98 00:06:29,800 --> 00:06:30,300 OK. 99 00:06:30,300 --> 00:06:32,720 So the last equation is OK now. 100 00:06:37,080 --> 00:06:41,880 And I can proceed to solve the two equations that are really 101 00:06:41,880 --> 00:06:44,160 there with four unknowns. 102 00:06:44,160 --> 00:06:48,660 OK, I, I, I want to do that, so this, this b is OK. 103 00:06:48,660 --> 00:06:51,700 It allows a solution. 104 00:06:51,700 --> 00:06:56,850 We're going to be, naturally, interested 105 00:06:56,850 --> 00:07:04,200 to keep track what are the conditions on b that 106 00:07:04,200 --> 00:07:06,550 make the equation solvable. 107 00:07:06,550 --> 00:07:11,020 So let me write down what we already see 108 00:07:11,020 --> 00:07:14,490 before I continue to solve it. 109 00:07:14,490 --> 00:07:17,286 Let me first -- solvability, solvability. 110 00:07:23,450 --> 00:07:31,015 So which -- so this is the condition on the right-hand 111 00:07:31,015 --> 00:07:31,515 sides. 112 00:07:34,170 --> 00:07:36,010 And what is that condition? 113 00:07:36,010 --> 00:07:39,440 This is solvability always of Ax=b. 114 00:07:39,440 --> 00:07:45,150 So Ax=b is solvable -- 115 00:07:45,150 --> 00:07:50,870 well, actually, we had an answer in the language of the column 116 00:07:50,870 --> 00:07:52,640 space. 117 00:07:52,640 --> 00:07:54,530 Can you remind me what that answer is? 118 00:07:54,530 --> 00:07:58,510 That, that was like our answer from earlier lecture. 119 00:07:58,510 --> 00:08:00,980 b had to be in the column space. 120 00:08:00,980 --> 00:08:10,140 Solvable if -- when -- exactly when b is in the column space 121 00:08:10,140 --> 00:08:13,560 of A. 122 00:08:13,560 --> 00:08:14,110 Right? 123 00:08:14,110 --> 00:08:17,840 That just says that b has to be a combination of the columns, 124 00:08:17,840 --> 00:08:21,890 and of course that's exactly what the equation is looking 125 00:08:21,890 --> 00:08:22,660 for. 126 00:08:22,660 --> 00:08:25,800 So that -- now I want to answer it -- 127 00:08:25,800 --> 00:08:30,050 the same answer but in different language. 128 00:08:30,050 --> 00:08:33,990 Another way to answer this -- 129 00:08:33,990 --> 00:08:52,980 if a combination of the rows of A gives the zero row, 130 00:08:52,980 --> 00:08:57,010 and this was an example where it happened, 131 00:08:57,010 --> 00:09:00,940 some combination of the rows of A produced the zero row -- 132 00:09:00,940 --> 00:09:04,560 then what's the requirement on b? 133 00:09:04,560 --> 00:09:07,130 Since we're going to do the same thing to both sides of all 134 00:09:07,130 --> 00:09:08,520 equations -- 135 00:09:08,520 --> 00:09:16,190 the same combination of the components of b has to give 0. 136 00:09:16,190 --> 00:09:16,740 Right? 137 00:09:16,740 --> 00:09:19,590 That's -- so if there's a combination of the rows that 138 00:09:19,590 --> 00:09:29,920 gives the zero row, then the same combination of the entries 139 00:09:29,920 --> 00:09:34,700 of b must give 0. 140 00:09:37,950 --> 00:09:40,475 And this isn't the zero row, that's the zero number. 141 00:09:43,650 --> 00:09:47,860 Tthis is another way of saying -- and it is not immediate, 142 00:09:47,860 --> 00:09:54,582 OK. right, that these two statements are equivalent. 143 00:09:54,582 --> 00:09:56,290 But somehow they must be, because they're 144 00:09:56,290 --> 00:09:59,961 both equivalent to the solvability of the system. 145 00:09:59,961 --> 00:10:00,460 OK. 146 00:10:00,460 --> 00:10:05,030 So we've got this, this sort of -- like question zero is, 147 00:10:05,030 --> 00:10:08,180 does the system have a solution? 148 00:10:08,180 --> 00:10:12,670 OK, I'll come back to discuss that further. 149 00:10:12,670 --> 00:10:17,310 Let's go forward when it does. 150 00:10:17,310 --> 00:10:19,470 When there is a solution. 151 00:10:19,470 --> 00:10:22,684 And so what's our job now? 152 00:10:22,684 --> 00:10:24,600 Abstractly we sit back and we say, OK, there's 153 00:10:24,600 --> 00:10:26,660 a solution, finished. 154 00:10:26,660 --> 00:10:27,570 It exists. 155 00:10:27,570 --> 00:10:29,450 But we want to construct it. 156 00:10:29,450 --> 00:10:34,500 So what's the algorithm, the sequence 157 00:10:34,500 --> 00:10:37,330 of steps to find the solution? 158 00:10:37,330 --> 00:10:38,670 That's what I -- 159 00:10:38,670 --> 00:10:42,130 and of course the quiz and the final, 160 00:10:42,130 --> 00:10:45,440 I'm going to give you a system Ax=b and I'm going to ask you 161 00:10:45,440 --> 00:10:48,350 for the solution, if there is one. 162 00:10:48,350 --> 00:10:54,430 And so this algorithm that you want to follow. 163 00:10:54,430 --> 00:10:58,290 OK, let's see. 164 00:10:58,290 --> 00:11:13,190 So what's the -- so now to find the complete solution to Ax=b. 165 00:11:13,190 --> 00:11:14,000 OK. 166 00:11:14,000 --> 00:11:17,150 Let me start by finding one solution, 167 00:11:17,150 --> 00:11:19,410 one particular solution. 168 00:11:22,030 --> 00:11:26,060 I'm expecting that I can, because my system of equations 169 00:11:26,060 --> 00:11:30,730 now, that last equation is zero equals zero, 170 00:11:30,730 --> 00:11:33,850 so that's all fine. 171 00:11:33,850 --> 00:11:36,720 I really have two equations -- 172 00:11:36,720 --> 00:11:38,910 actually I've got four unknowns, so I'm 173 00:11:38,910 --> 00:11:41,410 expecting to find not only a solution 174 00:11:41,410 --> 00:11:44,230 but a whole bunch of them. 175 00:11:44,230 --> 00:11:46,020 But let's just find one. 176 00:11:46,020 --> 00:11:50,930 So step one, a particular solution, x particular. 177 00:11:54,430 --> 00:11:57,010 How do I find one particular solution? 178 00:11:57,010 --> 00:12:00,740 Well, let me tell you how I, how I find it. 179 00:12:00,740 --> 00:12:02,160 So this is -- 180 00:12:02,160 --> 00:12:04,080 since there are lots of solutions, 181 00:12:04,080 --> 00:12:07,100 you could have your own way to find a particular one. 182 00:12:07,100 --> 00:12:10,780 But this is a pretty natural way. 183 00:12:10,780 --> 00:12:20,880 Set all free variables to zero. 184 00:12:20,880 --> 00:12:25,810 Since those free variables are the guys that can be anything, 185 00:12:25,810 --> 00:12:28,790 the most convenient choice is zero. 186 00:12:28,790 --> 00:12:37,975 And then solve Ax=b for the pivot variables. 187 00:12:41,170 --> 00:12:44,240 So what does that mean in this example? 188 00:12:44,240 --> 00:12:46,490 Which are the free variables? 189 00:12:46,490 --> 00:12:49,500 Which, which are the variables that we can assign freely 190 00:12:49,500 --> 00:12:52,500 and then there's one and only one way 191 00:12:52,500 --> 00:12:55,070 to find the pivot variables? 192 00:12:55,070 --> 00:13:01,120 They're x2 and -- so x2 is zero, because that's in a column 193 00:13:01,120 --> 00:13:04,280 without a pivot, the second column has no pivot. 194 00:13:04,280 --> 00:13:08,010 And the -- what's the other one? 195 00:13:08,010 --> 00:13:11,620 The fourth, x4 is zero. 196 00:13:11,620 --> 00:13:16,330 Because that, those are the, the free ones. 197 00:13:16,330 --> 00:13:18,870 Those are in the columns with no pivots. 198 00:13:18,870 --> 00:13:21,300 So you see what my -- so when I knock -- 199 00:13:21,300 --> 00:13:28,660 when x2 and x4 are zero, I'm left with the -- 200 00:13:28,660 --> 00:13:31,010 what I left with here? 201 00:13:31,010 --> 00:13:33,460 I'm just left with -- 202 00:13:33,460 --> 00:13:36,500 see, now I'm not using the two free 203 00:13:36,500 --> 00:13:37,020 columns. 204 00:13:37,020 --> 00:13:39,380 I'm only using the pivot columns. 205 00:13:39,380 --> 00:13:42,370 So I'm really left with x1 -- 206 00:13:42,370 --> 00:13:45,360 the first equation is just x1 and two 207 00:13:45,360 --> 00:13:48,720 x3s should be the right-hand side, which 208 00:13:48,720 --> 00:13:50,700 we picked to be a one. 209 00:13:50,700 --> 00:13:54,130 And the second equation is two x3s, 210 00:13:54,130 --> 00:13:57,735 as it happened, turned out to be, three. 211 00:14:02,090 --> 00:14:06,680 I just write it again here with the x2 and the x4 212 00:14:06,680 --> 00:14:09,420 knocked out, since we're set them to zero. 213 00:14:09,420 --> 00:14:14,150 And you see that we're back in the normal case of having back 214 00:14:14,150 --> 00:14:16,030 -- where back substitution will do it. 215 00:14:16,030 --> 00:14:21,640 So x3 is three halves, and then we go back up 216 00:14:21,640 --> 00:14:25,490 and x1 is one minus two x3. 217 00:14:25,490 --> 00:14:29,270 That's probably minus two. 218 00:14:29,270 --> 00:14:30,400 Good. 219 00:14:30,400 --> 00:14:34,210 So now we have the solution, x particular 220 00:14:34,210 --> 00:14:41,940 is the vector minus two zero three halves zero. 221 00:14:44,710 --> 00:14:46,790 OK, good. 222 00:14:46,790 --> 00:14:52,200 That's one particular solution, and we should and could plug it 223 00:14:52,200 --> 00:14:54,600 into the original system. 224 00:14:54,600 --> 00:14:57,010 Really if -- on the quiz, please, 225 00:14:57,010 --> 00:14:59,230 it's a good thing to do. 226 00:14:59,230 --> 00:15:03,650 So we did all this, these, row operations, 227 00:15:03,650 --> 00:15:06,960 but this is supposed to solve the original system, 228 00:15:06,960 --> 00:15:09,430 and I think it does. 229 00:15:09,430 --> 00:15:10,120 OK. 230 00:15:10,120 --> 00:15:14,810 So that's x particular which we've got. 231 00:15:14,810 --> 00:15:19,320 So that's like what's new today. 232 00:15:19,320 --> 00:15:23,780 The particular solution comes -- first you check that you have 233 00:15:23,780 --> 00:15:26,700 zero equals zero, so you're OK on the last 234 00:15:26,700 --> 00:15:27,920 equations. 235 00:15:27,920 --> 00:15:31,980 And then you set the free variables to zero, 236 00:15:31,980 --> 00:15:34,830 solve for the pivot variables, and you've 237 00:15:34,830 --> 00:15:38,500 got a particular solution, the particular solution that 238 00:15:38,500 --> 00:15:41,150 has zero free variables. 239 00:15:41,150 --> 00:15:42,040 OK. 240 00:15:42,040 --> 00:15:45,000 Now -- but that's only one solution, 241 00:15:45,000 --> 00:15:46,270 and now I'm looking for all. 242 00:15:49,020 --> 00:15:51,480 So how do I find the rest? 243 00:15:51,480 --> 00:15:58,770 The point is I can add on x -- anything out of the null space. 244 00:16:03,320 --> 00:16:06,190 We know how to find the vectors in the null space -- 245 00:16:06,190 --> 00:16:08,950 because we did it last time, but I'll remind you what we 246 00:16:08,950 --> 00:16:09,710 got. 247 00:16:09,710 --> 00:16:12,620 And then I'll add. 248 00:16:15,630 --> 00:16:20,650 So the final result will be that the complete solution -- 249 00:16:20,650 --> 00:16:23,620 this is now the complete guy -- 250 00:16:23,620 --> 00:16:27,980 the complete solution is this one particular solution 251 00:16:27,980 --> 00:16:34,930 plus any, any vector, all different vectors out 252 00:16:34,930 --> 00:16:37,600 of the null space. 253 00:16:37,600 --> 00:16:39,050 xn, OK. 254 00:16:39,050 --> 00:16:42,630 Well why, why this pattern, because this pattern shows up 255 00:16:42,630 --> 00:16:46,560 through all of mathematics, because it shows up everywhere 256 00:16:46,560 --> 00:16:48,690 we have linear equations. 257 00:16:48,690 --> 00:16:52,050 Let me just put here the, the reason. 258 00:16:52,050 --> 00:17:01,830 A xp, so that's x particular, so what does Ax particular give? 259 00:17:01,830 --> 00:17:05,410 That gives the correct right-hand side b. 260 00:17:05,410 --> 00:17:10,520 And what does A times an x in the null space give? 261 00:17:10,520 --> 00:17:11,710 Zero. 262 00:17:11,710 --> 00:17:17,920 So I add, and I put in parentheses. 263 00:17:17,920 --> 00:17:25,420 So xp plus xn is b plus zero, which is b. 264 00:17:25,420 --> 00:17:27,540 So -- oh, what I saying? 265 00:17:27,540 --> 00:17:30,450 Let me just say it in words. 266 00:17:30,450 --> 00:17:36,800 If I have one solution, I can add on anything 267 00:17:36,800 --> 00:17:40,600 in the null space, because anything in the null space 268 00:17:40,600 --> 00:17:43,910 has a zero right-hand side, and I still 269 00:17:43,910 --> 00:17:46,670 have the correct right-hand side B. 270 00:17:46,670 --> 00:17:47,850 So that's my system. 271 00:17:47,850 --> 00:17:50,290 That's my complete solution. 272 00:17:50,290 --> 00:17:54,620 Now let me write out what that will be for this example. 273 00:17:54,620 --> 00:18:02,070 So in this example, x general, x complete, 274 00:18:02,070 --> 00:18:07,440 the complete solution, is x particular, 275 00:18:07,440 --> 00:18:12,230 which is minus two zero three halves zero, 276 00:18:12,230 --> 00:18:15,900 with those zeroes in the free variable, plus -- 277 00:18:15,900 --> 00:18:18,410 you remember there were the special solutions in the null 278 00:18:18,410 --> 00:18:21,680 space that had a one in the free variables -- 279 00:18:21,680 --> 00:18:24,220 or one and zero in the free variables, 280 00:18:24,220 --> 00:18:29,880 and then we filled in to find I've forgotten what they were, 281 00:18:29,880 --> 00:18:32,020 but maybe it was that. the others? 282 00:18:32,020 --> 00:18:34,010 That was a special solution, and then 283 00:18:34,010 --> 00:18:36,950 there was another special solution that 284 00:18:36,950 --> 00:18:41,820 had that free variable zero and this free variable equal one, 285 00:18:41,820 --> 00:18:46,260 and I have to fill those in. 286 00:18:46,260 --> 00:18:48,420 Let's see, can I remember how those fill in? 287 00:18:48,420 --> 00:18:51,570 Maybe this was a minus two and this was a two, 288 00:18:51,570 --> 00:18:53,070 possibly? 289 00:18:53,070 --> 00:18:57,700 I think probably that's right. 290 00:18:57,700 --> 00:18:59,290 I'm not -- yeah. 291 00:18:59,290 --> 00:19:05,230 Does that look write to you? 292 00:19:05,230 --> 00:19:08,480 I would have to remember what are my equations. 293 00:19:08,480 --> 00:19:11,930 Can I, rather than go way back to that board, 294 00:19:11,930 --> 00:19:14,680 let me remember the first equation was 295 00:19:14,680 --> 00:19:19,520 two x3 plus two x4 equaling zero now, 296 00:19:19,520 --> 00:19:22,450 because I'm looking for the guys in the null space. 297 00:19:22,450 --> 00:19:28,510 So I set x4 to be one and the second equation, 298 00:19:28,510 --> 00:19:32,850 that I didn't copy again, gave me minus two for this and then 299 00:19:32,850 --> 00:19:35,090 -- yeah, so I think that's right. 300 00:19:35,090 --> 00:19:40,131 Two minus four and two gives zero, check. 301 00:19:40,131 --> 00:19:40,630 OK. 302 00:19:40,630 --> 00:19:43,830 Those were the special solutions. 303 00:19:43,830 --> 00:19:46,060 What do we do to get the complete solution? 304 00:19:49,860 --> 00:19:52,270 How do I get the complete solution now? 305 00:19:52,270 --> 00:19:57,020 I multiply this by anything, c1, say, 306 00:19:57,020 --> 00:19:58,930 and I multiply this by anything -- 307 00:19:58,930 --> 00:20:00,960 I take any combination. 308 00:20:00,960 --> 00:20:04,140 Remember that's how we described the null space? 309 00:20:04,140 --> 00:20:09,060 The null space consists of all combinations of -- 310 00:20:09,060 --> 00:20:10,760 so this is xn -- 311 00:20:10,760 --> 00:20:15,430 all combinations of the special solutions. 312 00:20:15,430 --> 00:20:18,410 There were two special solutions because there 313 00:20:18,410 --> 00:20:20,730 were two free variables. 314 00:20:20,730 --> 00:20:24,560 And we want to make that count -- 315 00:20:24,560 --> 00:20:26,660 carefully now. 316 00:20:26,660 --> 00:20:27,857 Just while I'm up here. 317 00:20:27,857 --> 00:20:30,190 So there's, that's what the -- that's the kind of answer 318 00:20:30,190 --> 00:20:31,230 I'm looking for. 319 00:20:31,230 --> 00:20:34,810 Is there a constant multiplying this guy? 320 00:20:34,810 --> 00:20:38,790 Is there a free constant that multiplies x particular? 321 00:20:38,790 --> 00:20:40,180 No way. 322 00:20:40,180 --> 00:20:44,430 Right? x particular solves A xp=b. 323 00:20:44,430 --> 00:20:47,450 I'm not allowed to multiply that by three. 324 00:20:47,450 --> 00:20:51,120 But Axn, I'm allowed to multiply xn by three, 325 00:20:51,120 --> 00:20:56,790 or add to another xn, because I keep getting zero on the right. 326 00:20:56,790 --> 00:20:57,290 OK. 327 00:20:57,290 --> 00:21:02,250 So, so again, xp is one particular guy. 328 00:21:02,250 --> 00:21:04,630 xn is a whole subspace. 329 00:21:04,630 --> 00:21:05,590 Right? 330 00:21:05,590 --> 00:21:09,380 It's one guy plus, plus anything from a subspace. 331 00:21:09,380 --> 00:21:11,120 Let me draw it. 332 00:21:11,120 --> 00:21:14,740 Let me try to -- oh. 333 00:21:14,740 --> 00:21:19,950 I want to draw, I want to graph all this -- 334 00:21:19,950 --> 00:21:25,910 I want to, I want to plot all solutions. 335 00:21:25,910 --> 00:21:29,020 Now x. 336 00:21:29,020 --> 00:21:32,850 So what dimension I in? 337 00:21:32,850 --> 00:21:35,950 This is a unfortunate point. 338 00:21:35,950 --> 00:21:38,650 How many components does x have? 339 00:21:38,650 --> 00:21:39,150 Four. 340 00:21:39,150 --> 00:21:40,280 There are four unknowns. 341 00:21:40,280 --> 00:21:47,150 So I have to draw a four dimensional picture on this MIT 342 00:21:47,150 --> 00:21:48,450 cheap blackboard. 343 00:21:48,450 --> 00:21:48,950 OK. 344 00:21:48,950 --> 00:21:50,970 So here we go. 345 00:21:50,970 --> 00:21:58,080 x1 -- Einstein could do it, but, this, this is -- 346 00:21:58,080 --> 00:22:06,090 those are four perpendicular axes in -- 347 00:22:06,090 --> 00:22:08,740 representing four dimensional space. 348 00:22:08,740 --> 00:22:09,720 OK. 349 00:22:09,720 --> 00:22:12,520 Where are my solutions? 350 00:22:12,520 --> 00:22:16,580 Do my solutions form a subspace? 351 00:22:16,580 --> 00:22:20,470 Does the set of solutions to Ax=b form a subspace? 352 00:22:20,470 --> 00:22:21,900 No way. 353 00:22:21,900 --> 00:22:23,890 What does it actually look like, though? 354 00:22:23,890 --> 00:22:26,530 A subspace is in this picture. 355 00:22:26,530 --> 00:22:30,250 This part is a subspace, right? 356 00:22:30,250 --> 00:22:33,060 That part is some, like, two dimensional, 357 00:22:33,060 --> 00:22:35,860 because I've got two parameters, so it's -- 358 00:22:35,860 --> 00:22:41,120 I'm thinking of this null space as a two dimensional subspace 359 00:22:41,120 --> 00:22:42,890 inside R^4. 360 00:22:42,890 --> 00:22:46,410 Now I have to tell you and will tell you next time, 361 00:22:46,410 --> 00:22:49,760 what does it mean to say a subspace, what's the dimension 362 00:22:49,760 --> 00:22:50,580 of a subspace. 363 00:22:50,580 --> 00:22:52,680 But you see what it's going to be. 364 00:22:52,680 --> 00:22:58,180 It's the number of free independent constants 365 00:22:58,180 --> 00:22:59,750 that we can choose. 366 00:22:59,750 --> 00:23:03,560 So somehow there'll be a two dimensional subspace, not 367 00:23:03,560 --> 00:23:07,860 a line, and not a three dimensional plane, but only 368 00:23:07,860 --> 00:23:10,110 a two dimensional guy. 369 00:23:10,110 --> 00:23:12,710 But it's doesn't go through the origin 370 00:23:12,710 --> 00:23:15,600 because it goes through this point. 371 00:23:15,600 --> 00:23:17,350 So there's x particular. 372 00:23:17,350 --> 00:23:19,970 x particular is somewhere here. 373 00:23:19,970 --> 00:23:21,580 x particular. 374 00:23:21,580 --> 00:23:25,750 So it's somehow a subspace -- can I try to draw it that way? 375 00:23:28,950 --> 00:23:36,070 It's a two dimensional subspace that goes through x particular 376 00:23:36,070 --> 00:23:39,970 and then onwards by -- so there's x particular, 377 00:23:39,970 --> 00:23:44,090 and I added on xn, and there's x. 378 00:23:44,090 --> 00:23:46,420 There's x=xp+xn. 379 00:23:46,420 --> 00:23:51,430 But the xn was anywhere in this subspace, 380 00:23:51,430 --> 00:23:56,700 so that filled out a plane. 381 00:23:56,700 --> 00:23:58,930 It's a subspace -- 382 00:23:58,930 --> 00:24:02,320 it's not a subspace, what I saying? 383 00:24:02,320 --> 00:24:05,220 It's like a flat thing, it's like a subspace, 384 00:24:05,220 --> 00:24:08,450 but it's been shifted, away from the origin. 385 00:24:08,450 --> 00:24:11,450 It doesn't contain zero. 386 00:24:11,450 --> 00:24:12,200 Thanks. 387 00:24:12,200 --> 00:24:13,150 OK. 388 00:24:13,150 --> 00:24:16,340 That's the picture, and that's the algorithm. 389 00:24:16,340 --> 00:24:20,940 So the algorithm is just go through elimination 390 00:24:20,940 --> 00:24:25,150 and, find the particular solution, 391 00:24:25,150 --> 00:24:27,290 and then find those special solutions. 392 00:24:27,290 --> 00:24:30,010 You can do that. 393 00:24:30,010 --> 00:24:34,620 Let me take our time here in the lecture to think, 394 00:24:34,620 --> 00:24:39,240 about the bigger picture. 395 00:24:39,240 --> 00:24:43,030 So let me think about -- 396 00:24:43,030 --> 00:24:46,050 so this is my pattern. 397 00:24:46,050 --> 00:24:47,040 Now I want to think -- 398 00:24:47,040 --> 00:24:54,410 I want to ask you about a question -- 399 00:24:54,410 --> 00:24:57,260 I want to ask you some questions. 400 00:24:57,260 --> 00:25:01,070 So when I mean think bigger, I mean I'll think about an m 401 00:25:01,070 --> 00:25:09,495 by n matrix A of rank r. 402 00:25:12,640 --> 00:25:13,140 OK. 403 00:25:15,820 --> 00:25:17,630 What's our definition of rank? 404 00:25:17,630 --> 00:25:22,910 Our current definition of rank is number of pivots. 405 00:25:22,910 --> 00:25:23,410 OK. 406 00:25:23,410 --> 00:25:26,400 First of all, how are these numbers related? 407 00:25:26,400 --> 00:25:31,050 Can you tell me a relation between r and m? 408 00:25:31,050 --> 00:25:35,860 If I have m rows in the matrix and R pivots, -- 409 00:25:35,860 --> 00:25:42,240 then I certainly know, always -- 410 00:25:42,240 --> 00:25:46,890 what relation do I know between r and m? 411 00:25:46,890 --> 00:25:49,810 r is less or equal, right? 412 00:25:49,810 --> 00:25:53,620 Because I've got m rows, I can't have more than m pivots, 413 00:25:53,620 --> 00:25:56,720 I might have m and I might have fewer. 414 00:25:56,720 --> 00:26:01,980 Also, I've got n columns. 415 00:26:01,980 --> 00:26:04,630 So what's the relation between r and n? 416 00:26:04,630 --> 00:26:10,480 It's the same, less or equal, because a column 417 00:26:10,480 --> 00:26:14,150 can't have more than one pivot. 418 00:26:14,150 --> 00:26:17,450 So I can't have more than n pivots altogether. 419 00:26:17,450 --> 00:26:18,740 OK, OK. 420 00:26:18,740 --> 00:26:22,360 So I have an m by n matrix of rank r. 421 00:26:22,360 --> 00:26:25,420 And I always know r less than or equal to m, r less than 422 00:26:25,420 --> 00:26:26,560 or equal to n. 423 00:26:26,560 --> 00:26:29,870 Now I'm specially interested in the case 424 00:26:29,870 --> 00:26:35,540 of full rank, when the rank r is as big as it can be. 425 00:26:35,540 --> 00:26:40,840 Well, I guess I've got two separate possibilities here, 426 00:26:40,840 --> 00:26:44,310 depending on what these numbers m and n are. 427 00:26:44,310 --> 00:26:49,970 So let me talk about the case of full column rank. 428 00:26:53,200 --> 00:26:54,958 And by that I mean r=n. 429 00:27:02,330 --> 00:27:11,960 And I want to ask you, what does that imply about our solutions? 430 00:27:11,960 --> 00:27:16,190 What does that tell us about the null space? 431 00:27:16,190 --> 00:27:21,240 What does that tell us about, the complete solution? 432 00:27:21,240 --> 00:27:22,860 OK, so what does that mean? 433 00:27:22,860 --> 00:27:28,520 So I want to ask you, well, OK, if the rank is 434 00:27:28,520 --> 00:27:31,580 n, what does that mean? 435 00:27:31,580 --> 00:27:35,570 That means there's a pivot in every column. 436 00:27:35,570 --> 00:27:39,280 So how many pivot variables are there? 437 00:27:39,280 --> 00:27:41,160 n. 438 00:27:41,160 --> 00:27:45,150 All the columns have pivots in this case. 439 00:27:45,150 --> 00:27:48,150 So how many free variables are there? 440 00:27:48,150 --> 00:27:50,460 None at all. 441 00:27:50,460 --> 00:27:52,770 So no free variables. 442 00:27:52,770 --> 00:27:54,770 r=n, no free variables. 443 00:27:57,820 --> 00:28:00,270 So what does that tell us about what's 444 00:28:00,270 --> 00:28:04,610 going to happen then in our, in our little algorithms? 445 00:28:04,610 --> 00:28:07,440 What will be in the null space? 446 00:28:07,440 --> 00:28:13,570 The null space of A has got what in it? 447 00:28:13,570 --> 00:28:15,990 Only the zero vector. 448 00:28:15,990 --> 00:28:20,590 There are no free variables to give other values to. 449 00:28:20,590 --> 00:28:23,780 So the null space is only the zero vector. 450 00:28:29,770 --> 00:28:33,580 And what about our solution to Ax=b? 451 00:28:33,580 --> 00:28:38,510 Solution to Ax=b? 452 00:28:38,510 --> 00:28:41,790 What, what's the story on that one? 453 00:28:41,790 --> 00:28:43,950 So now that's coming from today's lecture. 454 00:28:47,270 --> 00:28:51,020 The solution x is -- 455 00:28:51,020 --> 00:28:52,290 what's the complete solution? 456 00:28:55,920 --> 00:28:59,570 It's just x particular, right? 457 00:28:59,570 --> 00:29:02,830 If, if, if there is an x, if there is a solution. 458 00:29:02,830 --> 00:29:05,190 It's x equal x particular. 459 00:29:05,190 --> 00:29:08,900 There's nothing -- you know, there's just one solution. 460 00:29:08,900 --> 00:29:11,050 If there's one at all. 461 00:29:11,050 --> 00:29:13,920 So it's unique solution -- 462 00:29:13,920 --> 00:29:16,900 unique means only one -- 463 00:29:16,900 --> 00:29:22,945 unique solution if it exists, if it exists. 464 00:29:26,430 --> 00:29:29,880 In other words, I would say -- let me put it a different way. 465 00:29:29,880 --> 00:29:32,910 There're either zero or one solutions. 466 00:29:38,940 --> 00:29:40,920 This is all in this case r=n. 467 00:29:45,140 --> 00:29:50,400 So I'm -- because many, many applications in reality, 468 00:29:50,400 --> 00:29:55,693 the columns will be what I'll later call independent. 469 00:29:58,710 --> 00:30:04,340 And we'll have, nothing to look for in the null space, 470 00:30:04,340 --> 00:30:06,611 and we'll only have particular solutions. 471 00:30:06,611 --> 00:30:07,110 OK. 472 00:30:09,970 --> 00:30:13,500 Everybody see that possibility? 473 00:30:13,500 --> 00:30:15,920 But I need an example, right? 474 00:30:15,920 --> 00:30:18,590 So let me create an example. 475 00:30:18,590 --> 00:30:23,170 What sort of a matrix -- what's the shape of a matrix that has 476 00:30:23,170 --> 00:30:25,390 full column rank? 477 00:30:25,390 --> 00:30:30,140 So can I squeeze in an, an example here? 478 00:30:30,140 --> 00:30:35,080 If it exists. 479 00:30:35,080 --> 00:30:38,150 Let me put in an example, and it's just the right space 480 00:30:38,150 --> 00:30:41,140 to put in an example. 481 00:30:41,140 --> 00:30:45,330 Because the example will be like tall and thin. 482 00:30:45,330 --> 00:30:47,940 It will have -- 483 00:30:47,940 --> 00:30:54,140 well, I mean, here's an example, one two six five, three one 484 00:30:54,140 --> 00:30:55,130 one one. 485 00:30:55,130 --> 00:30:56,610 Brilliant example. 486 00:30:56,610 --> 00:30:57,590 OK. 487 00:30:57,590 --> 00:31:06,039 So there's a matrix A, and what's its rank? 488 00:31:06,039 --> 00:31:07,330 What's the rank of that matrix? 489 00:31:10,000 --> 00:31:12,235 How many pivots will I find if I do elimination? 490 00:31:14,810 --> 00:31:15,810 Two, right? 491 00:31:15,810 --> 00:31:16,990 Two. 492 00:31:16,990 --> 00:31:20,320 I see a pivot there -- 493 00:31:20,320 --> 00:31:23,730 oh certainly those two columns are headed off 494 00:31:23,730 --> 00:31:26,810 in different directions. 495 00:31:26,810 --> 00:31:29,870 When I do elimination, I'll certainly get another pivot 496 00:31:29,870 --> 00:31:35,270 here, fine, and I can use those to clean out below and above. 497 00:31:35,270 --> 00:31:43,960 So the -- actually, tell me what its row reduced row echelon 498 00:31:43,960 --> 00:31:45,360 form would be. 499 00:31:45,360 --> 00:31:49,410 Can you carry that, that elimination 500 00:31:49,410 --> 00:31:52,360 process to the bitter end? 501 00:31:52,360 --> 00:31:54,590 So what do, what does that mean? 502 00:31:54,590 --> 00:31:57,610 I subtract a multiple of this row from these rows. 503 00:31:57,610 --> 00:32:00,910 So I clean up, all zeros there. 504 00:32:00,910 --> 00:32:02,540 Then I've got some pivot here. 505 00:32:02,540 --> 00:32:04,140 What do I do with that? 506 00:32:04,140 --> 00:32:07,330 I go subtract it below and above, 507 00:32:07,330 --> 00:32:11,860 and then I divide through, and what's R for that example? 508 00:32:11,860 --> 00:32:14,450 Maybe I can -- you'll allow me to put that just here 509 00:32:14,450 --> 00:32:16,580 in the next board. 510 00:32:16,580 --> 00:32:21,150 What's the row reduced echelon form, just out of practice, 511 00:32:21,150 --> 00:32:25,300 for that matrix? 512 00:32:25,300 --> 00:32:28,800 It's got ones in the pivots. 513 00:32:28,800 --> 00:32:31,750 It's got the identity matrix, a little two by two identity 514 00:32:31,750 --> 00:32:34,270 matrix, and below it all zeros. 515 00:32:37,850 --> 00:32:43,940 That's a matrix that really has two independent rows, 516 00:32:43,940 --> 00:32:45,510 and they're the first two, actually. 517 00:32:45,510 --> 00:32:47,280 The first two rows are independent. 518 00:32:47,280 --> 00:32:49,190 They're not in the same direction. 519 00:32:49,190 --> 00:32:52,660 But the other rows are combinations of the first two, 520 00:32:52,660 --> 00:32:55,970 so -- 521 00:32:55,970 --> 00:32:59,850 is there always a solution to Ax=b? 522 00:32:59,850 --> 00:33:02,050 Tell me what's the picture here? 523 00:33:02,050 --> 00:33:06,880 For this matrix A, this is a case of full column rank. 524 00:33:06,880 --> 00:33:11,320 The two columns are -- give two pivots. 525 00:33:11,320 --> 00:33:13,090 There's nothing in the null space. 526 00:33:13,090 --> 00:33:15,540 There's no combination of those columns 527 00:33:15,540 --> 00:33:19,001 that gives the zero column except the zero zero 528 00:33:19,001 --> 00:33:19,500 combination. 529 00:33:22,430 --> 00:33:25,400 So there's nothing in the null space. 530 00:33:25,400 --> 00:33:29,830 But is there always a solution to A X equal B? 531 00:33:29,830 --> 00:33:31,530 What's up with A X equal B? 532 00:33:34,390 --> 00:33:38,220 I've got four, four equations here, and only two Xs. 533 00:33:40,840 --> 00:33:42,460 So the answer is certainly no. 534 00:33:42,460 --> 00:33:45,240 There's not always a solution. 535 00:33:45,240 --> 00:33:49,660 I may have zero solutions, and if I make a random choice, 536 00:33:49,660 --> 00:33:51,590 I'll have zero solutions. 537 00:33:51,590 --> 00:33:55,690 Or if I make a great particular choice of the right-hand side, 538 00:33:55,690 --> 00:33:59,160 which just happens to be a combination of those two guys 539 00:33:59,160 --> 00:34:01,500 -- like tell me one right-hand side that would have 540 00:34:01,500 --> 00:34:03,540 a solution. 541 00:34:03,540 --> 00:34:07,190 Tell me a right-hand side that would have a solution. 542 00:34:07,190 --> 00:34:09,800 Well, 0 0 0 0, OK. 543 00:34:09,800 --> 00:34:12,880 No prize for that one. 544 00:34:12,880 --> 00:34:14,250 Tell me another one. 545 00:34:14,250 --> 00:34:18,850 Another right-hand side that has a solution would be 4 3 7 6. 546 00:34:18,850 --> 00:34:21,900 I could add the two columns. 547 00:34:21,900 --> 00:34:25,070 What would be the total complete solution if the 548 00:34:25,070 --> 00:34:28,360 Right? right-hand side was 4 3 7 6? 549 00:34:28,360 --> 00:34:31,489 There would be the particular solution one 550 00:34:31,489 --> 00:34:34,067 one, one of that column plus one of that, 551 00:34:34,067 --> 00:34:35,150 and that would be the only 552 00:34:35,150 --> 00:34:36,429 solution. 553 00:34:36,429 --> 00:34:39,770 So there would be -- x particular would be one one 554 00:34:39,770 --> 00:34:43,560 in the case when the right side is the sum of those two 555 00:34:43,560 --> 00:34:46,850 columns, and that's it. 556 00:34:46,850 --> 00:34:50,670 So that would be a case with one solution. 557 00:34:50,670 --> 00:34:51,250 OK. 558 00:34:51,250 --> 00:34:55,469 That, this is the typical setup with full column rank. 559 00:34:55,469 --> 00:35:00,000 Now I go to full row rank. 560 00:35:00,000 --> 00:35:04,260 You see the sort of natural symmetry of this discussion. 561 00:35:04,260 --> 00:35:14,523 Full row rank means r=m. 562 00:35:17,390 --> 00:35:19,940 So this is what I'm interested in now, r=m. 563 00:35:23,420 --> 00:35:24,710 OK, what's up with that? 564 00:35:29,830 --> 00:35:31,010 How many pivots? 565 00:35:31,010 --> 00:35:33,000 m. 566 00:35:33,000 --> 00:35:40,060 So what happens when we do elimination in that case? 567 00:35:40,060 --> 00:35:42,520 I'm going to get m pivots. 568 00:35:42,520 --> 00:35:47,520 So every row has a pivot, right? 569 00:35:47,520 --> 00:35:48,855 Every row has a pivot. 570 00:35:52,120 --> 00:35:55,950 Then what about solvability? 571 00:35:55,950 --> 00:35:59,880 What about this business of -- for which right-hand sides can 572 00:35:59,880 --> 00:36:01,120 I solve it? 573 00:36:01,120 --> 00:36:02,970 So that's my question. 574 00:36:02,970 --> 00:36:14,180 I can solve Ax=b for which right-hand sides? 575 00:36:14,180 --> 00:36:18,450 Do you see what's coming? 576 00:36:18,450 --> 00:36:23,990 I do elimination, I don't get any zero rows. 577 00:36:23,990 --> 00:36:26,890 So there aren't any requirements on b. 578 00:36:26,890 --> 00:36:29,730 I can solve Ax=b for every b. 579 00:36:36,450 --> 00:36:39,990 I can solve Ax=b for every right-hand side. 580 00:36:39,990 --> 00:36:46,705 So this is the existence, exists a solution. 581 00:36:49,260 --> 00:36:57,180 Now tell me, so the, u- u- so every row has a pivot in it. 582 00:36:57,180 --> 00:37:00,820 So how many free variables are there? 583 00:37:00,820 --> 00:37:04,780 How many free variables in this case? 584 00:37:04,780 --> 00:37:07,560 If I had n variables to start with, 585 00:37:07,560 --> 00:37:11,180 how many are used up by pivot variables? 586 00:37:11,180 --> 00:37:13,890 r, which is m. 587 00:37:13,890 --> 00:37:25,600 So I'm left with, left with n-r free variables. 588 00:37:31,050 --> 00:37:31,910 OK. 589 00:37:31,910 --> 00:37:37,220 So this case of full row rank I can always solve, 590 00:37:37,220 --> 00:37:41,440 and then this tells me how many variables are free, 591 00:37:41,440 --> 00:37:43,400 and this is of course n-m. 592 00:37:43,400 --> 00:37:48,040 This is n-m free variables. 593 00:37:48,040 --> 00:37:48,980 Can I do an example? 594 00:37:52,310 --> 00:37:54,750 You know, the best way for me to do an example is just 595 00:37:54,750 --> 00:37:58,140 to transpose that example. 596 00:37:58,140 --> 00:38:01,950 So let me take, let me take that matrix that had column one 597 00:38:01,950 --> 00:38:05,970 two six five and make it a row. 598 00:38:05,970 --> 00:38:11,470 And let me take three one one one as the second row. 599 00:38:11,470 --> 00:38:18,700 And let me ask you, this is my matrix A, what's its rank? 600 00:38:18,700 --> 00:38:20,230 What's the rank of that matrix? 601 00:38:20,230 --> 00:38:24,560 Sorry to ask, but not sorry really, 602 00:38:24,560 --> 00:38:27,130 because we're just getting the idea of rank. 603 00:38:27,130 --> 00:38:29,770 What's the rank of that matrix? 604 00:38:29,770 --> 00:38:32,260 Two, exactly, two. 605 00:38:32,260 --> 00:38:33,770 There will be two pivots. 606 00:38:33,770 --> 00:38:36,770 What will the row reduced echelon form be? 607 00:38:36,770 --> 00:38:38,850 Anybody know that one? 608 00:38:38,850 --> 00:38:42,230 Actually, tell me not only -- you have to tell me not only 609 00:38:42,230 --> 00:38:45,110 the, there'll be two pivots but which will be the pivot 610 00:38:45,110 --> 00:38:46,550 columns. 611 00:38:46,550 --> 00:38:50,060 Which columns of this matrix will be pivot columns? 612 00:38:50,060 --> 00:38:53,140 So the first column is fine, and then 613 00:38:53,140 --> 00:38:55,720 I go on to the next column, and what do I get? 614 00:38:55,720 --> 00:38:57,640 Do I get a second pivot out of -- 615 00:38:57,640 --> 00:39:00,410 will I get a second pivot in this position? 616 00:39:00,410 --> 00:39:01,300 Yes. 617 00:39:01,300 --> 00:39:07,200 So the pivots, when I get all the way to R, will be there. 618 00:39:07,200 --> 00:39:13,860 And here will be some numbers. 619 00:39:13,860 --> 00:39:18,670 This is the part that I previously called F. 620 00:39:18,670 --> 00:39:23,720 This is the part that -- the pivot columns in R will be 621 00:39:23,720 --> 00:39:25,680 the identity matrix. 622 00:39:25,680 --> 00:39:31,300 There are no zero rows, no zero rows, because the rank is two. 623 00:39:31,300 --> 00:39:34,630 But there'll be stuff over here. 624 00:39:34,630 --> 00:39:42,530 And that will, enter the special solutions and the null space. 625 00:39:42,530 --> 00:39:43,230 OK. 626 00:39:43,230 --> 00:39:51,840 So this is a typical matrix with r=m smaller than n. 627 00:39:51,840 --> 00:39:56,135 Now finally I've got a space here for r=m=n. 628 00:40:01,540 --> 00:40:06,190 I'm off in the corner here with the most important case of all. 629 00:40:06,190 --> 00:40:08,910 So what's up with this matrix? 630 00:40:08,910 --> 00:40:10,970 So let me give an example. 631 00:40:10,970 --> 00:40:15,065 OK, brilliant example, 1 2 3 1. 632 00:40:19,860 --> 00:40:23,628 Tell me what -- how do I describe a matrix that has rank 633 00:40:23,628 --> 00:40:24,127 r=m=n? 634 00:40:26,930 --> 00:40:32,560 So the matrix is square, right, it's a square matrix. 635 00:40:32,560 --> 00:40:36,350 And if I know its rank is -- it's full rank, now. 636 00:40:36,350 --> 00:40:39,290 I don't have to say full column rank or full row rank -- 637 00:40:39,290 --> 00:40:43,800 I just say full rank, because the count, column count 638 00:40:43,800 --> 00:40:47,130 and the row count are the same, and the rank 639 00:40:47,130 --> 00:40:49,040 is as big as it can be. 640 00:40:49,040 --> 00:40:51,090 And what kind of a matrix have I got? 641 00:40:53,920 --> 00:40:56,670 It's invertible. 642 00:40:56,670 --> 00:41:01,510 So that's exactly the invertible matrices. 643 00:41:01,510 --> 00:41:06,310 r=m=n means the -- what's the row echelon form, 644 00:41:06,310 --> 00:41:10,920 the reduced row echelon form, for an invertible matrix? 645 00:41:10,920 --> 00:41:14,630 For a square, nice, square, invertible matrix? 646 00:41:14,630 --> 00:41:17,320 It's I. 647 00:41:17,320 --> 00:41:18,880 Right. 648 00:41:18,880 --> 00:41:25,530 So you see that the, the good matrices 649 00:41:25,530 --> 00:41:31,580 are the ones that kind of come out trivially in R. 650 00:41:31,580 --> 00:41:34,270 You reduce them all the way to the identity matrix. 651 00:41:34,270 --> 00:41:37,900 What's the null space for this, for this matrix? 652 00:41:37,900 --> 00:41:41,170 Can I just hammer away with questions? 653 00:41:41,170 --> 00:41:43,160 What's the null space for this matrix? 654 00:41:45,910 --> 00:41:51,570 The null space of that matrix is the zero vector only. 655 00:41:51,570 --> 00:41:54,660 The zero vector only. 656 00:41:54,660 --> 00:41:58,530 What are the conditions to solve Ax=b? 657 00:41:58,530 --> 00:42:01,950 Which right-hand sides b are OK? 658 00:42:01,950 --> 00:42:07,560 If I want to solve Ax=b for this example, so A is this, 659 00:42:07,560 --> 00:42:14,140 b is b1 b2, what are the conditions on b1 and b2? 660 00:42:14,140 --> 00:42:17,170 None at all, right. 661 00:42:17,170 --> 00:42:21,850 So this is the case, this is the case where I can solve -- 662 00:42:21,850 --> 00:42:25,660 so I've coming back here, I can -- since the rank equals m, 663 00:42:25,660 --> 00:42:28,460 I can solve for every b. 664 00:42:28,460 --> 00:42:33,840 And since the rank is also n, there's a unique solution. 665 00:42:33,840 --> 00:42:36,640 Let me summarize the whole picture here. 666 00:42:39,210 --> 00:42:41,210 Here's the whole picture. 667 00:42:41,210 --> 00:42:44,780 I could have r=m=n. 668 00:42:44,780 --> 00:42:51,190 This is the case where this is the identity matrix. 669 00:42:51,190 --> 00:42:53,595 And this is the case where there is one solution. 670 00:42:56,790 --> 00:43:02,300 That's the square invertible chapter two case. 671 00:43:02,300 --> 00:43:04,420 Now we're into chapter three. 672 00:43:04,420 --> 00:43:08,330 We could have r=m smaller than n. 673 00:43:11,920 --> 00:43:13,730 Now that's what we had over there, 674 00:43:13,730 --> 00:43:17,610 and the row echelon form looked like the identity 675 00:43:17,610 --> 00:43:18,960 with some zero rows. 676 00:43:21,960 --> 00:43:27,565 And that was the case where there are zero or one solution. 677 00:43:31,420 --> 00:43:34,420 When I say solution I mean to Ax=b. 678 00:43:37,540 --> 00:43:39,590 So this case, there's always one. 679 00:43:39,590 --> 00:43:42,130 This case there's zero or one. 680 00:43:42,130 --> 00:43:45,640 And now let me take the case of full column rank, 681 00:43:45,640 --> 00:43:55,860 but some, extra rows. 682 00:43:55,860 --> 00:44:00,000 So now R has -- 683 00:44:00,000 --> 00:44:04,700 well, the identity -- 684 00:44:04,700 --> 00:44:08,280 I'm almost tempted to write the identity matrix and then F, 685 00:44:08,280 --> 00:44:10,130 but that isn't necessarily right. 686 00:44:15,970 --> 00:44:20,360 I have -- is that right? 687 00:44:20,360 --> 00:44:24,310 Am I getting this correct here? 688 00:44:24,310 --> 00:44:25,430 Oh, I'm not! 689 00:44:25,430 --> 00:44:26,390 My God! 690 00:44:26,390 --> 00:44:33,330 This is the case R equals n, the columns, the columns are, 691 00:44:33,330 --> 00:44:34,390 are OK. 692 00:44:34,390 --> 00:44:38,770 That's the case that was on that board, r=n, full column rank. 693 00:44:38,770 --> 00:44:43,240 Now I want the case where m is smaller than n 694 00:44:43,240 --> 00:44:46,550 and I've got extra columns. 695 00:44:46,550 --> 00:44:47,050 OK. 696 00:44:47,050 --> 00:44:47,810 There we go. 697 00:44:52,030 --> 00:44:57,140 So this is now the case of full row rank, 698 00:44:57,140 --> 00:45:02,070 and it looks like I F except that I 699 00:45:02,070 --> 00:45:08,430 can't be sure that the pivot columns are the first columns. 700 00:45:08,430 --> 00:45:14,690 So the I and the F, could be partly mixed into the I. 701 00:45:14,690 --> 00:45:18,670 Can I write that with just like that? 702 00:45:18,670 --> 00:45:23,830 So the F could be sort of partly into the I 703 00:45:23,830 --> 00:45:28,450 if the first columns weren't the pivot columns. 704 00:45:28,450 --> 00:45:31,760 Now how many solutions in this case? 705 00:45:31,760 --> 00:45:34,330 There's always a solution. 706 00:45:34,330 --> 00:45:35,850 This is the existence case. 707 00:45:35,850 --> 00:45:37,020 There's always a solution. 708 00:45:37,020 --> 00:45:39,160 We're not getting any zero rows. 709 00:45:39,160 --> 00:45:41,430 There are no zero rows here. 710 00:45:41,430 --> 00:45:45,635 So there's always either one or infinitely many solutions. 711 00:45:50,061 --> 00:45:50,560 OK. 712 00:45:53,230 --> 00:45:55,600 Actually, I guess there's always an infinite number, 713 00:45:55,600 --> 00:46:02,760 because we always have some null space to deal with. 714 00:46:02,760 --> 00:46:06,150 Then the final case is where r is smaller than m 715 00:46:06,150 --> 00:46:08,880 and smaller than n. 716 00:46:08,880 --> 00:46:09,380 OK. 717 00:46:09,380 --> 00:46:14,830 Now that's the case where R is the identity 718 00:46:14,830 --> 00:46:19,920 with some free stuff but with some zero rows too. 719 00:46:19,920 --> 00:46:23,580 And that's the case where there's either no solution -- 720 00:46:23,580 --> 00:46:29,200 because we didn't get a zero equals zero for some bs, 721 00:46:29,200 --> 00:46:32,330 or infinitely many solutions. 722 00:46:37,100 --> 00:46:38,010 OK. 723 00:46:38,010 --> 00:46:44,310 Do you -- this board really summarizes the lecture, 724 00:46:44,310 --> 00:46:47,370 and this sentence summarizes the lecture. 725 00:46:47,370 --> 00:46:55,070 The rank tells you everything about the number of solutions. 726 00:46:55,070 --> 00:46:57,010 That number, the rank r, tells you 727 00:46:57,010 --> 00:47:01,620 all the information except the exact entries in the solutions. 728 00:47:01,620 --> 00:47:04,140 For that you go to the matrix. 729 00:47:04,140 --> 00:47:05,150 OK, good. 730 00:47:05,150 --> 00:47:08,770 Have a great weekend, and I'll see you on Monday.