1 00:00:11,410 --> 00:00:16,560 OK. Good morning, everyone. I think maybe we'll get started. So 2 00:00:16,560 --> 00:00:21,840 welcome to Linear Algebra two. If you weren't expecting announceable to either 3 00:00:21,840 --> 00:00:27,300 at least one of your eyes in the one game. So I'll start off just with a few 4 00:00:27,300 --> 00:00:39,050 practical bits of information. So this is your. 5 00:00:39,050 --> 00:00:44,240 Algebra two, which is booting on from 6 00:00:44,240 --> 00:00:49,420 an out. For one that you saw last term. Say 7 00:00:49,420 --> 00:00:57,180 we're meeting Monday is in Wednesdays 8 00:00:57,180 --> 00:01:02,700 from 10 to 11 a.m. and that's always in this room alone. 9 00:01:02,700 --> 00:01:08,160 So since you're here, you've probably on the set the timetable. Okay. 10 00:01:08,160 --> 00:01:14,250 And one thing to note is that this is just a short course. So we only have nachas 11 00:01:14,250 --> 00:01:19,350 between weeks. One in four. As you saw in 12 00:01:19,350 --> 00:01:28,490 many outbreed one, the idea of leaning out there. 13 00:01:28,490 --> 00:01:33,770 Is really to study maybe 14 00:01:33,770 --> 00:01:40,420 functions which have the property, the 15 00:01:40,420 --> 00:01:46,760 F of X plus B, Y is 16 00:01:46,760 --> 00:01:53,780 A times X plus B times 17 00:01:53,780 --> 00:01:58,970 Y and hey maybe I'm thinking of X and Y as being 18 00:01:58,970 --> 00:02:04,010 factors and and B, it's been scalars, but you grew so much in spite more 19 00:02:04,010 --> 00:02:09,830 general situations where this is true and any function that satisfies this, we call in Libya. 20 00:02:09,830 --> 00:02:15,050 And then linear algebra is the outback study 21 00:02:15,050 --> 00:02:20,090 of these sorts of functions. And it turns 22 00:02:20,090 --> 00:02:25,360 out that this is one of the most important and well understood areas of my Hartwick's 23 00:02:25,360 --> 00:02:30,920 say there's a very beautiful theory of this. And 24 00:02:30,920 --> 00:02:38,090 one quick summary is you can understand 25 00:02:38,090 --> 00:02:48,370 a lot about this by looking at matrices 26 00:02:48,370 --> 00:02:54,230 and vector spaces. So objects that we've encountered in Article one 27 00:02:54,230 --> 00:02:59,240 and actually a huge amount of modern mathematics is often 28 00:02:59,240 --> 00:03:04,450 to take some very complicated situation and to try and maybe 29 00:03:04,450 --> 00:03:10,220 approximate it in some way by some linear system. So when you do calculus, 30 00:03:10,220 --> 00:03:15,860 your approximating a more general function by something that looks linear on a small scale. 31 00:03:15,860 --> 00:03:22,100 And now it's really lies at the very heart of a huge amount of mathematics. 32 00:03:22,100 --> 00:03:27,110 You've maybe already seen that linear algebra has various different applications. If you want 33 00:03:27,110 --> 00:03:33,420 to solve simultaneous equations, you can cause this is a problem in linear algebra, 34 00:03:33,420 --> 00:03:40,130 just as you can understand linear algebra as looking at vector spaces and more geometric problems. 35 00:03:40,130 --> 00:03:45,290 And one philosophy for the way that I like to think 36 00:03:45,290 --> 00:03:54,710 about giving out that I think would be useful in this course. 37 00:03:54,710 --> 00:03:59,960 Is to when you're thinking about statements 38 00:03:59,960 --> 00:04:05,150 to you, always imagine them geometrically as questions 39 00:04:05,150 --> 00:04:10,370 about two by two matrices and transmissions of the plane. 40 00:04:10,370 --> 00:04:17,560 But to always write down proofs and arguments much more abstractly 41 00:04:17,560 --> 00:04:24,920 using out there. So think 42 00:04:24,920 --> 00:04:35,870 about it around the. 43 00:04:35,870 --> 00:04:41,180 Geometrically, in particular, as 44 00:04:41,180 --> 00:04:49,550 I say, transformations 45 00:04:49,550 --> 00:04:54,740 of the plane. So I'm not too 46 00:04:54,740 --> 00:04:59,810 good at visualising things in three dimensions, but I can visualise things okay in two dimensions. 47 00:04:59,810 --> 00:05:04,850 And often it's very easy to visualise things if you just imagine that everything is 48 00:05:04,850 --> 00:05:10,040 some operation on the plane. So I like to think about what's going 49 00:05:10,040 --> 00:05:15,710 on in a sort of geometric way. But we're going to prove the more general statements. 50 00:05:15,710 --> 00:05:20,780 And one of the big Paphos, if they bring out the powerful things about an out there is you can take this 51 00:05:20,780 --> 00:05:25,790 geometric intuition, then write down formally that work in arbitrary dimensions in a very 52 00:05:25,790 --> 00:05:31,280 nice way. And often the outbreak of doing the calculations is much, much more convenient 53 00:05:31,280 --> 00:05:37,190 than the geometric way. Even maybe we were actually motivated by the geometry, so 54 00:05:37,190 --> 00:05:43,640 on to do calculations 55 00:05:43,640 --> 00:05:53,720 and poufs. Abstractly. 56 00:05:53,720 --> 00:06:01,010 I using outerwear. 57 00:06:01,010 --> 00:06:06,050 So if you do it this way. The proof's in the statements often hold in much more 58 00:06:06,050 --> 00:06:14,830 generality, but also it's much more convenient to do the calculations in this way. 59 00:06:14,830 --> 00:06:19,990 And they say, I've baby said this 60 00:06:19,990 --> 00:06:25,270 in very vague terms. Maybe we can sort of think about a concrete 61 00:06:25,270 --> 00:06:30,550 example as to how this might work. And this gets us to 62 00:06:30,550 --> 00:06:44,320 the first thing that I'd like to talk about, a many algebra, which is determinants. 63 00:06:44,320 --> 00:06:54,040 But I'd like to demonstrate it to this sort of philosophy. So 64 00:06:54,040 --> 00:07:10,470 think of just some linear transformation of the plain. 65 00:07:10,470 --> 00:07:15,750 So what do I mean by this? Well, some combination of rotation 66 00:07:15,750 --> 00:07:20,920 or stretching in one direction, or maybe I could skew 67 00:07:20,920 --> 00:07:26,010 it if I could shift things, but a linear transformation is 68 00:07:26,010 --> 00:07:31,230 going to preserve Rhines. And I can think about just taking the 69 00:07:31,230 --> 00:07:36,330 unit box. And this will then be matched 70 00:07:36,330 --> 00:07:42,270 to some maybe skewed box. So be some sort of parallelogram. 71 00:07:42,270 --> 00:07:47,550 And maybe the axes won't be 72 00:07:47,550 --> 00:07:54,480 Zonta anymore. And I want to understand transformations like this. 73 00:07:54,480 --> 00:08:02,330 So this is just, uh. The Unit Square. 74 00:08:02,330 --> 00:08:12,980 An. Squares. And then this gets mapped to some poll. 75 00:08:12,980 --> 00:08:19,470 And. The Union Square has area. 76 00:08:19,470 --> 00:08:24,680 One. And this parallelogram, the area 77 00:08:24,680 --> 00:08:31,240 where Masoli represent so have some different area. See, 78 00:08:31,240 --> 00:08:36,550 but I can think about the linear transmission acting on other shapes. 79 00:08:36,550 --> 00:08:42,030 So I could have maybe a triangle and then 80 00:08:42,030 --> 00:08:49,790 set out or get map to some. Strange, skewed Trango or something. 81 00:08:49,790 --> 00:08:54,860 And maybe I can draw some 82 00:08:54,860 --> 00:09:00,380 try and go with area, too. 83 00:09:00,380 --> 00:09:05,780 And it's a fact that 84 00:09:05,780 --> 00:09:11,760 the image of the triangle would then have. 85 00:09:11,760 --> 00:09:16,900 A.C. So you couldn't. It's an exercise if you want to 86 00:09:16,900 --> 00:09:22,390 work this out. And in fact. Well, maybe we would guess 87 00:09:22,390 --> 00:09:28,840 just from doing a couple of examples like this, 88 00:09:28,840 --> 00:09:34,460 that every. 89 00:09:34,460 --> 00:09:40,580 Shape. I'm not going to be precise about what I mean by shape. 90 00:09:40,580 --> 00:09:45,640 Has the area. Scaled 91 00:09:45,640 --> 00:09:53,600 by a factor of say. 92 00:09:53,600 --> 00:09:58,760 So often it's unwise to make a general guess just from doing two examples, but you can do a few more examples 93 00:09:58,760 --> 00:10:03,860 if you like. And 94 00:10:03,860 --> 00:10:08,940 it turns out this guess is true. And this 95 00:10:08,940 --> 00:10:14,180 constancy is going to depend on the linear transformation. And this constancy 96 00:10:14,180 --> 00:10:19,850 is essentially the determinant of delay and transformation. So 97 00:10:19,850 --> 00:10:24,890 it's clearly going to be some important property of any in transmission that 98 00:10:24,890 --> 00:10:30,200 it turns out that even if you work in many dimensions, 99 00:10:30,200 --> 00:10:35,420 any reasonable shape is has its volume scaled by a constant factor. 100 00:10:35,420 --> 00:10:40,550 And this constant factor is called the determinant. So this is very much 101 00:10:40,550 --> 00:10:46,940 what I mean by thinking about things geometrically. But it turns out that 102 00:10:46,940 --> 00:10:52,070 it's really quite difficult to make lots of this precise in the calculations get a bit horrible if you 103 00:10:52,070 --> 00:10:57,200 try and think about things just in terms of volumes. So very 104 00:10:57,200 --> 00:11:02,330 much in the spirit of this philosophy. I want to do calculations 105 00:11:02,330 --> 00:11:07,640 trying to abstract out properties of this. So let's think about a few 106 00:11:07,640 --> 00:11:12,680 properties of this scaling areas would 107 00:11:12,680 --> 00:11:18,230 have in R-squared and try and think of 108 00:11:18,230 --> 00:11:23,270 if I can write these down as abstract properties, maybe I can understand the properties 109 00:11:23,270 --> 00:11:29,630 of this Lydian transmission just from these abstract ideas 110 00:11:29,630 --> 00:11:35,950 and say. Note 111 00:11:35,950 --> 00:11:43,660 that the 112 00:11:43,660 --> 00:11:48,790 area or volume. 113 00:11:48,790 --> 00:12:01,570 Of a linear transformation. 114 00:12:01,570 --> 00:12:06,730 Uh. Is limited 115 00:12:06,730 --> 00:12:11,750 in terms of the entries of the linear transformation. So maybe let's 116 00:12:11,750 --> 00:12:17,470 consider a matrix. I say that 117 00:12:17,470 --> 00:12:22,880 a b a matrix 118 00:12:22,880 --> 00:12:29,040 with columns. 119 00:12:29,040 --> 00:12:34,080 Two a n. Say, Lisa, if we can think 120 00:12:34,080 --> 00:12:40,450 of these are just factors in RVN, say Isaan and by a Matrix 121 00:12:40,450 --> 00:12:45,670 and the full name of the image of maybe 122 00:12:45,670 --> 00:12:50,740 the unique Cheban, RCN is 123 00:12:50,740 --> 00:13:00,860 living in the columns. 124 00:13:00,860 --> 00:13:10,910 Say the for you. Of. 125 00:13:10,910 --> 00:13:16,340 So maybe a nice way for me to write this is I whenever Matrixes columns, 126 00:13:16,340 --> 00:13:21,560 I want to I n. All right. This is a sequel to square brackets. 127 00:13:21,560 --> 00:13:27,760 A one of two I. N. And the volume 128 00:13:27,760 --> 00:13:32,830 of some columns and even the great column 129 00:13:32,830 --> 00:13:38,560 I have fact I've a to B.J. procedure Jay 130 00:13:38,560 --> 00:13:44,060 of the Cube is equal to the volume. 131 00:13:44,060 --> 00:13:53,560 Well, I have the same columns with B.J. 132 00:13:53,560 --> 00:13:58,790 Plus, the volume when I have C.J. is the 28th 133 00:13:58,790 --> 00:14:04,040 column, and this 134 00:14:04,040 --> 00:14:10,550 is basically based on the idea that 135 00:14:10,550 --> 00:14:15,620 the if I have to vote is. 136 00:14:15,620 --> 00:14:20,900 Here, A and B something, and this is or maybe I 137 00:14:20,900 --> 00:14:29,770 say consistent, I call this B and C. The area of this. 138 00:14:29,770 --> 00:14:36,500 It is equal to just the air. 139 00:14:36,500 --> 00:14:42,150 Well, I have to be policy here. 140 00:14:42,150 --> 00:14:47,690 So that was high then I'd have just a hair. 141 00:14:47,690 --> 00:14:54,260 So that's one property. A 142 00:14:54,260 --> 00:14:59,270 second and easy policy is that 143 00:14:59,270 --> 00:15:04,960 if a E.J. is equal to 144 00:15:04,960 --> 00:15:10,050 ACA for some. Jay, not equal 145 00:15:10,050 --> 00:15:15,320 to Kay. So to. The columns are the same. Then 146 00:15:15,320 --> 00:15:20,610 the image of the uniquely gave to cops. So maybe it's easiest 147 00:15:20,610 --> 00:15:28,270 if we just think about aside for two consecutive columns. 148 00:15:28,270 --> 00:15:33,300 And then we don't need to worry about. Yes. 149 00:15:33,300 --> 00:15:39,460 So if I have two consecutive columns, which are exactly the same, 150 00:15:39,460 --> 00:15:44,530 then the volume of. Of the 151 00:15:44,530 --> 00:15:53,490 image. Of the Unit Q. 152 00:15:53,490 --> 00:15:59,030 Is equal to zero. And 153 00:15:59,030 --> 00:16:04,100 so, again, I'm really thinking about. 154 00:16:04,100 --> 00:16:10,200 This and if I have two vectors being the same, then I would just have. 155 00:16:10,200 --> 00:16:15,510 Two columns. Ajanta J plus one and 156 00:16:15,510 --> 00:16:33,420 the whole unit Kubel just degenerate down to a line. 157 00:16:33,420 --> 00:16:39,420 And then a third property that we get from just looking at transmission's on 158 00:16:39,420 --> 00:16:45,760 R-squared, is that the. 159 00:16:45,760 --> 00:16:52,120 If you see I didn't see Matrix, 160 00:16:52,120 --> 00:17:00,910 then it preserves the unique Q. 161 00:17:00,910 --> 00:17:07,780 In the end and the volume. 162 00:17:07,780 --> 00:17:13,960 Equals walking. So I haven't necessarily proved any of these statements, but maybe they're all believable. 163 00:17:13,960 --> 00:17:19,800 And so what I want to say then is that these are properties 164 00:17:19,800 --> 00:17:27,860 that this determine I think should satisfy. And these are all outback properties. 165 00:17:27,860 --> 00:17:32,900 And therefore, I want to study the determinant, just 166 00:17:32,900 --> 00:17:39,620 the outback properties. So this is going to. 167 00:17:39,620 --> 00:17:44,750 To study the term, we we're then going to define what it means for any map 168 00:17:44,750 --> 00:17:49,940 to be determined. The term mental, which means it's going to satisfy the equivalent 169 00:17:49,940 --> 00:17:55,150 of these properties one, two and three. So we're not going to start 170 00:17:55,150 --> 00:18:03,780 doing some concrete mathematics maybe before with some geometric motivation. 171 00:18:03,780 --> 00:18:11,890 So we're going to say if I take any function. 172 00:18:11,890 --> 00:18:17,500 So I'm thinking of A Function B and it's going to take his input. 173 00:18:17,500 --> 00:18:24,160 And by and matrices with real entries. And it's going to output some 174 00:18:24,160 --> 00:18:30,120 real value. And I'm going to call this map. 175 00:18:30,120 --> 00:18:35,490 I'm going to say it has the property been determined, determined until meaning 176 00:18:35,490 --> 00:18:41,770 they pay you smoke. What I expect this the tenants behave like if it satisfies the charges 177 00:18:41,770 --> 00:18:46,770 properties. So if I think about my 178 00:18:46,770 --> 00:18:51,780 map acting on some matrix while I like it in column form and in the 179 00:18:51,780 --> 00:18:56,970 JF column, it has the 180 00:18:56,970 --> 00:19:02,210 Vector B.J. Policy J. 181 00:19:02,210 --> 00:19:10,650 Then I want it to be equal to the determinant 182 00:19:10,650 --> 00:19:18,130 B.J., plus the determinant at. 183 00:19:18,130 --> 00:19:23,820 C.J. 184 00:19:23,820 --> 00:19:29,010 For any choice column in the exchange between one 185 00:19:29,010 --> 00:19:34,500 man and simile, I wanted to scale 186 00:19:34,500 --> 00:19:40,810 so servi I scale one column. If I factor Alanda 187 00:19:40,810 --> 00:19:49,600 then I just wanted this to scale the determining factor. Let's 188 00:19:49,600 --> 00:19:55,410 say this is going to be the first popsy and I want this to be D 189 00:19:55,410 --> 00:20:06,320 is linear in each column. 190 00:20:06,320 --> 00:20:11,360 So this is basically the poverty that we guessed, the volume of the 191 00:20:11,360 --> 00:20:16,940 image of the unit Bush should satisfy. Funny late in translation. 192 00:20:16,940 --> 00:20:24,290 The second thing is that. 193 00:20:24,290 --> 00:20:29,890 I want. These have. According to the second property here, 194 00:20:29,890 --> 00:20:37,250 that if. 195 00:20:37,250 --> 00:20:42,830 I evaluated some matrix 196 00:20:42,830 --> 00:20:48,110 and two of the columns are equal to each other, 197 00:20:48,110 --> 00:20:53,170 maybe two consecutive columns. Then I want the determined 198 00:20:53,170 --> 00:20:59,790 to be equal to zero. 199 00:20:59,790 --> 00:21:05,050 And then the final thing is, if I evaluate 200 00:21:05,050 --> 00:21:12,560 the discernment of the identity matrix, I just want this to be equal to what? 201 00:21:12,560 --> 00:21:17,660 So this is a completely abstract definition of what it means for any function 202 00:21:17,660 --> 00:21:23,150 on matrices to be determined to two, and at least based on this motivation, 203 00:21:23,150 --> 00:21:28,190 we expect that the volume of the image of the unit cube should satisfy this. And 204 00:21:28,190 --> 00:21:33,320 so we think it should be an important property of 205 00:21:33,320 --> 00:21:40,570 any living man. But now this applies in general to matrices, that outright call. 206 00:21:40,570 --> 00:21:47,350 So the first thing I want to do is to say that 207 00:21:47,350 --> 00:21:52,690 if any linear maps are strikes, these properties that are actually satisfies slightly stronger 208 00:21:52,690 --> 00:22:00,670 versions of these properties. So we can automatically upgrade. 209 00:22:00,670 --> 00:22:05,790 So proposition. 210 00:22:05,790 --> 00:22:17,850 If the is a determined mental maths. 211 00:22:17,850 --> 00:22:23,760 Then one. 212 00:22:23,760 --> 00:22:29,070 I can swap two columns if I 213 00:22:29,070 --> 00:22:34,920 switch the sign of the determinates, say. 214 00:22:34,920 --> 00:22:47,460 There's alternating and the columns 215 00:22:47,460 --> 00:22:53,020 say the 216 00:22:53,020 --> 00:22:58,470 name of. Some 217 00:22:58,470 --> 00:23:03,750 matrix with columns, B.J. and B.J. plus one 218 00:23:03,750 --> 00:23:18,940 is equal to minus the determent of the same matrix if I swapped the columns. 219 00:23:18,940 --> 00:23:31,330 Say, swapping to. 220 00:23:31,330 --> 00:23:38,410 Just change is a sign. 221 00:23:38,410 --> 00:23:44,050 A second property is that. 222 00:23:44,050 --> 00:23:54,120 OK. I said the deterrent is equal to zero. Two consecutive columns the same. But actually they don't have to be consecutive. 223 00:23:54,120 --> 00:24:00,930 So if I evaluated it some matrix 224 00:24:00,930 --> 00:24:06,330 with two columns, B.J. and B.I, this is zero 225 00:24:06,330 --> 00:24:23,060 if any two of the different columns the same. 226 00:24:23,060 --> 00:24:28,970 And then a final property is a stronger version 227 00:24:28,970 --> 00:24:34,820 of it being alternating. 228 00:24:34,820 --> 00:24:39,950 The first one we said, and it's also 18 consecutive columns. But actually I can do it for 229 00:24:39,950 --> 00:25:09,050 general columns, so if I swap any two columns. 230 00:25:09,050 --> 00:25:14,090 I say whenever I switch and into columns, I just get exactly the same answer. But with the 231 00:25:14,090 --> 00:25:20,970 opposite side. 232 00:25:20,970 --> 00:25:26,450 So let's go ahead and prove this proposition. Based on the properties that we've defined 233 00:25:26,450 --> 00:25:35,570 for the term mentally. 234 00:25:35,570 --> 00:25:40,690 So. Let's start off with part one, because that really holds the 235 00:25:40,690 --> 00:25:46,840 key to the whole thing, and what I want to do is to 236 00:25:46,840 --> 00:25:52,080 think about being evaluated where I'm evaluating 237 00:25:52,080 --> 00:25:57,260 the JF Common column and the J plus one column, APJ plus PJ 238 00:25:57,260 --> 00:26:19,580 one. 239 00:26:19,580 --> 00:26:24,890 So here this is the JF column. 240 00:26:24,890 --> 00:26:31,970 And this is the the past first column. 241 00:26:31,970 --> 00:26:37,190 Some say by many 242 00:26:37,190 --> 00:26:43,870 Anstee, I can expand the sounds. 243 00:26:43,870 --> 00:26:49,660 Say, what do I get? Well, I'll get one term 244 00:26:49,660 --> 00:26:58,100 with B.J. and B.J. 245 00:26:58,100 --> 00:27:03,720 I'll get one term with B.J. 246 00:27:03,720 --> 00:27:12,130 and B.J. plus one. I'll get one term 247 00:27:12,130 --> 00:27:19,600 with Jerry Jay column, B.J. plus one and J plus first column B J. 248 00:27:19,600 --> 00:27:25,570 And then I'll get a final term 249 00:27:25,570 --> 00:27:30,700 with J. Colin, B.J. Paswan and J. Plus first columns. Also, 250 00:27:30,700 --> 00:27:38,770 B.J. plus one. 251 00:27:38,770 --> 00:27:52,420 So this is probably one. 252 00:27:52,420 --> 00:28:00,030 But by property, too, 253 00:28:00,030 --> 00:28:05,030 we have. That, 254 00:28:05,030 --> 00:28:20,580 uh, the left hand side. 255 00:28:20,580 --> 00:28:28,520 Is equal to zero. Because it has the J. Column in the J plus first column being the same. 256 00:28:28,520 --> 00:28:34,410 So I'm just using this property, but moreover, we also have 257 00:28:34,410 --> 00:28:39,460 that in my expansion into four terms on the right hand side. The first time in the last term 258 00:28:39,460 --> 00:29:07,060 must also be zero because these also have two columns which are the same. 259 00:29:07,060 --> 00:29:14,120 And so if I substitute these building codes zero, I therefore find. 260 00:29:14,120 --> 00:29:19,580 Yes, this is it. This time is. This time you. 261 00:29:19,580 --> 00:29:25,400 And so I just got some. 262 00:29:25,400 --> 00:29:31,070 Well, I have great column B, Jane Deprez, first column B.J. plus one 263 00:29:31,070 --> 00:29:42,310 is equal to the negative of exactly the same thing when I stopped the two columns. 264 00:29:42,310 --> 00:29:51,260 And so this gives the first property. 265 00:29:51,260 --> 00:29:56,350 So for part two. I can 266 00:29:56,350 --> 00:30:13,550 now just repeatedly use part one. 267 00:30:13,550 --> 00:30:19,030 To move column, Jay. 268 00:30:19,030 --> 00:30:24,450 Next to. Column 269 00:30:24,450 --> 00:30:46,090 I and I changed the determine just by a factor of plus or minus one. 270 00:30:46,090 --> 00:31:00,700 But then the determined has to be zero, because I have. We'll have two consecutive columns being the same. 271 00:31:00,700 --> 00:31:05,970 And so that instantly gives the final property safe and now 272 00:31:05,970 --> 00:31:11,110 for the third part of part three. There's 273 00:31:11,110 --> 00:31:16,150 a few different ways I could do it, but I can essentially follow exactly the same 274 00:31:16,150 --> 00:31:23,430 proofers. Part one. I can 275 00:31:23,430 --> 00:31:29,030 think about a valley right in. My matrix, 276 00:31:29,030 --> 00:31:34,190 where I are now evaluating it with columns, B.I 277 00:31:34,190 --> 00:31:40,060 and. B.J. in. Position. 278 00:31:40,060 --> 00:31:46,060 I and P.J. plus B.I also 279 00:31:46,060 --> 00:31:54,980 in position, Jay. 280 00:31:54,980 --> 00:32:06,100 And I can follow exactly the same argument. 281 00:32:06,100 --> 00:32:11,480 You men as four Part 282 00:32:11,480 --> 00:32:16,610 II. I can expand this out four terms. This 283 00:32:16,610 --> 00:32:22,520 left hand side must vanish because two of the columns are the same. 284 00:32:22,520 --> 00:32:28,820 And by part two, I've seen that whenever two columns are the same, the whole thing vanishes. 285 00:32:28,820 --> 00:32:38,050 But then when I think trying to get out the first term, the last term must also vanish. 286 00:32:38,050 --> 00:32:47,320 Using. Part two. 287 00:32:47,320 --> 00:32:52,410 OK. So so far, what we've seen is that we've maybe 288 00:32:52,410 --> 00:32:58,230 had this geometric guess that there's an important property of any 289 00:32:58,230 --> 00:33:03,300 matrix or linear transformation, which is how it scales volumes. And this 290 00:33:03,300 --> 00:33:09,040 is a important concept that you associate to any matrix. I've then 291 00:33:09,040 --> 00:33:14,140 defined this abstract property of. Any 292 00:33:14,140 --> 00:33:19,450 function being determinant, too, if it satisfies first of the factors that we expect this 293 00:33:19,450 --> 00:33:25,180 genuine constant to satisfy. And I've seen 294 00:33:25,180 --> 00:33:30,220 that if it satisfies these properties, then feel this proposition I actually to slightly stronger 295 00:33:30,220 --> 00:33:37,000 versions of those properties. So we've now 296 00:33:37,000 --> 00:33:42,880 done enough of the basic important properties of the determinant. 297 00:33:42,880 --> 00:33:50,040 But we haven't seen that actually there's any such map that satisfies these properties. 298 00:33:50,040 --> 00:33:55,050 And so that's what I'd like to do now. I want to show you that there is at 299 00:33:55,050 --> 00:34:00,180 least some function which satisfies all these properties in the back of my mind. I'm 300 00:34:00,180 --> 00:34:05,220 thinking that a function that satisfies these properties is the volume of the image 301 00:34:05,220 --> 00:34:10,760 of the unit cube. But again, I don't want to do any calculations geometrically. I want to do them all in 302 00:34:10,760 --> 00:34:16,410 an outback way because that makes everything much, much clean. And so this is maybe the first 303 00:34:16,410 --> 00:34:23,070 important result of the cool. So let's call it a film. 304 00:34:23,070 --> 00:34:34,810 Are you determined to map exists? 305 00:34:34,810 --> 00:34:40,810 On and by and matrices 306 00:34:40,810 --> 00:34:45,880 exists for each 307 00:34:45,880 --> 00:34:51,520 and beginning to one 308 00:34:51,520 --> 00:34:58,810 say. 309 00:34:58,810 --> 00:35:04,550 It's not totally obvious that something should exist from this. And 310 00:35:04,550 --> 00:35:09,560 to prove this, I want to prove by induction where I'm going to induct 311 00:35:09,560 --> 00:35:21,730 on the science, the matrices. 312 00:35:21,730 --> 00:35:27,250 OK, so let's first of all, start thinking about the case when N equals one. 313 00:35:27,250 --> 00:35:32,480 So we just we at one by one matrices. 314 00:35:32,480 --> 00:35:38,690 And I can define the of just a one by one matrix, a 315 00:35:38,690 --> 00:35:44,320 to be maybe the most obvious thing, just essentially a. And since 316 00:35:44,320 --> 00:35:49,450 the function, the identity function is certainly linear, it's very easy to cheque 317 00:35:49,450 --> 00:36:04,140 that dissatisfies the properties one, two and three here. 318 00:36:04,140 --> 00:36:09,220 OK. So you're saying that something exists for 319 00:36:09,220 --> 00:36:14,810 one by one waitress's? And now we want to 320 00:36:14,810 --> 00:36:20,300 consider the case when this began one. And we're going to assume 321 00:36:20,300 --> 00:36:25,460 that something exists. Some determine mental map exists for and minus one by 322 00:36:25,460 --> 00:36:53,680 minus one seeds. 323 00:36:53,680 --> 00:36:59,950 So we somehow want to construct a function on and by and matrices 324 00:36:59,950 --> 00:37:08,150 out of this function that we're assuming to exist on and minus one Byan minus one matrices. 325 00:37:08,150 --> 00:37:13,160 So I want to come up with some. I 326 00:37:13,160 --> 00:37:20,190 want to cook up some way of coming up with a determined to map the satisfies these three properties. 327 00:37:20,190 --> 00:37:26,720 But I want to somehow use this and minus one by minus one. 328 00:37:26,720 --> 00:37:33,530 So I want to think of some Matrix A. 329 00:37:33,530 --> 00:37:39,170 which is an end by an all matrix. So let's say 330 00:37:39,170 --> 00:37:44,960 it has entry's little a J. 331 00:37:44,960 --> 00:37:50,450 And if I'm going to use this Dearne minus one, which is clearly gonna be important in the argument, 332 00:37:50,450 --> 00:37:55,730 I have to think about how can I construct a minus one by minus one 333 00:37:55,730 --> 00:38:03,630 matrix out of this and by an matrix? Maybe there's one easy way to do this. 334 00:38:03,630 --> 00:38:22,250 Which is just to fill out some loans, some colleton. 335 00:38:22,250 --> 00:38:27,330 So I want a IJA capley IJA to be 336 00:38:27,330 --> 00:38:35,530 the N minus one and minus one matrix formed by removing the. 337 00:38:35,530 --> 00:38:43,830 I threw a and and. Colin. 338 00:38:43,830 --> 00:38:49,110 OK. So this is one way 339 00:38:49,110 --> 00:38:54,330 of constructing various different and minus one by minus one matrices for my. 340 00:38:54,330 --> 00:38:59,420 And by Matrix A and are now 341 00:38:59,420 --> 00:39:07,950 going to consider. So let's choose 342 00:39:07,950 --> 00:39:13,520 some. I between 343 00:39:13,520 --> 00:39:18,860 one and N and I'm going to define my attempt. 344 00:39:18,860 --> 00:39:25,230 And a determined man to function on and by and make sees 345 00:39:25,230 --> 00:39:30,230 as follows. 346 00:39:30,230 --> 00:39:35,720 So I would put minus one to the five plus one a 347 00:39:35,720 --> 00:39:42,020 I won and then I'm going to use the N minus one. Five minus one function 348 00:39:42,020 --> 00:39:47,440 on the Matrix. I won. 349 00:39:47,440 --> 00:39:53,200 But maybe say this is some function that would work on environment sees, 350 00:39:53,200 --> 00:39:58,880 but is ignoring lots of the elements 351 00:39:58,880 --> 00:40:03,880 in the i3. So instead, I want to add lots of these similar versions of this together 352 00:40:03,880 --> 00:40:09,070 to get something that sensitive to every single element in The Matrix. 353 00:40:09,070 --> 00:40:14,440 And it turns out that a good way to do this is to keep on multiplying by 354 00:40:14,440 --> 00:40:20,440 minus one. And just going through all the different possible choices 355 00:40:20,440 --> 00:40:42,070 in the i3. 356 00:40:42,070 --> 00:40:47,140 OK, so I've just plucked from thin air this matrix here. But I think if you play 357 00:40:47,140 --> 00:40:52,570 around with a little bit and you decide that you want to come up with some way of defining 358 00:40:52,570 --> 00:40:57,580 a function on environmental crises in terms of function by minus one by one and minus one makes these. 359 00:40:57,580 --> 00:41:02,800 That depends on each element. You see that actually this isn't such an unnatural thing 360 00:41:02,800 --> 00:41:07,960 that I just picked out of the air. Maybe you wouldn't have guessed these minus one factors, 361 00:41:07,960 --> 00:41:14,500 but it should be clear once you go through a bit more the. Why these Manasquan factors turn off. 362 00:41:14,500 --> 00:41:20,410 Anyway, I've defined this function and this is a function on and by and matrices. 363 00:41:20,410 --> 00:41:25,510 And I want to show you that it's now the term mental. So I want to verify that it satisfies properties one, 364 00:41:25,510 --> 00:41:46,660 two and three. And if I done this, I have completed the proof. 365 00:41:46,660 --> 00:41:51,690 OK, so I'm just going to verify the properties in order. I got to move 366 00:41:51,690 --> 00:41:57,200 on to the middle. 367 00:41:57,200 --> 00:42:02,670 I guess maybe would be useful to keep the sport up so that we can see for ourselves. And maybe I'll move 368 00:42:02,670 --> 00:42:07,790 after I say I start off with property 369 00:42:07,790 --> 00:42:13,280 one. 370 00:42:13,280 --> 00:42:19,200 So we'd like to show you these línea in each column. 371 00:42:19,200 --> 00:42:24,440 So since the 372 00:42:24,440 --> 00:42:32,270 is a sum of terms 373 00:42:32,270 --> 00:42:39,020 ahj the N minus one. Hey ija 374 00:42:39,020 --> 00:42:44,270 it is sufficient 375 00:42:44,270 --> 00:42:53,750 to show each of these. 376 00:42:53,750 --> 00:43:05,730 Is Minear. In the columns. 377 00:43:05,730 --> 00:43:11,230 OK. So I just need to show you that each of these different. 378 00:43:11,230 --> 00:43:17,010 Quantity is AIG, the minus one of Charolais IJA is 379 00:43:17,010 --> 00:43:23,950 in the columns. And so let's concentrate 380 00:43:23,950 --> 00:43:31,340 on the case column. 381 00:43:31,340 --> 00:43:36,470 Of. And biometrics. And I want to consider 382 00:43:36,470 --> 00:43:41,780 these functions. So if Jay 383 00:43:41,780 --> 00:43:46,850 is equal to Kay, then 384 00:43:46,850 --> 00:43:54,740 AIG doesn't depend. 385 00:43:54,740 --> 00:44:02,540 On the case column of A, 386 00:44:02,540 --> 00:44:07,850 because this is formed by removing the case column. But 387 00:44:07,850 --> 00:44:13,390 little IJA. It's just the 388 00:44:13,390 --> 00:44:35,170 ice elements of the cave column. And so certainly Línea. 389 00:44:35,170 --> 00:44:42,950 And so I think these two things together, we see the AIG. 390 00:44:42,950 --> 00:44:54,560 The N minus one of A.J. is living the case. Him. 391 00:44:54,560 --> 00:45:01,190 When James equal to Kay. 392 00:45:01,190 --> 00:45:07,040 What if Jay is not equal? OK. 393 00:45:07,040 --> 00:45:12,290 Well, then we have the 394 00:45:12,290 --> 00:45:18,080 AIG does not depend on the cave column at all because it's just some other entry 395 00:45:18,080 --> 00:45:32,980 in the J column. 396 00:45:32,980 --> 00:45:39,760 But. We have the the minus one 397 00:45:39,760 --> 00:45:49,440 AHJ is linnear. In the cave column. 398 00:45:49,440 --> 00:45:54,520 Of a sense, we we're assuming that this function is the term mental and socially 399 00:45:54,520 --> 00:46:10,240 and the continent's. 400 00:46:10,240 --> 00:46:15,400 And therefore, in this case, we have the AIG deal, 401 00:46:15,400 --> 00:46:21,190 minus one of AIG is Línea 402 00:46:21,190 --> 00:46:26,270 in the Case column. 403 00:46:26,270 --> 00:46:31,280 And therefore, regardless of what are your charges, these functions are always living in the cave column. And 404 00:46:31,280 --> 00:46:40,660 so D William Lillian column. 405 00:46:40,660 --> 00:46:46,310 Hmm hmm hmm hmm, 406 00:46:46,310 --> 00:46:51,410 OK. So we've therefore verified Property One for this function 407 00:46:51,410 --> 00:46:56,490 D that I've written down. So we're left to verify 408 00:46:56,490 --> 00:47:06,710 properties. Two and three. 409 00:47:06,710 --> 00:47:11,850 So maybe we have just enough space here. 410 00:47:11,850 --> 00:47:20,860 So let's go for probably to. 411 00:47:20,860 --> 00:47:25,860 Imagine that 412 00:47:25,860 --> 00:47:31,200 in my matrix, hey, I have a veejays column is equal 413 00:47:31,200 --> 00:47:36,350 to the Japes first column. And so I want to show you that this determined 414 00:47:36,350 --> 00:47:41,400 has to be zero then. So, again, it's going to look a little bit 415 00:47:41,400 --> 00:47:49,470 similar to this. 416 00:47:49,470 --> 00:47:56,340 Hey, I. J will have 417 00:47:56,340 --> 00:48:09,100 two columns, the same. Two consecutive columns, the same 418 00:48:09,100 --> 00:48:14,530 because it essentially has the columns in Hatchet from A.J., and I just put one unless we removed 419 00:48:14,530 --> 00:48:19,770 one of these two. OK. So 420 00:48:19,770 --> 00:48:24,950 my notations slightly panned. Let's imagine that AK is equal to 421 00:48:24,950 --> 00:48:31,050 a K plus one just because otherwise I'll get confused by Wiggins's apologies. 422 00:48:31,050 --> 00:48:36,150 So I imagine that the 28th column is equal to the Kapos first column. And now I'm looking at one 423 00:48:36,150 --> 00:48:41,190 of these matrices I ija we are moving the Arthur in the great column. If I 424 00:48:41,190 --> 00:48:46,230 haven't removed either the case column or the K plus first column, then whatever corresponds to these two 425 00:48:46,230 --> 00:48:51,790 will still be the same. And so we'll have two consecutive columns, the same. And therefore 426 00:48:51,790 --> 00:48:56,880 our have the D minus one of AIG is 427 00:48:56,880 --> 00:49:06,720 equal to zero. But this is only true if I haven't removed 428 00:49:06,720 --> 00:49:12,180 one of those two columns. So if I didn't remove either those two columns, then two columns would be the same. 429 00:49:12,180 --> 00:49:18,150 And so the determinate will vanish on these and minus one by minus one matrices. 430 00:49:18,150 --> 00:49:23,160 And so this means that when I'm evaluating DNA, I would have 431 00:49:23,160 --> 00:49:28,200 loads and I see terms. But virtually all of them say I'm only gonna get the terms 432 00:49:28,200 --> 00:49:43,950 from a K. And I keep us warm 433 00:49:43,950 --> 00:49:51,120 so I can write out precisely what this is. 434 00:49:51,120 --> 00:49:57,420 And I just get the two terms from when I have a K and 435 00:49:57,420 --> 00:50:08,310 I keep us one. 436 00:50:08,310 --> 00:50:13,360 But then. If the case column 437 00:50:13,360 --> 00:50:19,300 was equal to the Kapos first column, I have that 438 00:50:19,300 --> 00:50:24,810 a K is equal to a I keep us one and 439 00:50:24,810 --> 00:50:32,100 little I k is equal to a little a. I keep Wasswa. 440 00:50:32,100 --> 00:50:47,740 And so take these two terms are equal to each other. And so the determining functions. 441 00:50:47,740 --> 00:50:59,450 So now, finally, I just need to cheque property, see? 442 00:50:59,450 --> 00:51:05,740 So if A is equal to I can 443 00:51:05,740 --> 00:51:11,750 then. A IJA is equal to zero 444 00:51:11,750 --> 00:51:16,770 unless Jay Eco's by. And 445 00:51:16,770 --> 00:51:23,640 so all but one term in this huge expansion vanishes. 446 00:51:23,640 --> 00:51:28,770 And so I have the dysfunction valuated eight is minus one 447 00:51:28,770 --> 00:51:34,680 to the I plus I, I, I, I the 448 00:51:34,680 --> 00:51:39,730 minus one of a I. But this is equal 449 00:51:39,730 --> 00:51:44,800 to one because minus one to the iPod, sorry, is one I 450 00:51:44,800 --> 00:51:49,900 is one whenever I is the identity matrix and by assumption 451 00:51:49,900 --> 00:51:55,000 of the N minus one being the term rental. This also evaluates to one at the identity 452 00:51:55,000 --> 00:52:07,770 matrix, which is what I get when I remove the fifth column and the i3. 453 00:52:07,770 --> 00:52:18,730 Yes. 454 00:52:18,730 --> 00:52:24,090 And so this means the function de 455 00:52:24,090 --> 00:52:29,850 satisfies poverties one, two and three. And so I've shown it's determined to. 456 00:52:29,850 --> 00:52:34,890 And so, therefore, by induction, we know that 457 00:52:34,890 --> 00:52:40,080 there exists at least one different determinant to map on environment. Sources 458 00:52:40,080 --> 00:52:50,430 say. Therefore, these term rental. 459 00:52:50,430 --> 00:53:02,330 And therefore determine from exit tests. 460 00:53:02,330 --> 00:53:07,400 Okay, so I'll stop here. What we've 461 00:53:07,400 --> 00:53:12,500 seen is the this we guess that there's this important geometric 462 00:53:12,500 --> 00:53:17,500 property of matrices, which is how they scale volumes. But it turns out that is 463 00:53:17,500 --> 00:53:22,880 very messy to work with. So we've abstract this out to some outback properties and then working 464 00:53:22,880 --> 00:53:28,020 with those outback properties. We've shown that there's something that exists that behaves 465 00:53:28,020 --> 00:53:34,780 the exact same way. And next time will show that this is, in fact, a unique map that has all these properties. 466 00:53:34,780 --> 00:53:54,680 OK. Thanks a lot.