1 00:00:00,070 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:18,230 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,230 --> 00:00:18,730 ocw.mit.edu. 8 00:00:23,330 --> 00:00:27,357 PROFESSOR: OK, so let's get started. 9 00:00:27,357 --> 00:00:28,940 I just wanted to make one announcement 10 00:00:28,940 --> 00:00:31,340 before we start the lecture. 11 00:00:31,340 --> 00:00:32,130 So Prof. 12 00:00:32,130 --> 00:00:36,320 Zwiebach is a way again today, which is why I'm lecturing. 13 00:00:36,320 --> 00:00:40,290 And his office hours he's obviously not going to have, 14 00:00:40,290 --> 00:00:41,350 but Prof. 15 00:00:41,350 --> 00:00:44,030 Harrow has kindly agreed to take them over. 16 00:00:44,030 --> 00:00:48,235 So today I'll have office hours four to five, and then Prof. 17 00:00:48,235 --> 00:00:52,850 Harrow will have office hours afterwards five to six. 18 00:00:52,850 --> 00:00:54,960 So feel free to come and talk to us. 19 00:00:54,960 --> 00:00:58,410 So today we're going to try and cover a few things. 20 00:00:58,410 --> 00:01:02,770 So we're going to spend a little bit of time talking about 21 00:01:02,770 --> 00:01:10,760 eigenvalues and vectors, which we've-- finishing this 22 00:01:10,760 --> 00:01:12,230 discussion from last time. 23 00:01:12,230 --> 00:01:19,110 Then we'll talk about inner products and inner product 24 00:01:19,110 --> 00:01:21,150 spaces. 25 00:01:21,150 --> 00:01:23,360 And then we'll talk about-- we'll 26 00:01:23,360 --> 00:01:35,340 introduce Dirac's notation, some of which 27 00:01:35,340 --> 00:01:37,360 we've already been using. 28 00:01:37,360 --> 00:01:39,480 And then, depending on time, we'll 29 00:01:39,480 --> 00:01:44,795 also talk a little bit more about linear operators. 30 00:01:51,380 --> 00:01:53,200 OK? 31 00:01:53,200 --> 00:01:58,470 So let's start with where we were last time. 32 00:01:58,470 --> 00:02:02,120 So we were talking about T-invariant subspaces. 33 00:02:02,120 --> 00:02:18,230 So we had U is a T-invariant subspace 34 00:02:18,230 --> 00:02:22,190 if the following is satisfied. 35 00:02:22,190 --> 00:02:30,180 If T of U is equal to-- if this thing, which 36 00:02:30,180 --> 00:02:36,860 is all vectors that are generated by T from vectors 37 00:02:36,860 --> 00:02:42,020 that live in U. So if T is inside U itself. 38 00:02:47,639 --> 00:02:48,139 OK? 39 00:02:51,520 --> 00:02:59,170 And we can define this in general for any U. However, 40 00:02:59,170 --> 00:03:02,770 one class of these invariant subspaces are very useful. 41 00:03:02,770 --> 00:03:06,815 So if we take U to be one dimensional. 42 00:03:13,900 --> 00:03:27,770 OK/ and so that really means that U I can write as some 43 00:03:27,770 --> 00:03:31,000 whatever field I'm defining my vector space over. 44 00:03:31,000 --> 00:03:35,180 Every element of this subspace U is just some scalar multiple 45 00:03:35,180 --> 00:03:41,890 of a single vector U. So this is a one dimensional thing. 46 00:03:41,890 --> 00:03:51,730 Now if we have a T-invariant subspace of this one-- 47 00:03:51,730 --> 00:03:54,310 if this is going to be a T-invariant objective, 48 00:03:54,310 --> 00:03:59,716 then we get a very simple equation 49 00:03:59,716 --> 00:04:01,720 that you've seen before. 50 00:04:05,390 --> 00:04:09,910 So we're taking all vectors in U acting on them with T, 51 00:04:09,910 --> 00:04:11,500 and if it stays within U, then it 52 00:04:11,500 --> 00:04:14,500 has to be able to be written like this. 53 00:04:14,500 --> 00:04:18,550 So we have some operator acting on our vector space 54 00:04:18,550 --> 00:04:21,000 producing something in the same vector space, 55 00:04:21,000 --> 00:04:22,310 just rescaling it. 56 00:04:27,170 --> 00:04:28,520 OK? 57 00:04:28,520 --> 00:04:31,670 For sum lambda, which we haven't specified. 58 00:04:35,260 --> 00:04:37,020 And you've seen this equation before 59 00:04:37,020 --> 00:04:38,860 in terms of matrices and vectors, right? 60 00:04:38,860 --> 00:04:40,920 This is an eigenvalue equation. 61 00:04:40,920 --> 00:04:48,085 So these are eigenvalues and these are eigenvectors. 62 00:04:53,090 --> 00:04:55,060 But now they're just an abstract version 63 00:04:55,060 --> 00:04:58,300 of what you've discussed before. 64 00:04:58,300 --> 00:05:02,420 And we'll come back to this in a moment. 65 00:05:02,420 --> 00:05:06,370 One thing that we just defined at the end 66 00:05:06,370 --> 00:05:08,470 is the spectrum of an operator. 67 00:05:13,970 --> 00:05:18,482 The spectrum of T is equal to all eigenvalues 68 00:05:18,482 --> 00:05:19,190 of that operator. 69 00:05:32,610 --> 00:05:35,350 And so later on these will become-- 70 00:05:35,350 --> 00:05:37,690 this object will become important. 71 00:05:37,690 --> 00:05:42,250 Let's just concentrate on this and ask what does it mean. 72 00:05:42,250 --> 00:05:54,722 So if we have lambda being an eigenvalues, 73 00:05:54,722 --> 00:05:55,680 what does that tell us? 74 00:05:55,680 --> 00:05:57,160 What does this equation tell us? 75 00:05:57,160 --> 00:06:07,250 Well, it tells us that on U. So all I'm doing 76 00:06:07,250 --> 00:06:11,050 is taking this term over to the other side of the equation 77 00:06:11,050 --> 00:06:14,370 and inserting the identity operator I. 78 00:06:14,370 --> 00:06:18,540 So this is in itself an operator now, right? 79 00:06:31,500 --> 00:06:40,650 And so this tells us also that this operator, because it maps 80 00:06:40,650 --> 00:06:46,170 something that's non-zero to the null vector, this 81 00:06:46,170 --> 00:06:55,810 is not injective, OK? 82 00:06:55,810 --> 00:07:04,110 And you can even write the null space of T-- of T minus I 83 00:07:04,110 --> 00:07:14,600 lambda is equal to all eigenvectors 84 00:07:14,600 --> 00:07:17,400 with eigenvalue lambda. 85 00:07:22,210 --> 00:07:22,710 OK? 86 00:07:22,710 --> 00:07:28,440 So every eigenvector with eigenvalue lambda, 87 00:07:28,440 --> 00:07:31,250 T acting on it is just going to give me lambda times 88 00:07:31,250 --> 00:07:33,820 the eigenvector again, and so this will vanish. 89 00:07:33,820 --> 00:07:38,430 So for all eigenvectors with that eigenvalue. 90 00:07:40,950 --> 00:07:48,790 And we've previously seen that, if something is not injective, 91 00:07:48,790 --> 00:07:55,980 it's also not invertible, right? 92 00:08:05,690 --> 00:08:13,125 So this lets us write something quite nice down. 93 00:08:13,125 --> 00:08:14,000 So there's a theorem. 94 00:08:17,880 --> 00:08:19,050 Let me write it out. 95 00:08:21,970 --> 00:08:27,160 So if we let T is in the space of linear operators acting 96 00:08:27,160 --> 00:08:33,309 on this vector space v, and we have 97 00:08:33,309 --> 00:08:41,409 a set of eigenvalues, lambda 1, lambda 2, lambda n, 98 00:08:41,409 --> 00:08:54,970 distinct eigenvalues, eigenvalues of T, 99 00:08:54,970 --> 00:09:06,580 and the corresponding eigenvectors, which 100 00:09:06,580 --> 00:09:14,080 we will call U. OK, so the sum set U1, U2, up to Un 101 00:09:14,080 --> 00:09:17,600 with the correspondence by their label. 102 00:09:21,070 --> 00:09:30,740 So then we know that this list is actually 103 00:09:30,740 --> 00:09:32,770 a linearly independent set. 104 00:09:46,750 --> 00:09:49,500 So we can prove this one very quickly. 105 00:09:49,500 --> 00:09:50,330 So let's do that. 106 00:09:52,980 --> 00:09:56,660 So let's assume it's false. 107 00:09:56,660 --> 00:10:00,300 So the proof is y contradiction, so assume it's false. 108 00:10:03,130 --> 00:10:04,130 And what does that mean? 109 00:10:04,130 --> 00:10:07,900 Well, that means that there is a non-trivial relation. 110 00:10:07,900 --> 00:10:27,180 I could write down some relation C1U1 plus C2U2 plus CkUk 111 00:10:27,180 --> 00:10:32,162 equals 0 without all the C's being 0. 112 00:10:32,162 --> 00:10:33,620 And what we'll do is we'll actually 113 00:10:33,620 --> 00:10:37,920 say OK, let's do let-- so we'll let there 114 00:10:37,920 --> 00:10:46,280 be a k, a value of k that's less than or equal to n, such 115 00:10:46,280 --> 00:10:53,630 that this holds for Ci not equal to 0. 116 00:10:53,630 --> 00:10:57,590 So we're postulating that there is some linear dependence 117 00:10:57,590 --> 00:11:00,856 on some of these things. 118 00:11:00,856 --> 00:11:07,570 So what we can do is then act on this vector 119 00:11:07,570 --> 00:11:16,140 here with T minus lambda k times the identity acting on this. 120 00:11:16,140 --> 00:11:20,992 So this is C1U1 plus dot dot dot plus CkUk. 121 00:11:24,440 --> 00:11:25,030 OK? 122 00:11:25,030 --> 00:11:28,050 And what do we get here? 123 00:11:28,050 --> 00:11:31,700 So we're going to get, if act on this piece of it, 124 00:11:31,700 --> 00:11:35,660 this is an eigenvector, so T acting on this one 125 00:11:35,660 --> 00:11:38,080 would just give us lambda 1, right? 126 00:11:38,080 --> 00:11:40,860 And so we're going to get products of lambda 1 minus 127 00:11:40,860 --> 00:11:43,850 lambda k for this piece, et cetera. 128 00:11:43,850 --> 00:11:55,695 So this will give us C1 lambda 1 minus lambda k U1 plus dot dot 129 00:11:55,695 --> 00:12:05,310 dot up to the Ck minus 1 lambda k minus 1 130 00:12:05,310 --> 00:12:10,500 minus lambda k Uk minus 1. 131 00:12:10,500 --> 00:12:13,400 And then when we act on this one here, 132 00:12:13,400 --> 00:12:18,300 so this one has an eigenvalue-- the eigenvalue corresponding 133 00:12:18,300 --> 00:12:22,690 to the eigenvector is lambda k, so that last term gets killed. 134 00:12:22,690 --> 00:12:26,355 So we get plus 0 lots of Uk. 135 00:12:29,220 --> 00:12:33,120 And we know this is still 0. 136 00:12:33,120 --> 00:12:35,950 And now we've established, in fact, 137 00:12:35,950 --> 00:12:40,030 these things here are just numbers. 138 00:12:40,030 --> 00:12:41,270 All of these things. 139 00:12:41,270 --> 00:12:43,570 So we've actually written down a relation 140 00:12:43,570 --> 00:12:49,444 that involves less than k. 141 00:12:49,444 --> 00:12:50,860 Actually, I should have said this. 142 00:12:54,080 --> 00:12:59,030 Let there be a least k less than or equal to, 143 00:12:59,030 --> 00:13:01,910 and such that we have linear dependence. 144 00:13:01,910 --> 00:13:04,640 But what we've just shown is that, in fact, there's 145 00:13:04,640 --> 00:13:08,240 a smaller space that's also linear independent, right? 146 00:13:08,240 --> 00:13:15,780 So we've contradicted what we assumed to start with. 147 00:13:15,780 --> 00:13:18,470 And you can just repeat this procedure, OK? 148 00:13:18,470 --> 00:13:20,765 And so this is a contradiction. 149 00:13:25,230 --> 00:13:32,300 And so, in fact, there must be no non-trivial relation 150 00:13:32,300 --> 00:13:36,280 even for k equals n between these vectors, OK? 151 00:13:40,270 --> 00:13:42,960 Another brief theorem that we won't 152 00:13:42,960 --> 00:13:56,480 prove, although we'll sort of see why it works in a moment, 153 00:13:56,480 --> 00:14:08,120 is, again, for T in linear operators on v with v 154 00:14:08,120 --> 00:14:15,140 being a finite dimensional complex vector space. 155 00:14:28,160 --> 00:14:29,360 OK? 156 00:14:29,360 --> 00:14:30,850 There is at least one eigenvalue. 157 00:14:33,515 --> 00:14:43,670 So I guess for this-- so T has at least one eigenvalue. 158 00:14:49,720 --> 00:14:51,600 Now remember, in the last lecture, 159 00:14:51,600 --> 00:14:54,390 we looked at a matrix, two by two matrix, that 160 00:14:54,390 --> 00:14:58,780 was rotations in the xy-plane and found 161 00:14:58,780 --> 00:15:00,910 there were, in fact, no eigenvalues. 162 00:15:00,910 --> 00:15:05,780 But that's because we were looking at a real vector space. 163 00:15:05,780 --> 00:15:11,570 So we were looking at rotations of the real plane. 164 00:15:11,570 --> 00:15:14,650 So this is something that you can prove. 165 00:15:14,650 --> 00:15:17,560 We will see why it's true, but we won't prove it. 166 00:15:20,660 --> 00:15:24,580 And so one way of saying this is to go to a basis. 167 00:15:24,580 --> 00:15:27,235 And so everything we've said so far about eigenvalues 168 00:15:27,235 --> 00:15:30,590 and eigenvectors has not been referring 169 00:15:30,590 --> 00:15:32,720 to any particular basis. 170 00:15:32,720 --> 00:15:35,495 And in fact, eigenvalues are basis independent. 171 00:15:39,880 --> 00:15:41,660 But we can use a basis. 172 00:15:41,660 --> 00:15:44,220 And then we have matrix representations of operators 173 00:15:44,220 --> 00:15:47,030 that we've talked about. 174 00:15:47,030 --> 00:15:49,900 And sort of this operator equation, 175 00:15:49,900 --> 00:15:56,250 or the operator statement that T minus lambda 176 00:15:56,250 --> 00:16:10,030 I-- so as operator statement T minus lambda I U equals 0 177 00:16:10,030 --> 00:16:17,757 is equivalent to saying that-- well, we've said it here. 178 00:16:17,757 --> 00:16:19,340 We said that it's equivalent to saying 179 00:16:19,340 --> 00:16:20,740 that it's not invertible. 180 00:16:20,740 --> 00:16:22,110 This operator is not invertible. 181 00:16:29,730 --> 00:16:31,330 But that's also equivalent to saying 182 00:16:31,330 --> 00:16:34,285 that the matrix representation of it in any basis 183 00:16:34,285 --> 00:16:35,035 is not invertible. 184 00:17:01,310 --> 00:17:05,170 And by here we just mean inverses as in the inverses 185 00:17:05,170 --> 00:17:07,380 that you've taken the many matrices in your lives. 186 00:17:10,410 --> 00:17:15,550 And so what that means then, if-- I'm sure you remember. 187 00:17:15,550 --> 00:17:17,329 If a matrix is not invertible, that 188 00:17:17,329 --> 00:17:19,619 means it has a vanishing determinant. 189 00:17:19,619 --> 00:17:24,069 So it has debt of this. 190 00:17:24,069 --> 00:17:26,849 Now you can think of this as a matrix. 191 00:17:26,849 --> 00:17:30,460 This determinant has to be 0. 192 00:17:30,460 --> 00:17:34,120 And remembering we can write this thing out. 193 00:17:34,120 --> 00:17:36,250 And so it has lambdas on the diagonal, 194 00:17:36,250 --> 00:17:39,510 and then whatever entries T has wherever it wants. 195 00:17:39,510 --> 00:17:42,150 This just gives us a polynomial in lambda, right? 196 00:17:42,150 --> 00:17:48,090 So this gives us some f of lambda, which is a polynomial. 197 00:17:48,090 --> 00:17:50,800 And if you remember, this is called 198 00:17:50,800 --> 00:17:52,370 the characteristic polynomial. 199 00:17:52,370 --> 00:17:52,995 Characteristic. 200 00:18:01,710 --> 00:18:02,350 Right? 201 00:18:02,350 --> 00:18:11,410 And so we can write it, if we want, as just some f of lambda 202 00:18:11,410 --> 00:18:15,050 is equal to just, in this case, it's 203 00:18:15,050 --> 00:18:18,565 going to be just lambda minus some lambda 1. 204 00:18:25,930 --> 00:18:28,030 I have to be able to write it like this. 205 00:18:28,030 --> 00:18:33,650 I can just break it up into these terms 206 00:18:33,650 --> 00:18:40,830 here, where the lambda I's, the 0's of this polynomial 207 00:18:40,830 --> 00:18:44,115 are, in general, complex and can be repeated. 208 00:18:52,920 --> 00:18:57,362 Now what can happen is that you have, in the worst case -- 209 00:18:57,362 --> 00:18:59,570 I don't know if it's the worst case, but in one case, 210 00:18:59,570 --> 00:19:03,330 you could have all of the singularities-- 211 00:19:03,330 --> 00:19:07,200 all of the the 0's being at the same place. 212 00:19:07,200 --> 00:19:12,230 And you could have a eigenvalue that is in full degenerate 213 00:19:12,230 --> 00:19:12,841 here. 214 00:19:12,841 --> 00:19:13,340 Right? 215 00:19:13,340 --> 00:19:16,780 So if we, say, have lambda 1 occurring twice 216 00:19:16,780 --> 00:19:21,690 in this sequence, then we set out to a degenerate eigenvalue. 217 00:19:21,690 --> 00:19:25,420 And in principle, you could have just a single eigenvalue 218 00:19:25,420 --> 00:19:27,170 that's in full degenerate. 219 00:19:27,170 --> 00:19:29,040 But you can always write this. 220 00:19:29,040 --> 00:19:33,040 There has to be one lambda there at least, 221 00:19:33,040 --> 00:19:34,820 one lambda I there at least. 222 00:19:34,820 --> 00:19:38,220 And so you can see why this is true, right? 223 00:19:38,220 --> 00:19:43,395 Now if you're in a real vector space, 224 00:19:43,395 --> 00:19:47,230 you don't get to say that, because this polynomial may 225 00:19:47,230 --> 00:19:49,630 only have complex roots, and they're not 226 00:19:49,630 --> 00:19:53,440 part of the space you're talking about. 227 00:19:53,440 --> 00:19:53,940 OK? 228 00:19:57,232 --> 00:19:59,440 So it can be repeated, and this is called degeneracy. 229 00:20:06,620 --> 00:20:09,717 OK, so are there any questions? 230 00:20:09,717 --> 00:20:10,800 AUDIENCE: Can you turn it? 231 00:20:10,800 --> 00:20:13,871 It should be lambda I minus T, just so 232 00:20:13,871 --> 00:20:15,650 that it matches the next line. 233 00:20:15,650 --> 00:20:17,932 PROFESSOR: Thank you, OK. 234 00:20:23,540 --> 00:20:25,870 Thank you. 235 00:20:25,870 --> 00:20:29,762 I could have flipped the sign on the next line as well. 236 00:20:29,762 --> 00:20:30,720 So any other questions? 237 00:20:34,440 --> 00:20:34,940 No? 238 00:20:34,940 --> 00:20:40,710 OK, so let's move on and we can talk about inner products. 239 00:20:40,710 --> 00:20:43,750 And so first, what is an inner product? 240 00:20:48,180 --> 00:20:57,170 So an inner product is a map, but it's a very specific map. 241 00:21:05,720 --> 00:21:16,210 So an inner product on a vector space V 242 00:21:16,210 --> 00:21:33,110 is a map from V cross V to the field, F. 243 00:21:33,110 --> 00:21:35,590 And that's really what it's going to be. 244 00:21:35,590 --> 00:21:39,740 Now who has seen an inner product somewhere? 245 00:21:39,740 --> 00:21:43,647 OK, what do we call it? 246 00:21:43,647 --> 00:21:44,701 AUDIENCE: Dot product. 247 00:21:44,701 --> 00:21:45,950 PROFESSOR: Dot product, right. 248 00:21:45,950 --> 00:21:48,110 So we can learn a lot from thinking 249 00:21:48,110 --> 00:21:53,820 about this simple case, so the motivation for thinking 250 00:21:53,820 --> 00:22:00,170 about this is really the dot product. 251 00:22:00,170 --> 00:22:05,880 So we have a vector space Rn. 252 00:22:05,880 --> 00:22:06,510 OK? 253 00:22:06,510 --> 00:22:15,050 And on that vector space, we might have two vectors, a, 254 00:22:15,050 --> 00:22:17,260 which I'm going to write as a1. 255 00:22:17,260 --> 00:22:18,010 a2 dot dot dot. 256 00:22:18,010 --> 00:22:21,900 a2 dot dot dot. an, and b. 257 00:22:25,980 --> 00:22:30,990 So we have two vectors, and these are in vector space V. 258 00:22:30,990 --> 00:22:36,999 Then we can define the dot product, which 259 00:22:36,999 --> 00:22:38,790 is an example of one of these in a product. 260 00:22:41,766 --> 00:22:45,200 So a dot b. 261 00:22:45,200 --> 00:22:49,120 We can even put little vectors over these. 262 00:22:49,120 --> 00:22:53,030 And so our definitions that we've used for many years 263 00:22:53,030 --> 00:23:04,020 is that this is a1 b1 plus a2 b2 plus dot dot dot an bn. 264 00:23:04,020 --> 00:23:06,230 And you see that this does what we want it. 265 00:23:06,230 --> 00:23:09,560 So it takes two vectors which live in our vector space. 266 00:23:09,560 --> 00:23:12,650 And from that, you get a number, right? 267 00:23:12,650 --> 00:23:17,370 So this lives in R. 268 00:23:17,370 --> 00:23:20,300 So this is a nice example in a product. 269 00:23:20,300 --> 00:23:24,700 And we can look at what properties it gives us. 270 00:23:33,390 --> 00:23:37,110 So what do we know about this dot product? 271 00:23:37,110 --> 00:23:45,890 Well, we know some properties that it has is that a dot b. 272 00:23:50,425 --> 00:23:52,800 So it doesn't care which order you give the arguments in, 273 00:23:52,800 --> 00:23:53,299 all right? 274 00:23:56,880 --> 00:24:02,350 Also, if I take the same vector, I 275 00:24:02,350 --> 00:24:07,650 know that this is got to be greater than or equal to 0, 276 00:24:07,650 --> 00:24:08,250 right? 277 00:24:08,250 --> 00:24:10,560 Because this is going to be our length. 278 00:24:10,560 --> 00:24:13,910 And the only case where it's 0 is when the vector is 0. 279 00:24:33,990 --> 00:24:38,910 Well we can write this. a dotted in to, say, 280 00:24:38,910 --> 00:24:45,440 b to 1 b1 plus b to 1 b2. 281 00:24:45,440 --> 00:24:50,540 So this b2's are real numbers, and these b's are vectors, 282 00:24:50,540 --> 00:24:51,400 right? 283 00:24:51,400 --> 00:24:56,370 So this thing we can just write is equal to b to one 284 00:24:56,370 --> 00:25:03,000 a dot b1 plus b to 2 a do b2. 285 00:25:06,070 --> 00:25:09,220 And make them vectors everywhere. 286 00:25:09,220 --> 00:25:12,614 OK, so we've got three nice properties. 287 00:25:12,614 --> 00:25:14,280 And you can write down more if you want, 288 00:25:14,280 --> 00:25:18,817 but this will be enough for us. 289 00:25:18,817 --> 00:25:20,900 And the other thing that we can do with this is we 290 00:25:20,900 --> 00:25:23,300 can define the length of a vector, right? 291 00:25:23,300 --> 00:25:28,540 So we can say this is for this defines a length. 292 00:25:40,540 --> 00:25:45,370 And more generally, we only call this the norm of the vector. 293 00:25:52,850 --> 00:26:00,520 And that, of course, you know is that mod a squared is 294 00:26:00,520 --> 00:26:04,710 just equal to a dot a, all right? 295 00:26:04,710 --> 00:26:06,475 So this is our definition of the norm. 296 00:26:10,850 --> 00:26:18,750 OK so this definition over here is really by no means 297 00:26:18,750 --> 00:26:21,330 unique in satisfying these properties. 298 00:26:21,330 --> 00:26:31,960 So if I wrote down something where, instead of just say 299 00:26:31,960 --> 00:26:37,600 a1 b1 plus a1 b2 et cetera, I wrote down some positive number 300 00:26:37,600 --> 00:26:42,620 times a1 b1 times some other positive number a2 b2, 301 00:26:42,620 --> 00:26:45,650 et cetera, that would also satisfy all of these properties 302 00:26:45,650 --> 00:26:47,774 up here. 303 00:26:47,774 --> 00:26:48,565 So it's not unique. 304 00:26:52,700 --> 00:26:57,280 And so you could consider another 305 00:26:57,280 --> 00:27:03,880 the dot product, which we would write 306 00:27:03,880 --> 00:27:17,420 as just c1 a1 b1 plus c2 a2 b2 plus some cn an bn, where 307 00:27:17,420 --> 00:27:23,080 the c's are just positive real numbers. 308 00:27:23,080 --> 00:27:25,510 That would satisfy all of the things 309 00:27:25,510 --> 00:27:27,580 that we know about our standard dot product. 310 00:27:31,180 --> 00:27:33,870 But for obvious reasons, we don't choose to do this, 311 00:27:33,870 --> 00:27:36,540 because it's not a very natural definition 312 00:27:36,540 --> 00:27:40,050 to put these random positive numbers along here. 313 00:27:42,560 --> 00:27:44,460 But we could. 314 00:27:44,460 --> 00:27:46,070 And I guess one other thing that we 315 00:27:46,070 --> 00:27:49,155 have is the Schwarz inequality. 316 00:27:57,650 --> 00:28:03,700 And so this is the a dot b. 317 00:28:06,850 --> 00:28:11,420 So the absolute value of the dot product of a dot b 318 00:28:11,420 --> 00:28:17,090 is less than or equal to the product 319 00:28:17,090 --> 00:28:21,150 of the norms of the vectors, right? 320 00:28:24,830 --> 00:28:27,540 And so one of the problems in the piece 321 00:28:27,540 --> 00:28:29,960 is to consider this in the more abstract sense, 322 00:28:29,960 --> 00:28:34,050 but this is very easy to show for real vectors, right? 323 00:28:37,480 --> 00:28:39,640 So this is all very nice. 324 00:28:39,640 --> 00:28:42,560 So we've talked about Rn. 325 00:28:42,560 --> 00:28:44,890 What we really are going to worry about 326 00:28:44,890 --> 00:28:47,460 is complex vector spaces. 327 00:28:47,460 --> 00:28:49,290 And so there we have a little problem. 328 00:28:49,290 --> 00:28:53,580 And the problem comes in defining 329 00:28:53,580 --> 00:28:55,220 what we mean by a normal, right? 330 00:28:55,220 --> 00:29:01,570 Because if I say now that this vector has complex components 331 00:29:01,570 --> 00:29:03,530 and write this thing here, I'm not 332 00:29:03,530 --> 00:29:06,710 guaranteed that this is a real number, right? 333 00:29:06,710 --> 00:29:09,250 And so I need to be a little bit careful. 334 00:29:09,250 --> 00:29:19,375 So let's just talk about complex spaces. 335 00:29:25,130 --> 00:29:28,360 And we really want to have a useful definition of a length. 336 00:29:28,360 --> 00:29:36,680 So let's let z be in this thing, in interdimensional complex 337 00:29:36,680 --> 00:29:37,180 space. 338 00:29:37,180 --> 00:29:44,990 So really my z is equal to z1 z2 zn, where 339 00:29:44,990 --> 00:29:49,850 the zI line as being in c, right? 340 00:29:49,850 --> 00:29:51,905 So how can define a link for this object? 341 00:29:54,570 --> 00:29:57,015 Well, we have to do it sort of in two steps. 342 00:30:00,070 --> 00:30:02,080 So already know how to define the length 343 00:30:02,080 --> 00:30:04,450 for a complex number, right? 344 00:30:04,450 --> 00:30:08,100 It's just the absolute value, the distance 345 00:30:08,100 --> 00:30:10,750 from the origin in the complex plane. 346 00:30:10,750 --> 00:30:12,690 But now we need to do this in terms 347 00:30:12,690 --> 00:30:15,670 of a more complicated vector space. 348 00:30:15,670 --> 00:30:19,720 And so we can really think of this as 349 00:30:19,720 --> 00:30:28,350 equal to the sum of the squares of z1, of the absolute values 350 00:30:28,350 --> 00:30:29,420 of these complex numbers. 351 00:30:36,670 --> 00:30:37,170 OK? 352 00:30:37,170 --> 00:30:47,195 Which if we write it out, looks like z1 star z1 plus. 353 00:30:54,456 --> 00:30:55,430 OK? 354 00:30:55,430 --> 00:31:02,340 And so we should now, thinking about the inner product, 355 00:31:02,340 --> 00:31:05,130 we should be thinking that the appearance 356 00:31:05,130 --> 00:31:09,510 of complex conjugation is not entirely unnatural. 357 00:31:09,510 --> 00:31:13,370 So if we ask about the length of a vector here, 358 00:31:13,370 --> 00:31:16,900 then that's going to arise from an inner product. 359 00:31:16,900 --> 00:31:17,510 OK? 360 00:31:17,510 --> 00:31:21,090 This object we want to arise from our inner product. 361 00:31:21,090 --> 00:31:29,561 So we can now define our general in a product with the following 362 00:31:29,561 --> 00:31:30,060 axioms. 363 00:31:33,920 --> 00:31:38,475 So firstly, we want to basically maintain the properties 364 00:31:38,475 --> 00:31:40,600 that we've written down here, because we don't want 365 00:31:40,600 --> 00:31:43,831 to make our dot product not being in an inner product 366 00:31:43,831 --> 00:31:44,330 anymore. 367 00:31:44,330 --> 00:31:45,880 That'd be kind of silly. 368 00:31:45,880 --> 00:31:50,640 So let's define our inner product in the following way. 369 00:31:54,000 --> 00:31:55,750 I'm going to write it in a particular way. 370 00:31:55,750 --> 00:32:03,060 So the inner product is going to be, again, a map. 371 00:32:03,060 --> 00:32:06,290 And it's going to take our vector space, two elements 372 00:32:06,290 --> 00:32:11,520 of the vector space to the field. 373 00:32:11,520 --> 00:32:13,200 And I'm in a complex vector space. 374 00:32:20,620 --> 00:32:24,030 So it's a map that I'm going to right like 375 00:32:24,030 --> 00:32:29,835 this that takes v cross v to c. 376 00:32:32,570 --> 00:32:34,730 OK? 377 00:32:34,730 --> 00:32:39,300 And what I mean here is you put the two elements of your vector 378 00:32:39,300 --> 00:32:42,680 space in these positions in this thing, OK? 379 00:32:42,680 --> 00:32:49,530 And so really a b is what I mean by this. 380 00:32:49,530 --> 00:32:54,940 Where a and b-- so let me write it this way. 381 00:32:54,940 --> 00:33:03,800 So this thing is in c for a and b are in the v, right? 382 00:33:03,800 --> 00:33:05,420 So these things dots are where I'm 383 00:33:05,420 --> 00:33:08,720 going to plug-in my vectors. 384 00:33:08,720 --> 00:33:11,005 And so this inner product should satisfy some axioms. 385 00:33:15,200 --> 00:33:19,040 And they look very much like what we've written here. 386 00:33:19,040 --> 00:33:23,060 So the first one is a slight modification. 387 00:33:23,060 --> 00:33:31,600 We want that a b is equal not to b a, 388 00:33:31,600 --> 00:33:35,690 but to its complex conjugate, OK? 389 00:33:35,690 --> 00:33:38,245 And this is related to what I was discussing here. 390 00:33:43,730 --> 00:33:48,510 But from this, we can see that the product of a with itself 391 00:33:48,510 --> 00:33:52,440 is always real, because it and its complex conjugate 392 00:33:52,440 --> 00:33:54,350 are the same. 393 00:33:54,350 --> 00:33:57,450 So we know that a a is real. 394 00:33:57,450 --> 00:34:00,180 And we're also going to demand a definition 395 00:34:00,180 --> 00:34:01,920 of this inner a product that this 396 00:34:01,920 --> 00:34:03,980 is greater than or equal to 0. 397 00:34:03,980 --> 00:34:13,170 And it's only 0 if a equals 0. 398 00:34:13,170 --> 00:34:14,820 Right? 399 00:34:14,820 --> 00:34:17,230 So that's pretty much unchanged. 400 00:34:17,230 --> 00:34:22,060 And then we want the same sort of distributivity. 401 00:34:22,060 --> 00:34:34,260 We do want to have that a inner producted with B to 1 b 402 00:34:34,260 --> 00:34:49,082 plus B to 2 b2 should be equal to B to 1 a b1 plus B to 2 403 00:34:49,082 --> 00:34:58,213 a b2 where the [INAUDIBLE] are just complex numbers, right? 404 00:35:03,420 --> 00:35:07,270 And that's what we need to ask of this. 405 00:35:07,270 --> 00:35:12,270 And then we can make a sensible definitions of it 406 00:35:12,270 --> 00:35:18,360 that will give us a useful norm as well. 407 00:35:18,360 --> 00:35:19,900 Now I'll just make one remark. 408 00:35:19,900 --> 00:35:26,410 This notation here, this is due to Dirac. 409 00:35:31,890 --> 00:35:35,210 And so it's very prevalent in physics. 410 00:35:35,210 --> 00:35:40,480 You will see in most purely mathematical literature 411 00:35:40,480 --> 00:35:48,210 you will see this written just like this. 412 00:35:48,210 --> 00:35:52,210 So let me write it as a b and put these things in explicitly. 413 00:35:59,470 --> 00:36:04,710 And sometimes you'll even see a combination of these written 414 00:36:04,710 --> 00:36:09,520 like this, OK/ But they all mean the same thing. 415 00:36:18,540 --> 00:36:21,730 Compared to what we've written up here, 416 00:36:21,730 --> 00:36:24,610 this seems a little asymmetric between the two items, right? 417 00:36:27,780 --> 00:36:29,510 Well firstly, these are isometric. 418 00:36:29,510 --> 00:36:31,660 And then down here we've shown something 419 00:36:31,660 --> 00:36:34,840 about that we demand something about the second argument, 420 00:36:34,840 --> 00:36:39,090 but we don't demand the same thing about the first argument. 421 00:36:39,090 --> 00:36:41,260 So why not? 422 00:36:44,062 --> 00:36:47,570 Can anyone see? 423 00:36:47,570 --> 00:36:52,561 I guess what we would demand is exactly the same thing 424 00:36:52,561 --> 00:36:53,436 the other way around. 425 00:36:59,780 --> 00:37:06,520 So we would demand another thing that 426 00:37:06,520 --> 00:37:18,140 would be sort of alpha 1 a plus alpha 2 a2 b 427 00:37:18,140 --> 00:37:22,340 is equal to-- well, something like this. 428 00:37:26,230 --> 00:37:30,595 Well, we would actually demand this. 429 00:37:30,595 --> 00:37:32,675 a1 b. 430 00:37:41,830 --> 00:37:44,120 But I don't actually need to demand that, 431 00:37:44,120 --> 00:37:48,020 because that follows from number one, right? 432 00:37:48,020 --> 00:37:50,720 I take axiom one, apply it to this, 433 00:37:50,720 --> 00:37:55,262 and I automatically get this thing here. 434 00:37:55,262 --> 00:37:56,610 OK? 435 00:37:56,610 --> 00:38:02,350 And notice what's arisen is-- actually 436 00:38:02,350 --> 00:38:04,540 let's just go through that, because you really 437 00:38:04,540 --> 00:38:08,286 do want to see these complex conjugates appearing here, 438 00:38:08,286 --> 00:38:09,515 because they are important. 439 00:38:13,010 --> 00:38:16,160 So this follows. 440 00:38:16,160 --> 00:38:20,630 So 1 plus 3, imply this. 441 00:38:20,630 --> 00:38:21,770 Let's just do this. 442 00:38:21,770 --> 00:38:25,570 So let's start with this expression 443 00:38:25,570 --> 00:38:28,050 and start with this piece. 444 00:38:28,050 --> 00:38:29,940 And we know that this will then be 445 00:38:29,940 --> 00:38:40,310 given by axiom one by b alpha 1 a 1 plus alpha 2 a2 complex 446 00:38:40,310 --> 00:38:44,570 conjugate, all right? 447 00:38:44,570 --> 00:38:48,200 And then by this linearity of the second argument 448 00:38:48,200 --> 00:38:51,640 we can now distribute this piece, right? 449 00:38:51,640 --> 00:39:06,260 We can write this is alpha 1 b a1 plus alpha 2 b a2, 450 00:39:06,260 --> 00:39:07,350 all complex conjugated. 451 00:39:10,670 --> 00:39:13,080 Which let's put all the steps in. 452 00:39:13,080 --> 00:39:16,190 Is alpha 1 star and this one star. 453 00:39:26,630 --> 00:39:30,640 And then, again, by the first argument, by the first axiom, 454 00:39:30,640 --> 00:39:33,810 we can flip these and get rid of the complex conjugation. 455 00:39:33,810 --> 00:39:39,330 And that gives us this one up here, right? 456 00:39:39,330 --> 00:39:42,370 So we only need to define this linearity, distributive 457 00:39:42,370 --> 00:39:44,130 property on one side of this thing. 458 00:39:44,130 --> 00:39:46,463 We could have chosen to define it here and wouldn't have 459 00:39:46,463 --> 00:39:49,200 needed that one, but we didn't. 460 00:39:49,200 --> 00:39:54,685 OK, so let's look at a couple of examples. 461 00:39:59,600 --> 00:40:03,520 And the first one is a finite dimensional example. 462 00:40:03,520 --> 00:40:07,660 And we're going to take v is equal to cn. 463 00:40:07,660 --> 00:40:09,440 And our definition is going to be 464 00:40:09,440 --> 00:40:11,960 a pretty natural generalization of what 465 00:40:11,960 --> 00:40:13,335 we've written down before. 466 00:40:17,400 --> 00:40:19,980 So a and b are elements of cn. 467 00:40:19,980 --> 00:40:25,200 And this is just going to be a1 star 468 00:40:25,200 --> 00:40:34,975 b1 plus a2 star b2 plus an star bn. 469 00:40:38,800 --> 00:40:41,590 Another piece of chalk. 470 00:40:41,590 --> 00:40:46,650 So the only difference from dot product in real vector space 471 00:40:46,650 --> 00:40:49,540 is that we've put this complex conjugates here. 472 00:40:49,540 --> 00:40:51,785 And that you can check satisfies all of these axioms. 473 00:40:56,240 --> 00:40:58,590 Another example is actually an example 474 00:40:58,590 --> 00:41:02,890 of an infinite dimensional vector space. 475 00:41:02,890 --> 00:41:15,960 Let's take v is the set of all complex functions, 476 00:41:15,960 --> 00:41:27,180 all f of x that are in c with x living in some finite interval. 477 00:41:30,275 --> 00:41:30,775 OK? 478 00:41:35,540 --> 00:41:39,760 And so a natural norm to define on this space-- and this 479 00:41:39,760 --> 00:41:42,260 is something that we can certainly talk about 480 00:41:42,260 --> 00:41:49,260 in recitations-- is that if I have f and g in this vector 481 00:41:49,260 --> 00:41:56,990 space v, then f g I'm going to define-- 482 00:41:56,990 --> 00:42:00,126 this is my definition of what the dot product is-- 483 00:42:00,126 --> 00:42:06,961 is the integral from 0 to l f star of x g of x dx. 484 00:42:09,916 --> 00:42:10,416 OK? 485 00:42:15,200 --> 00:42:19,100 If you think of this as arising from evaluating 486 00:42:19,100 --> 00:42:21,330 f at a set of discrete points where you've 487 00:42:21,330 --> 00:42:23,210 got a finite dimensional vector space, 488 00:42:23,210 --> 00:42:26,390 and then letting the space between those points go to 0, 489 00:42:26,390 --> 00:42:30,090 this is kind of the natural thing to arise. 490 00:42:30,090 --> 00:42:32,730 It's really an integral as a limit of a sum. 491 00:42:32,730 --> 00:42:35,610 And over here, of course, I could write this one 492 00:42:35,610 --> 00:42:40,850 as just the sum over i of ai star bi. 493 00:42:40,850 --> 00:42:44,500 i equals 1 to n. 494 00:42:44,500 --> 00:42:48,560 And so this is the integral is infinite dimensional 495 00:42:48,560 --> 00:42:51,320 generalization of the sum, and so we have this. 496 00:42:53,960 --> 00:42:58,720 And that might be something to talk about in recitations. 497 00:42:58,720 --> 00:42:59,220 OK? 498 00:43:02,110 --> 00:43:04,370 So we've gone from having just a vector 499 00:43:04,370 --> 00:43:08,820 space to having a vector space where we've 500 00:43:08,820 --> 00:43:14,270 added this new operation on it, this inner product operation. 501 00:43:14,270 --> 00:43:18,770 And that lets us do things that we couldn't do before. 502 00:43:18,770 --> 00:43:29,825 So firstly, it lets us talk about orthogonality. 503 00:43:39,680 --> 00:43:41,720 Previously we couldn't ask any question 504 00:43:41,720 --> 00:43:43,967 about two objects within our vector space. 505 00:43:43,967 --> 00:43:45,925 This let's us ask a question about two objects. 506 00:43:49,350 --> 00:44:07,240 So if we have the inner product a b in some vector space V, 507 00:44:07,240 --> 00:44:15,570 then if this is 0, we say they're orthogonal. 508 00:44:15,570 --> 00:44:21,690 We say that the vectors a and b are orthogonal. 509 00:44:31,440 --> 00:44:33,800 And I'm sure you know what orthogonal means 510 00:44:33,800 --> 00:44:37,340 in terms of Rn, but this is just the statement 511 00:44:37,340 --> 00:44:42,700 of what it means in a abstract vector space. 512 00:44:42,700 --> 00:44:44,413 This is the definition of orthogonality. 513 00:44:53,790 --> 00:44:56,170 So this is one thing if we have a set of vectors. 514 00:45:00,610 --> 00:45:16,950 e1 e2 en, such that ei ej is equal to delta ij, chronic 515 00:45:16,950 --> 00:45:21,280 to delta ij, this set is orthonormal. 516 00:45:31,590 --> 00:45:33,810 Again, a word you've seen many times. 517 00:45:40,300 --> 00:45:55,940 OK, so we can also define the components of vectors 518 00:45:55,940 --> 00:46:00,195 now in basis dependent way. 519 00:46:11,650 --> 00:46:15,450 We're going to choose ei to be a set of vectors in our vector 520 00:46:15,450 --> 00:46:24,030 space V. We previously had things that 521 00:46:24,030 --> 00:46:35,460 form a basis, a basis of V. And if we also 522 00:46:35,460 --> 00:46:49,670 demand that they're orthonormal, then we can-- well, 523 00:46:49,670 --> 00:46:51,670 we can always decompose any vector in V 524 00:46:51,670 --> 00:46:53,530 in terms of its basis, right? 525 00:46:53,530 --> 00:46:55,770 But if it's also orthonormal, then we 526 00:46:55,770 --> 00:47:01,970 can write a, which is a is some vector in V. 527 00:47:01,970 --> 00:47:12,460 a is equal to the sum over i equals 1 to n of some ai ei. 528 00:47:12,460 --> 00:47:14,740 So we can do that for any basis. 529 00:47:14,740 --> 00:47:20,940 But then we can take this vector and form its inner product 530 00:47:20,940 --> 00:47:23,090 with the basis vectors. 531 00:47:23,090 --> 00:47:31,350 So we can look at what ek a is, right? 532 00:47:31,350 --> 00:47:34,880 So we have our basis vectors ek, and we take one of them 533 00:47:34,880 --> 00:47:37,470 and we dot product it into this vector here. 534 00:47:37,470 --> 00:47:39,700 And this is straightforward to c. 535 00:47:39,700 --> 00:47:44,160 This is going to be equal to the sum over i equals 1 to n ai. 536 00:47:44,160 --> 00:47:48,091 And then it's going to be the inner product of ek with ei, 537 00:47:48,091 --> 00:47:48,590 right? 538 00:47:48,590 --> 00:47:53,919 Because of this distributive property here. 539 00:47:53,919 --> 00:47:56,190 OK? 540 00:47:56,190 --> 00:48:01,540 But we also know that, because this is an orthonormal basis, 541 00:48:01,540 --> 00:48:07,600 this thing here is a delta function, delta ik, right? 542 00:48:10,380 --> 00:48:13,540 And so I can, in fact, do this sum, and I get 543 00:48:13,540 --> 00:48:17,580 and this is equal to ak. 544 00:48:17,580 --> 00:48:19,420 And so we've defined what we mean 545 00:48:19,420 --> 00:48:25,042 by a component of this vector in this basis ei. 546 00:48:25,042 --> 00:48:26,625 They're defined by this inner product. 547 00:48:33,880 --> 00:48:41,130 So we can also talk about the norm, which 548 00:48:41,130 --> 00:48:49,910 I think, unsurprisingly, we are going to take the norm 549 00:48:49,910 --> 00:48:53,160 to be, again, equal to this, just as we did in Rn, 550 00:48:53,160 --> 00:48:58,860 but now it's the more general definition of my inner product 551 00:48:58,860 --> 00:48:59,950 that defines our norm. 552 00:49:04,700 --> 00:49:07,330 And because of our axiom-- so because 553 00:49:07,330 --> 00:49:11,260 of number two in particular, this is a sensible norm. 554 00:49:11,260 --> 00:49:14,000 It's always going to be greater than or equal to 0. 555 00:49:18,460 --> 00:49:18,960 OK? 556 00:49:18,960 --> 00:49:25,000 And conveniently we can also change this Schwarz inequality. 557 00:49:25,000 --> 00:49:30,550 So instead of the one that's specific to Rn, 558 00:49:30,550 --> 00:49:37,095 that becomes a b. 559 00:49:40,110 --> 00:49:45,990 All right, so let's cross that one out. 560 00:49:45,990 --> 00:49:47,750 This is what it becomes. 561 00:49:47,750 --> 00:49:50,020 And in the current p set, you've got 562 00:49:50,020 --> 00:49:54,130 to prove this is true, right? 563 00:49:54,130 --> 00:50:03,190 We can also write down a triangle inequality, 564 00:50:03,190 --> 00:50:07,810 which is really something that norms should satisfy. 565 00:50:07,810 --> 00:50:13,510 So the norm of a plus b should be less than 566 00:50:13,510 --> 00:50:18,060 or equal to the norm of a plus the norm of b. 567 00:50:18,060 --> 00:50:24,610 And the R3 version of this is the longest side 568 00:50:24,610 --> 00:50:27,830 of a triangle is shorter than the two shorter sides, right? 569 00:50:27,830 --> 00:50:32,800 So this is fine. 570 00:50:32,800 --> 00:50:40,320 OK, so you might ask why we're doing 571 00:50:40,320 --> 00:50:43,900 all of this seemingly abstract mathematics. 572 00:50:43,900 --> 00:50:47,290 Well, so now we're in a place where we can actually 573 00:50:47,290 --> 00:50:54,120 talk about the space where all of our quantum states 574 00:50:54,120 --> 00:50:56,590 are going to live. 575 00:50:56,590 --> 00:50:58,900 And so these inner product space-- 576 00:50:58,900 --> 00:51:02,767 these vector spaces that we've given an inner product. 577 00:51:02,767 --> 00:51:04,350 We can call them inner product spaces. 578 00:51:07,210 --> 00:51:22,590 So we have a vector space with an inner product 579 00:51:22,590 --> 00:51:27,065 is actually we call a Hilbert space. 580 00:51:32,270 --> 00:51:35,010 And so this needs a little qualifier. 581 00:51:35,010 --> 00:51:37,220 So if this is a finite dimensional vector space, 582 00:51:37,220 --> 00:51:39,100 then this is straight. 583 00:51:42,900 --> 00:51:45,940 It is just a Hilbert space. 584 00:51:45,940 --> 00:51:47,560 Let me write it here. 585 00:51:47,560 --> 00:51:55,320 So let's write it as a finite dimensional vector 586 00:51:55,320 --> 00:51:59,660 space with an inner product is a Hilbert space. 587 00:51:59,660 --> 00:52:02,092 But if we have an infinite dimensional vector space, 588 00:52:02,092 --> 00:52:03,550 we need to be a little bit careful. 589 00:52:11,370 --> 00:52:16,510 For an infinite dimensional vector space, 590 00:52:16,510 --> 00:52:17,945 we again need an inner product. 591 00:52:26,130 --> 00:52:30,400 We need to make sure that this space is complete. 592 00:52:30,400 --> 00:52:32,510 OK? 593 00:52:32,510 --> 00:52:34,560 And this is a kind of technical point 594 00:52:34,560 --> 00:52:37,680 that I don't want to spend too much time on, 595 00:52:37,680 --> 00:52:42,960 but if you think about-- well, let me just write it down. 596 00:52:42,960 --> 00:52:43,780 Vector space. 597 00:52:48,260 --> 00:52:49,250 Let me write it here. 598 00:52:52,670 --> 00:52:55,945 And I haven't defined what this complete vector space means. 599 00:52:55,945 --> 00:52:57,820 But if we have an infinite dimensional vector 600 00:52:57,820 --> 00:53:00,420 space that is complete or we make it complete, 601 00:53:00,420 --> 00:53:01,670 then we have an inner product. 602 00:53:01,670 --> 00:53:02,836 We also get a Hilbert space. 603 00:53:07,320 --> 00:53:11,147 And all of quantum mechanical states live in a Hilbert space. 604 00:53:11,147 --> 00:53:12,063 AUDIENCE: [INAUDIBLE]. 605 00:53:20,970 --> 00:53:22,700 PROFESSOR: Yes, that's true. 606 00:53:22,700 --> 00:53:24,940 So how's that? 607 00:53:29,330 --> 00:53:32,170 So we need to define what we mean by complete though, right? 608 00:53:32,170 --> 00:53:35,285 So I don't want to spend much time on this. 609 00:53:35,285 --> 00:53:36,535 But we can just do an example. 610 00:53:41,260 --> 00:53:54,350 If we take the space of-- let V equal the space of polynomials 611 00:53:54,350 --> 00:54:01,270 on an interval 0 to L, say. 612 00:54:07,570 --> 00:54:10,510 So this means I've got all pn's. 613 00:54:14,760 --> 00:54:21,300 P0 plus p1x plus pn xn. 614 00:54:26,430 --> 00:54:31,430 There are things that will live in the completed vector 615 00:54:31,430 --> 00:54:34,750 space that are not of this form here. 616 00:54:34,750 --> 00:54:38,822 So for example, if I take n larger and larger, 617 00:54:38,822 --> 00:54:40,280 I could write down this polynomial. 618 00:54:40,280 --> 00:54:52,945 I could write pn of x is the sum over i equals 1 up to n of i-- 619 00:54:52,945 --> 00:54:58,470 x to the i over i factorial, right? 620 00:54:58,470 --> 00:55:03,860 And all of pn's live in this space of polynomials. 621 00:55:03,860 --> 00:55:06,350 But their limit, as n becomes large, 622 00:55:06,350 --> 00:55:12,210 there's a sequence of these call it a cushy sequence 623 00:55:12,210 --> 00:55:17,840 that, as n goes to infinity, I generate something 624 00:55:17,840 --> 00:55:19,790 that's actually not a polynomial. 625 00:55:19,790 --> 00:55:25,570 So I generate e of x, which lives 626 00:55:25,570 --> 00:55:31,490 in the completion of this, but itself not a polynomial. 627 00:55:35,070 --> 00:55:38,120 Don't worry about this too much, but in order 628 00:55:38,120 --> 00:55:39,570 to really define a Hilbert space, 629 00:55:39,570 --> 00:55:40,986 we have to be a little bit careful 630 00:55:40,986 --> 00:55:42,445 for infinite dimensional cases. 631 00:55:45,760 --> 00:55:53,580 OK, so a few more things that we can do to talk about. 632 00:55:53,580 --> 00:56:02,300 Well how do we make a orthonormal basis? 633 00:56:02,300 --> 00:56:08,440 So I presume you've all heard of Gram-Schmidt? 634 00:56:08,440 --> 00:56:10,126 The Gram-Schmidt procedure? 635 00:56:10,126 --> 00:56:11,020 Yep. 636 00:56:11,020 --> 00:56:16,340 OK, so that's how we make a orthonormal basis. 637 00:56:16,340 --> 00:56:20,130 And just the way you do it in R3, 638 00:56:20,130 --> 00:56:22,527 you do it the same way in your arbitrary vector space. 639 00:56:22,527 --> 00:56:24,110 So we have the Gram-Schmidt procedure. 640 00:56:37,990 --> 00:56:47,950 So you can define this-- so we have a list v1, v2, vn are just 641 00:56:47,950 --> 00:56:52,365 vectors in our vector space that are linearly independent. 642 00:56:59,630 --> 00:57:05,425 So we can construct another list. 643 00:57:24,130 --> 00:57:27,270 There's also orthonormal, so it's a very useful thing for us 644 00:57:27,270 --> 00:57:28,880 to have. 645 00:57:28,880 --> 00:57:31,530 And so you could define this recursively. 646 00:57:31,530 --> 00:57:39,500 You can just write that ej is equal to vj 647 00:57:39,500 --> 00:57:47,360 minus the sum over i less than j of ei. 648 00:58:04,030 --> 00:58:07,250 So this thing divided by its length. 649 00:58:07,250 --> 00:58:14,750 And so by the sum, you're orthogonalizing 650 00:58:14,750 --> 00:58:17,410 ej versus all of the previous ei's that you've already 651 00:58:17,410 --> 00:58:18,550 defined. 652 00:58:18,550 --> 00:58:22,650 And then you normalize it by dividing by its length, right? 653 00:58:22,650 --> 00:58:28,200 So that's something that's very useful. 654 00:58:28,200 --> 00:58:32,240 And the last thing I want to say about these inner product 655 00:58:32,240 --> 00:58:37,600 spaces is that we can use them-- these inner products at least, 656 00:58:37,600 --> 00:58:40,910 is that we can use them to find the orthogonal complement 657 00:58:40,910 --> 00:58:45,390 of something, of anything really. 658 00:58:50,490 --> 00:59:01,760 So let's let u-- so we have a vector space V, 659 00:59:01,760 --> 00:59:04,700 and I can just choose some things in that and make a set. 660 00:59:07,860 --> 00:59:16,358 So u is the set of v that are in V. 661 00:59:16,358 --> 00:59:18,030 So it doesn't need to be a subspace. 662 00:59:18,030 --> 00:59:18,900 It's just a set. 663 00:59:23,770 --> 00:59:27,030 For example, if v Rn, I could just 664 00:59:27,030 --> 00:59:31,510 choose vectors pointing along two directions, 665 00:59:31,510 --> 00:59:32,760 and that would give me my set. 666 00:59:32,760 --> 00:59:37,490 But that's not a subspace, because it doesn't contain 667 00:59:37,490 --> 00:59:39,490 some multiple of this vector times some multiple 668 00:59:39,490 --> 00:59:42,110 of this vector, which would be pointing over here. 669 00:59:42,110 --> 00:59:44,180 So this is just a set so far. 670 00:59:47,630 --> 00:59:56,050 We can define u perpendicular, which 671 00:59:56,050 --> 01:00:09,240 we'll call the orthogonal complement of u. 672 01:00:09,240 --> 01:00:12,310 And this is defined as u perpendicular 673 01:00:12,310 --> 01:00:20,200 is equal to the set of v's in V such 674 01:00:20,200 --> 01:00:34,270 that v u is equal to 0 for all u in U. All of the things 675 01:00:34,270 --> 01:00:37,690 that live in this space are orthogonal to everything 676 01:00:37,690 --> 01:00:40,560 that lives in U. OK 677 01:00:40,560 --> 01:00:45,705 And in fact, this one is a subspace automatically. 678 01:00:51,381 --> 01:00:52,380 So it is a vector space. 679 01:00:55,460 --> 01:00:59,440 So if I took my example of choosing the x direction and y 680 01:00:59,440 --> 01:01:03,910 direction for my set here, then everything perpendicular 681 01:01:03,910 --> 01:01:05,480 to the x direction and y direction 682 01:01:05,480 --> 01:01:09,100 is actually everything perpendicular to the xy-plane, 683 01:01:09,100 --> 01:01:14,190 and so that is actually a subspace of R3. 684 01:01:18,900 --> 01:01:23,070 And so there's a nice theorem that you can think about, 685 01:01:23,070 --> 01:01:26,630 but it's actually kind of obvious. 686 01:01:26,630 --> 01:01:35,050 So if u is a subspace, then I can actually 687 01:01:35,050 --> 01:01:39,360 write that V is equal to the direct sum of U 688 01:01:39,360 --> 01:01:43,766 plus its orthogonal complement, OK? 689 01:01:48,930 --> 01:01:52,940 So that one's kind of fairly straightforward to prove, 690 01:01:52,940 --> 01:01:54,810 but we won't do it now. 691 01:01:54,810 --> 01:01:58,110 OK, so in the last little bit, I want 692 01:01:58,110 --> 01:02:04,354 to talk more about this notation that I've introduced, 693 01:02:04,354 --> 01:02:05,270 that Dirac introduced. 694 01:02:08,110 --> 01:02:09,650 What can we say? 695 01:02:09,650 --> 01:02:14,206 If I can find a [INAUDIBLE] here. 696 01:02:14,206 --> 01:02:16,170 Are there any questions about this? 697 01:02:16,170 --> 01:02:17,505 Yep. 698 01:02:17,505 --> 01:02:22,024 AUDIENCE: So when we find space and the idea of basis balance, 699 01:02:22,024 --> 01:02:24,730 why is that [INAUDIBLE] decompose things 700 01:02:24,730 --> 01:02:27,436 into plane waves when we're not actually [INAUDIBLE]? 701 01:02:31,600 --> 01:02:37,860 PROFESSOR: So it's because it's-- basically it works. 702 01:02:37,860 --> 01:02:39,980 Mathematically, we're doing things 703 01:02:39,980 --> 01:02:41,970 that are not quite legitimate. 704 01:02:44,480 --> 01:02:54,360 And so we can generalize the Hilbert space a little bit, 705 01:02:54,360 --> 01:02:56,990 such that these non normalizable things can 706 01:02:56,990 --> 01:03:00,060 live in this generalized space. 707 01:03:00,060 --> 01:03:02,880 But really the answer is that it works, 708 01:03:02,880 --> 01:03:05,630 but no physical system is going to correspond to something 709 01:03:05,630 --> 01:03:06,190 like that. 710 01:03:06,190 --> 01:03:07,890 So if I take plane waves, that's not 711 01:03:07,890 --> 01:03:11,980 a physically realizable thing. 712 01:03:11,980 --> 01:03:14,440 It gives us an easy way to, instead 713 01:03:14,440 --> 01:03:17,580 of talking about some wave packet 714 01:03:17,580 --> 01:03:20,140 that some superposition of plane waves, 715 01:03:20,140 --> 01:03:22,250 we can talk about the plane waves by themselves 716 01:03:22,250 --> 01:03:24,715 and then form the wave packet afterwards, for example. 717 01:03:29,165 --> 01:03:32,040 Does that answer the question a little bit at least? 718 01:03:36,270 --> 01:03:37,160 Yep. 719 01:03:37,160 --> 01:03:41,489 AUDIENCE: If p could be written as a sum of U [INAUDIBLE], 720 01:03:41,489 --> 01:03:45,430 why is U not [INAUDIBLE]? 721 01:03:45,430 --> 01:03:47,480 PROFESSOR: Well, just think about the case 722 01:03:47,480 --> 01:03:48,615 that I was talking about. 723 01:03:55,480 --> 01:03:58,720 So if we're looking at R3 and we take 724 01:03:58,720 --> 01:04:04,700 U to be the set the unit vector in the x direction, 725 01:04:04,700 --> 01:04:06,550 the unit vector in the y direction, 726 01:04:06,550 --> 01:04:08,210 that's not a subspace, as I said, 727 01:04:08,210 --> 01:04:10,964 because I can take the unit vector in the x direction 728 01:04:10,964 --> 01:04:12,630 plus the unit vector in the y direction. 729 01:04:12,630 --> 01:04:15,520 It goes in the 45 degree direction. 730 01:04:15,520 --> 01:04:19,620 And it's not in the things I've written down originally. 731 01:04:19,620 --> 01:04:22,780 So then if I talk about the subspace, 732 01:04:22,780 --> 01:04:30,200 the things spanned by x hat and y hat, then I have a subspace. 733 01:04:30,200 --> 01:04:33,580 It's the whole xy-plane. 734 01:04:33,580 --> 01:04:38,050 And the things are orthogonal to it 735 01:04:38,050 --> 01:04:43,505 in R3 are just the things proportion to z hat. 736 01:04:46,670 --> 01:04:55,080 And so then I've got the things in this x hat and y hat, 737 01:04:55,080 --> 01:04:57,260 and the thing that's in here is z hat. 738 01:04:57,260 --> 01:05:03,730 And so that really is the basis for my R3 that I started with. 739 01:05:03,730 --> 01:05:07,060 That contains everything. 740 01:05:07,060 --> 01:05:13,680 And more generally, the reason I need to make this a subspace 741 01:05:13,680 --> 01:05:23,200 is just because-- so I define U by some set of vectors 742 01:05:23,200 --> 01:05:25,260 that I'm putting into it. 743 01:05:25,260 --> 01:05:27,020 The things that are orthogonal to that 744 01:05:27,020 --> 01:05:31,382 are automatically already everything 745 01:05:31,382 --> 01:05:32,840 that's orthogonal to it, so there's 746 01:05:32,840 --> 01:05:36,720 no combination of the things in the orthogonal complement 747 01:05:36,720 --> 01:05:39,690 that's not already in that complement. 748 01:05:39,690 --> 01:05:44,480 Because I'm saying that this is everything in V 749 01:05:44,480 --> 01:05:50,610 that's orthogonal to these things in this subspace. 750 01:05:50,610 --> 01:05:53,720 So I could write down some arbitrary vector v, 751 01:05:53,720 --> 01:05:59,590 and I could aways write it as a projection 752 01:05:59,590 --> 01:06:01,590 onto things that live in here and things 753 01:06:01,590 --> 01:06:04,850 that don't live in this one, right? 754 01:06:04,850 --> 01:06:09,940 And what I'm doing by defining this complement is 755 01:06:09,940 --> 01:06:15,200 I'm getting rid of the bits that are proportional to things 756 01:06:15,200 --> 01:06:21,150 in this, OK? 757 01:06:21,150 --> 01:06:22,219 All right any-- yep? 758 01:06:22,219 --> 01:06:23,760 AUDIENCE: So an orthogonal complement 759 01:06:23,760 --> 01:06:25,206 is automatically a subspace? 760 01:06:25,206 --> 01:06:25,830 PROFESSOR: Yes. 761 01:06:25,830 --> 01:06:27,862 AUDIENCE: But that doesn't necessarily 762 01:06:27,862 --> 01:06:32,437 mean that any random collection of vectors is a subspace. 763 01:06:32,437 --> 01:06:33,020 PROFESSOR: No. 764 01:06:41,680 --> 01:06:44,400 All right, so let's move on and talk 765 01:06:44,400 --> 01:06:48,600 about the Dirac's notation. 766 01:06:48,600 --> 01:06:51,610 And let's do it here. 767 01:06:56,710 --> 01:07:01,670 So three or four lectures ago, we 768 01:07:01,670 --> 01:07:05,310 started talking about these objects, 769 01:07:05,310 --> 01:07:09,210 and we were calling them kets, right? 770 01:07:09,210 --> 01:07:12,570 And they were things that live in our vector space V. 771 01:07:12,570 --> 01:07:17,000 So these are just a way of writing down our vectors, OK? 772 01:07:23,200 --> 01:07:25,640 So when I write down the inner product, 773 01:07:25,640 --> 01:07:29,210 which we have on the wall above, one of the bits of it 774 01:07:29,210 --> 01:07:31,810 looks lot like this. 775 01:07:31,810 --> 01:07:40,450 So we can really think of a b, the b being a ket. 776 01:07:40,450 --> 01:07:42,600 We know that b is a vector, and here we're 777 01:07:42,600 --> 01:07:45,240 writing in a particular way of writing things 778 01:07:45,240 --> 01:07:47,670 in terms of a ket. 779 01:07:47,670 --> 01:07:50,720 And what we can do is actually think 780 01:07:50,720 --> 01:07:54,810 about breaking this object, this inner product 781 01:07:54,810 --> 01:07:58,970 up into two pieces. 782 01:07:58,970 --> 01:08:02,910 So remember the dot product is taking two vectors, a and b. 783 01:08:02,910 --> 01:08:06,240 One of them, we already have written it like a vector, 784 01:08:06,240 --> 01:08:09,500 because a ket is a vector. 785 01:08:09,500 --> 01:08:12,450 What Dirac did in breaking this up is he said, 786 01:08:12,450 --> 01:08:19,330 OK, well this thing is a bracket, 787 01:08:19,330 --> 01:08:25,740 and so he's going to call this one a ket, and this is a bra. 788 01:08:25,740 --> 01:08:29,874 So this object with something it. 789 01:08:29,874 --> 01:08:31,290 The things inside these you should 790 01:08:31,290 --> 01:08:33,960 think of as just labeling these things. 791 01:08:33,960 --> 01:08:34,760 OK? 792 01:08:34,760 --> 01:08:37,700 Now we already know this thing here. 793 01:08:37,700 --> 01:08:39,859 So these kets are things that live 794 01:08:39,859 --> 01:08:46,119 in-- I should say this is direct notation. 795 01:08:51,040 --> 01:08:53,450 OK, so we already know these kets 796 01:08:53,450 --> 01:08:56,819 are things that live in the vector space. 797 01:08:56,819 --> 01:09:01,000 But what are the bras? 798 01:09:01,000 --> 01:09:09,250 Well, they're not vectors in V. So b is a vector, 799 01:09:09,250 --> 01:09:10,779 so maybe I should've called this one 800 01:09:10,779 --> 01:09:13,050 b to be a little less confusing. 801 01:09:13,050 --> 01:09:15,510 So b is a ket, and this is something 802 01:09:15,510 --> 01:09:19,170 that lives in our vector space V. 803 01:09:19,170 --> 01:09:23,950 This inner product we're writing in terms of bra and a ket. 804 01:09:23,950 --> 01:09:30,060 The bra, what does it actually do? 805 01:09:30,060 --> 01:09:35,200 So I'm going to use it to make this inner product. 806 01:09:35,200 --> 01:09:40,060 And so what it's doing is it's taking a vector 807 01:09:40,060 --> 01:09:42,454 and returning a complex number. 808 01:09:45,689 --> 01:09:51,080 The inner product takes v cross v goes to c. 809 01:09:51,080 --> 01:09:56,000 But if I think of it as the action of this bra on this ket, 810 01:09:56,000 --> 01:09:59,740 then the action is that this bra eats a vector 811 01:09:59,740 --> 01:10:02,400 and spits back a complex number, OK? 812 01:10:02,400 --> 01:10:04,610 So a is actually a map. 813 01:10:13,510 --> 01:10:14,010 OK? 814 01:10:14,010 --> 01:10:17,950 So these bras live in a very different place 815 01:10:17,950 --> 01:10:20,898 than the kets do. 816 01:10:20,898 --> 01:10:23,740 Although they are going to be very closely related. 817 01:10:23,740 --> 01:10:30,980 So firstly, it's not in V. You should be careful if you ever 818 01:10:30,980 --> 01:10:32,790 say that, because it's not right. 819 01:10:32,790 --> 01:10:45,720 We actually say that it belongs to a dual space, which 820 01:10:45,720 --> 01:10:52,590 we label as V star, because it is very dependent on V, right? 821 01:10:52,590 --> 01:10:54,510 It's mapped from V to c. 822 01:10:57,950 --> 01:11:00,110 And I shouldn't even say this is a linear map. 823 01:11:10,330 --> 01:11:13,410 Now what is V star? 824 01:11:13,410 --> 01:11:16,230 Well, at the moment it's just the space 825 01:11:16,230 --> 01:11:19,040 of all linear maps from V to c. 826 01:11:19,040 --> 01:11:22,630 Me But it itself is a vector space. 827 01:11:22,630 --> 01:11:31,405 So we can define addition of these maps. 828 01:11:34,760 --> 01:11:39,990 We can define addition on V star and also a scalar modification 829 01:11:39,990 --> 01:11:40,745 of these maps. 830 01:11:45,960 --> 01:11:51,430 And so what that means is that I can define some bra 831 01:11:51,430 --> 01:12:01,210 w That's equal to alpha lots of another one plus B to b. 832 01:12:01,210 --> 01:12:03,950 And all of these live in this V star space. 833 01:12:09,064 --> 01:12:10,950 Let me write that explicitly. 834 01:12:10,950 --> 01:12:22,120 So a b and w live in V star, OK? 835 01:12:22,120 --> 01:12:23,950 And the way we define this is actually 836 01:12:23,950 --> 01:12:27,750 through the inner product. 837 01:12:27,750 --> 01:12:48,850 We define it such that-- so I take 838 01:12:48,850 --> 01:12:53,270 all vectors v in the vector space big V, 839 01:12:53,270 --> 01:12:57,510 and the definition of w is that this holds. 840 01:13:00,660 --> 01:13:04,060 And then basically from the properties 841 01:13:04,060 --> 01:13:08,180 of the inner product, you inherit the vector structure, 842 01:13:08,180 --> 01:13:11,120 the vector space structure. 843 01:13:11,120 --> 01:13:14,345 So this tells us V star is a vector space. 844 01:13:31,260 --> 01:13:32,290 Let's go over here. 845 01:13:36,480 --> 01:13:38,030 And there's actually a correspondence 846 01:13:38,030 --> 01:13:42,000 between the objects in the original vector space V 847 01:13:42,000 --> 01:13:43,450 and those that live in V star. 848 01:13:46,230 --> 01:13:54,070 So we can say for any v in V, there's 849 01:13:54,070 --> 01:13:57,740 a unique-- I should write it like this. 850 01:13:57,740 --> 01:14:02,190 Any ket v in the vector space, there 851 01:14:02,190 --> 01:14:09,230 is a unique bra, which I'm also going to label by v, 852 01:14:09,230 --> 01:14:11,265 and this lives in V star. 853 01:14:14,340 --> 01:14:20,080 And so we can show uniqueness by assuming it doesn't work. 854 01:14:20,080 --> 01:14:37,870 So let's assume that there exists a v and a v 855 01:14:37,870 --> 01:14:57,050 prime in here such that v-- so we'll 856 01:14:57,050 --> 01:15:01,200 assume that this one is not unique, 857 01:15:01,200 --> 01:15:04,250 but there are two things, v and v prime. 858 01:15:04,250 --> 01:15:10,200 And then we can construct-- from this, 859 01:15:10,200 --> 01:15:11,830 I can take this over to this side here, 860 01:15:11,830 --> 01:15:18,090 and I just get that v w minus v prime w 861 01:15:18,090 --> 01:15:30,260 is equal to 0, which I can then use the skew 862 01:15:30,260 --> 01:15:32,530 symmetry of these objects to write 863 01:15:32,530 --> 01:15:45,920 as w v minus w v prime star. 864 01:15:45,920 --> 01:15:49,020 So I've just changed the order of both of them. 865 01:15:49,020 --> 01:15:53,590 And then I can use the property of kets. 866 01:15:53,590 --> 01:15:56,130 I can combine them linearly. 867 01:15:56,130 --> 01:16:06,700 So I know this is equal to w v minus v prime star. 868 01:16:06,700 --> 01:16:10,730 And essentially, that's it, because I 869 01:16:10,730 --> 01:16:15,280 know this has to be true for every w in the vector space V. 870 01:16:15,280 --> 01:16:16,800 So this thing is equal to 0. 871 01:16:20,360 --> 01:16:25,510 And so the only thing that can annihilate every other vector 872 01:16:25,510 --> 01:16:29,290 is going to be 0 for my definition, in fact, 873 01:16:29,290 --> 01:16:31,950 of the inner product. 874 01:16:31,950 --> 01:16:36,390 So this implies that v minus v prime 875 01:16:36,390 --> 01:16:42,980 equals 0, the null vector, which implies that v equals v prime. 876 01:16:42,980 --> 01:16:49,370 So our assumption was wrong, and so this is unique. 877 01:16:49,370 --> 01:16:52,080 OK, let's see. 878 01:16:54,890 --> 01:16:58,540 And so we actually have really a one 879 01:16:58,540 --> 01:17:01,840 to one correspondence between things in the vector space 880 01:17:01,840 --> 01:17:07,440 and things in the joule space, OK? 881 01:17:07,440 --> 01:17:12,370 And so we can actually label the bras 882 01:17:12,370 --> 01:17:16,050 by the same thing that's labeling the kets. 883 01:17:16,050 --> 01:17:24,510 So I can really do what I've done in the top line up there 884 01:17:24,510 --> 01:17:27,040 and have something-- everything is labeled 885 01:17:27,040 --> 01:17:31,540 by the same little v. Both the thing in the big vector 886 01:17:31,540 --> 01:17:33,510 space, big V, and the thing in V star 887 01:17:33,510 --> 01:17:36,750 are label by the same thing. 888 01:17:36,750 --> 01:17:48,050 And more generally, I could say that v-- 889 01:17:48,050 --> 01:17:52,480 so there's a correspondence between this thing 890 01:17:52,480 --> 01:18:04,790 and this thing. 891 01:18:08,080 --> 01:18:12,240 And notice the stars appearing here. 892 01:18:12,240 --> 01:18:16,680 They came out of how we define the inner product. 893 01:18:21,030 --> 01:18:27,410 OK, so really, in fact, any linear map you write down, 894 01:18:27,410 --> 01:18:33,100 any linear map like this defines one of these bras, 895 01:18:33,100 --> 01:18:35,040 because every linear map that takes 896 01:18:35,040 --> 01:18:36,860 to V to c lives in V star. 897 01:18:36,860 --> 01:18:39,872 So there has to be an element that corresponds to it. 898 01:18:43,110 --> 01:18:45,960 And just if you want to think about kind 899 01:18:45,960 --> 01:18:48,930 of a concrete way of talking about these, 900 01:18:48,930 --> 01:19:04,420 I can think of-- if I think of this as a column vector, 901 01:19:04,420 --> 01:19:10,411 v1 to vn, the way I should think about the bras 902 01:19:10,411 --> 01:19:19,390 is that they are really what you want to write as row vectors. 903 01:19:19,390 --> 01:19:26,010 And they may have to have the conjugates of the thing. 904 01:19:26,010 --> 01:19:27,260 The components are conjugated. 905 01:19:33,990 --> 01:19:37,030 OK, and now you can ask what the dot product looks like. 906 01:19:40,730 --> 01:19:48,370 Alpha v is then just this matrix multiplication. 907 01:19:48,370 --> 01:19:53,790 But it's matrix multiplication of an n by 1 thing 908 01:19:53,790 --> 01:19:54,910 by 1 by n thing. 909 01:20:02,300 --> 01:20:14,130 Alpha 1 star alpha 2 star alpha n star times this one here. 910 01:20:14,130 --> 01:20:18,384 So v1 vn. 911 01:20:18,384 --> 01:20:20,175 And this is now just matrix multiplication. 912 01:20:33,412 --> 01:20:34,915 I guess I can write it like this. 913 01:20:44,080 --> 01:20:48,980 So they're really, really quite concrete. 914 01:20:48,980 --> 01:20:51,710 They're as concrete as the kets are. 915 01:20:51,710 --> 01:20:54,370 So you can construct them as vectors 916 01:20:54,370 --> 01:20:58,450 like strings of numbers in this way. 917 01:20:58,450 --> 01:21:01,602 So I guess we should finish. 918 01:21:01,602 --> 01:21:03,560 So I didn't get to talk about linear operators, 919 01:21:03,560 --> 01:21:06,070 but we will resume there next week. 920 01:21:06,070 --> 01:21:09,750 Are there any questions about this last stuff or anything? 921 01:21:14,200 --> 01:21:14,710 No? 922 01:21:14,710 --> 01:21:15,210 OK. 923 01:21:15,210 --> 01:21:17,380 So you next week, or see you tomorrow, some of you. 924 01:21:17,380 --> 01:21:17,880 Thanks. 925 01:21:17,880 --> 01:21:19,230 [APPLAUSE]