1 00:00:00,050 --> 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 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,282 at ocw.mit.edu. 8 00:00:21,630 --> 00:00:26,540 PROFESSOR: Last time we talked about the spin operator 9 00:00:26,540 --> 00:00:30,810 pointing in some particular direction. 10 00:00:30,810 --> 00:00:32,740 There were questions. 11 00:00:32,740 --> 00:00:35,160 In fact, there was a useful question 12 00:00:35,160 --> 00:00:42,110 that I think I want to begin the lecture by going back to it. 13 00:00:42,110 --> 00:00:47,740 And this, you received an email from me. 14 00:00:47,740 --> 00:00:53,880 The notes have an extra section added to it that is stuff 15 00:00:53,880 --> 00:00:56,140 that I didn't do in class last time, 16 00:00:56,140 --> 00:01:02,570 but I was told in fact some of the recitation instructors 17 00:01:02,570 --> 00:01:06,100 did discuss this matter And I'm going 18 00:01:06,100 --> 00:01:07,920 to say a few words about it. 19 00:01:07,920 --> 00:01:11,990 Now, I do expect you to read the notes. 20 00:01:11,990 --> 00:01:16,780 So things that you will need for the homework, all the material 21 00:01:16,780 --> 00:01:20,500 that is in the notes is material that I kind of 22 00:01:20,500 --> 00:01:23,690 assume you're familiar with. 23 00:01:23,690 --> 00:01:27,118 And you've read it and understood it. 24 00:01:27,118 --> 00:01:31,560 And I probably don't cover all what is in the notes, 25 00:01:31,560 --> 00:01:34,440 especially examples or some things 26 00:01:34,440 --> 00:01:36,000 don't go into so much detail. 27 00:01:36,000 --> 00:01:39,540 But the notes should really be helping you understand things 28 00:01:39,540 --> 00:01:40,670 well. 29 00:01:40,670 --> 00:01:47,490 So the remark I want to make is that-- there was a question 30 00:01:47,490 --> 00:01:53,330 last time that better that we think about it more 31 00:01:53,330 --> 00:01:57,760 deliberately in which we saw there that Pauli matrices, 32 00:01:57,760 --> 00:02:03,170 sigma 1 squared was equal to sigma 2 squared equal 33 00:02:03,170 --> 00:02:07,360 to 2 sigma 3 squared was equal to 1. 34 00:02:07,360 --> 00:02:09,810 Well, that, indeed, tells you something 35 00:02:09,810 --> 00:02:17,170 important about the eigenvalues of this matrices. 36 00:02:17,170 --> 00:02:20,280 And it's a general fact. 37 00:02:20,280 --> 00:02:26,585 If you have some matrix M that satisfies an equation. 38 00:02:26,585 --> 00:02:28,800 Now, let me write an equation. 39 00:02:28,800 --> 00:02:35,560 The matrix M squared plus alpha M plus beta times the identity 40 00:02:35,560 --> 00:02:37,570 is equal to 0. 41 00:02:37,570 --> 00:02:39,230 This is a matrix equation. 42 00:02:39,230 --> 00:02:43,210 It takes the whole matrix, square it, add alpha times 43 00:02:43,210 --> 00:02:46,390 the matrix, and then beta times the identity matrix 44 00:02:46,390 --> 00:02:47,470 is equal to 0. 45 00:02:47,470 --> 00:02:52,440 Suppose you discover that such an equation holds 46 00:02:52,440 --> 00:02:57,050 for that matrix M. Then, suppose you are also 47 00:02:57,050 --> 00:03:01,010 asked to find eigenvalues of this matrix M. So suppose 48 00:03:01,010 --> 00:03:04,200 there is a vector-- that is, an eigenvector 49 00:03:04,200 --> 00:03:06,850 with eigenvalue lambda. 50 00:03:06,850 --> 00:03:09,130 That's what having an eigenvector 51 00:03:09,130 --> 00:03:12,880 with eigenvalue lambda means. 52 00:03:12,880 --> 00:03:17,300 And you're supposed to calculate these values of lambda. 53 00:03:17,300 --> 00:03:21,280 So what you do here is let this equation, 54 00:03:21,280 --> 00:03:25,240 this matrix on the left, act on the vector v. 55 00:03:25,240 --> 00:03:32,970 So you have M squared plus alpha M plus beta 1 act 56 00:03:32,970 --> 00:03:41,630 on v. Since the matrix is 0, it should be 0. 57 00:03:41,630 --> 00:03:44,410 And now you come and say, well, let's see. 58 00:03:44,410 --> 00:03:48,920 Beta times 1 on v. Well, that's just beta times 59 00:03:48,920 --> 00:03:53,140 v, the vector v. 60 00:03:53,140 --> 00:03:58,020 Alpha M on v, but M on v is lambda v. So this 61 00:03:58,020 --> 00:04:06,400 is alpha lambda v. And M squared on v, as you can imagine, 62 00:04:06,400 --> 00:04:09,100 you act with another M here. 63 00:04:09,100 --> 00:04:10,500 Then you go to this side. 64 00:04:10,500 --> 00:04:14,130 You get lambda Mv, which is, again, another lambda 65 00:04:14,130 --> 00:04:19,300 times v. So M squared on v is lambda squared 66 00:04:19,300 --> 00:04:25,960 v. If acts two times on v. 67 00:04:25,960 --> 00:04:28,540 Therefore, this is 0. 68 00:04:28,540 --> 00:04:31,190 And here you have, for example, that lambda 69 00:04:31,190 --> 00:04:40,020 squared plus alpha lambda plus beta on v is equal to 0. 70 00:04:40,020 --> 00:04:43,460 Well, v cannot be 0. 71 00:04:43,460 --> 00:04:49,500 Any eigenvector-- by definition, eigenvectors are not 0 vectors. 72 00:04:49,500 --> 00:04:54,179 You can have 0 eigenvalues but not 0 eigenvectors. 73 00:04:54,179 --> 00:04:54,970 That doesn't exist. 74 00:04:54,970 --> 00:04:58,720 An eigenvector that is 0 is a crazy thing 75 00:04:58,720 --> 00:05:01,240 because this would be 0, and then it 76 00:05:01,240 --> 00:05:05,780 would be-- the eigenvalue would not be determined. 77 00:05:05,780 --> 00:05:07,190 It just makes no sense. 78 00:05:07,190 --> 00:05:09,190 So v is different from 0. 79 00:05:09,190 --> 00:05:16,300 So you see that lambda squared plus alpha lambda plus beta is 80 00:05:16,300 --> 00:05:17,820 equal to 0. 81 00:05:17,820 --> 00:05:22,490 And the eigenvalues, any eigenvalue of this matrix, 82 00:05:22,490 --> 00:05:26,660 must satisfy this equation. 83 00:05:26,660 --> 00:05:29,410 So the eigenvalues of sigma 1, you 84 00:05:29,410 --> 00:05:33,270 have sigma 1 squared, for example, is equal to 1. 85 00:05:33,270 --> 00:05:38,880 So the eigenvalues, any lambda squared 86 00:05:38,880 --> 00:05:42,170 must be equal to 1, the number 1. 87 00:05:44,790 --> 00:05:47,810 And therefore, the eigenvalues of sigma 1 88 00:05:47,810 --> 00:05:51,260 are possibly plus or minus 1. 89 00:05:51,260 --> 00:05:54,080 We don't know yet. 90 00:05:54,080 --> 00:06:00,730 Could be two 1's, 2 minus 1's, one 1 and one minus 1. 91 00:06:00,730 --> 00:06:07,420 But there's another nice thing, the trace of sigma 1. 92 00:06:07,420 --> 00:06:09,970 We'll study more the trace, don't worry. 93 00:06:09,970 --> 00:06:12,430 If you are not that familiar with it, 94 00:06:12,430 --> 00:06:15,620 it will become more familiar soon. 95 00:06:15,620 --> 00:06:18,290 The trace of sigma 1 or any matrix 96 00:06:18,290 --> 00:06:20,540 is the sum of elements in the diagonal. 97 00:06:20,540 --> 00:06:25,290 Sigma 1, if you remember, was of this form. 98 00:06:25,290 --> 00:06:27,580 Therefore, the trace is 0. 99 00:06:27,580 --> 00:06:33,910 And in fact, the traces of any of the Pauli matrices are 0. 100 00:06:33,910 --> 00:06:37,010 Another little theorem of linear algebra 101 00:06:37,010 --> 00:06:43,430 shows that the trace of a matrix is 102 00:06:43,430 --> 00:06:45,540 equal to the sum of eigenvalues. 103 00:06:45,540 --> 00:06:49,020 So whatever two eigenvlaues sigma 1 has, 104 00:06:49,020 --> 00:06:50,970 they must add up to 0. 105 00:06:50,970 --> 00:06:55,810 Because the trace is 0 and it's equal to the sum 106 00:06:55,810 --> 00:06:57,015 of eigenvalues. 107 00:07:01,310 --> 00:07:04,930 And therefore, if the eigenvalues can only 108 00:07:04,930 --> 00:07:10,510 be plus or minus 1, you have the result 109 00:07:10,510 --> 00:07:13,920 that one eigenvalue must be plus 1. 110 00:07:13,920 --> 00:07:16,290 The other eigenvalue must be minus 1, 111 00:07:16,290 --> 00:07:19,480 is the only way you can get that to work. 112 00:07:19,480 --> 00:07:33,451 So two sigma 1 eigenvalues of sigma 1 are plus 1 and minus 1. 113 00:07:33,451 --> 00:07:34,700 Those are the two eigenvalues. 114 00:07:38,390 --> 00:07:43,980 So in that section as well, there's 115 00:07:43,980 --> 00:07:48,260 some discussion about properties of the Pauli matrices. 116 00:07:48,260 --> 00:07:54,390 And two basic properties of Pauli matrices 117 00:07:54,390 --> 00:07:56,680 are the following. 118 00:07:56,680 --> 00:08:00,780 Remember that the spin matrices, the spin operators, 119 00:08:00,780 --> 00:08:05,220 are h bar over 2 times the Pauli matrices. 120 00:08:05,220 --> 00:08:09,850 And the spin operators had the algebra for angular momentum. 121 00:08:09,850 --> 00:08:12,510 So from the algebra of angular momentum 122 00:08:12,510 --> 00:08:23,820 that says that Si Sj is equal to i h bar epsilon i j k Sk, 123 00:08:23,820 --> 00:08:29,860 you deduce after plugging this that sigma i sigma 124 00:08:29,860 --> 00:08:36,320 j is 2i epsilon i j k sigma k. 125 00:08:45,130 --> 00:08:47,480 Moreover, there's another nice property 126 00:08:47,480 --> 00:08:52,800 of the Pauli matrices having to deal with anticommutators. 127 00:08:52,800 --> 00:08:58,150 If you do experimentally try multiplying 128 00:08:58,150 --> 00:09:01,710 Pauli matrices, sigma 1 and sigma 2, 129 00:09:01,710 --> 00:09:04,970 you will find out that if you compare it with sigma 2 sigma 130 00:09:04,970 --> 00:09:06,750 1, it's different. 131 00:09:06,750 --> 00:09:09,950 Of course, it's not the same. 132 00:09:09,950 --> 00:09:11,460 These matrices don't commute. 133 00:09:11,460 --> 00:09:15,000 But they actually-- while they fail to commute, 134 00:09:15,000 --> 00:09:17,470 they still fail to commute in a nice way. 135 00:09:17,470 --> 00:09:21,530 Actually, these are minus each other. 136 00:09:21,530 --> 00:09:28,130 So in fact, sigma 1 sigma 2 plus sigma 2 sigma 1 is equal to 0. 137 00:09:28,130 --> 00:09:31,470 And by this, we mean that they anticommute. 138 00:09:31,470 --> 00:09:35,050 And we have a brief way of calling this. 139 00:09:35,050 --> 00:09:38,580 When this sign was a minus, it was called the commutator. 140 00:09:38,580 --> 00:09:42,460 When this is a plus, it's called an anticommutator. 141 00:09:42,460 --> 00:09:48,930 So the anticommutator of sigma 1 with sigma 2 is equal to 0. 142 00:09:48,930 --> 00:09:53,730 Anticommutator defined in general by A, 143 00:09:53,730 --> 00:09:58,090 B. Two operators is AB plus BA. 144 00:10:01,100 --> 00:10:03,540 And as you will read in the notes, 145 00:10:03,540 --> 00:10:06,955 a little more analysis shows that, in fact, 146 00:10:06,955 --> 00:10:10,290 the anticommutator of sigma i and sigma j 147 00:10:10,290 --> 00:10:16,300 has a nice formula, which is 2 delta ij times 148 00:10:16,300 --> 00:10:19,490 the unit matrix, the 2 by 2 unit matrix. 149 00:10:25,640 --> 00:10:29,860 With this result, you get a general formula. 150 00:10:29,860 --> 00:10:36,190 Any product of two operators, AB, you can write as 1/2 151 00:10:36,190 --> 00:10:41,569 of the anticommutator plus 1-- no, 1/2 of the commutator 152 00:10:41,569 --> 00:10:42,860 plus 1/2 of the anticommutator. 153 00:10:46,510 --> 00:10:50,450 Expand it out, that right-hand side, 154 00:10:50,450 --> 00:10:52,520 and you will see quite quickly this 155 00:10:52,520 --> 00:10:55,980 is true for any two operators. 156 00:10:55,980 --> 00:11:00,450 This has AB minus BA and this has AB plus BA. 157 00:11:00,450 --> 00:11:05,370 The BA term cancels and the AB terms are [INAUDIBLE]. 158 00:11:05,370 --> 00:11:13,140 So sigma i sigma j would be equal to 1/2. 159 00:11:13,140 --> 00:11:15,450 And then they put down the anticommutator first. 160 00:11:15,450 --> 00:11:19,700 So you get delta ij times the identity, which 161 00:11:19,700 --> 00:11:22,960 is 1/2 of the anticommutator plus 1/2 162 00:11:22,960 --> 00:11:30,129 of the commutator, which is i epsilon i j k sigma k. 163 00:11:35,510 --> 00:11:37,530 It's a very useful formula. 164 00:11:40,930 --> 00:11:46,130 In order to make those formulas look neater, 165 00:11:46,130 --> 00:11:54,400 we invent a notation in which we think of sigma as a triplet-- 166 00:11:54,400 --> 00:11:58,440 sigma 1, sigma 2, and sigma 3. 167 00:11:58,440 --> 00:12:04,470 And then we have vectors, like a-- normal vectors, 168 00:12:04,470 --> 00:12:06,170 components a1, a2, a3. 169 00:12:09,080 --> 00:12:17,520 And then we have a dot sigma must be defined. 170 00:12:17,520 --> 00:12:19,480 Well, there's an obvious definition 171 00:12:19,480 --> 00:12:22,180 of what this should mean, but it's not 172 00:12:22,180 --> 00:12:24,390 something you're accustomed to. 173 00:12:24,390 --> 00:12:27,340 And one should pause before saying this. 174 00:12:27,340 --> 00:12:31,440 You're having a normal vector, a triplet of numbers, 175 00:12:31,440 --> 00:12:34,780 multiplied by a triplet of matrices, 176 00:12:34,780 --> 00:12:38,370 or a triplet of operators. 177 00:12:38,370 --> 00:12:41,930 Since numbers commute with matrices, 178 00:12:41,930 --> 00:12:45,000 the order in which you write this doesn't matter. 179 00:12:45,000 --> 00:12:49,135 But this is defined to be a1 sigma 1 180 00:12:49,135 --> 00:12:52,930 plus a2 sigma 2 plus a3 sigma 3. 181 00:12:56,250 --> 00:13:02,990 This can be written as ai sigma i with our repeated index 182 00:13:02,990 --> 00:13:06,550 convention that you sum over the possibilities. 183 00:13:06,550 --> 00:13:10,360 So here is what you're supposed to do here 184 00:13:10,360 --> 00:13:14,070 to maybe interpret this equation nicely. 185 00:13:14,070 --> 00:13:18,850 You multiply this equation n by ai bj. 186 00:13:21,480 --> 00:13:22,930 Now, these are numbers. 187 00:13:22,930 --> 00:13:24,210 These are matrices. 188 00:13:24,210 --> 00:13:27,810 I better not change this order, but I can certainly, 189 00:13:27,810 --> 00:13:34,590 by multiplying that way, I have ai sigma i bj sigma j 190 00:13:34,590 --> 00:13:46,070 equals 2 ai bj delta ij times the matrix 1 plus i 191 00:13:46,070 --> 00:13:52,740 epsilon i j k ai bj sigma k. 192 00:14:01,660 --> 00:14:02,860 Now, what? 193 00:14:02,860 --> 00:14:06,490 Well, write it in terms of things that look neat. 194 00:14:06,490 --> 00:14:11,420 a dot sigma, that's a matrix. 195 00:14:11,420 --> 00:14:15,010 This whole thing is a matrix multiplied by the matrix 196 00:14:15,010 --> 00:14:20,750 b dot sigma gives you-- 197 00:14:20,750 --> 00:14:29,580 Well, ai bj delta ij, this delta ij forces j to become i. 198 00:14:29,580 --> 00:14:34,460 In other words, you can replace these two terms by just bi. 199 00:14:34,460 --> 00:14:36,980 And then you have ai bi. 200 00:14:36,980 --> 00:14:39,934 So this is twice. 201 00:14:39,934 --> 00:14:42,640 I don't know why I have a 2. 202 00:14:42,640 --> 00:14:44,780 No 2. 203 00:14:44,780 --> 00:14:48,300 There was no 2 there, sorry. 204 00:14:48,300 --> 00:14:49,650 So what do we get here? 205 00:14:49,650 --> 00:14:54,300 We get a dot b, the dot product. 206 00:14:54,300 --> 00:14:57,050 This is a normal dot product. 207 00:14:57,050 --> 00:15:02,690 This is just a number times 1 plus i. 208 00:15:02,690 --> 00:15:05,580 Now, what is this thing? 209 00:15:05,580 --> 00:15:09,740 You should try to remember how the epsilon tensor can 210 00:15:09,740 --> 00:15:11,880 be used to do cross products. 211 00:15:11,880 --> 00:15:16,820 This, there's just one free index, the index k. 212 00:15:16,820 --> 00:15:19,000 So this must be some sort of vector. 213 00:15:19,000 --> 00:15:24,090 And in fact, if you try the definition of epsilon and look 214 00:15:24,090 --> 00:15:26,670 in detail what this is, you will find 215 00:15:26,670 --> 00:15:35,410 that this is nothing but the k component of a dot b. 216 00:15:35,410 --> 00:15:37,280 The k-- so I'll write it here. 217 00:15:37,280 --> 00:15:42,710 This is a cross b sub k. 218 00:15:42,710 --> 00:15:47,230 But now you have a cross b sub k times sigma k. 219 00:15:47,230 --> 00:15:52,255 So this is the same as a cross b dot sigma. 220 00:15:55,960 --> 00:16:03,375 And here you got a pretty nice equation for Pauli matrices. 221 00:16:06,490 --> 00:16:10,930 It expresses the general product of Pauli matrices 222 00:16:10,930 --> 00:16:14,090 in somewhat geometric terms. 223 00:16:14,090 --> 00:16:25,490 So if you take, for example here, an operator. 224 00:16:25,490 --> 00:16:26,420 No. 225 00:16:26,420 --> 00:16:29,640 If you take, for example, a equals 226 00:16:29,640 --> 00:16:38,680 b equal to a unit vector, then what do we get? 227 00:16:38,680 --> 00:16:41,717 You get n dot sigma squared. 228 00:16:46,250 --> 00:16:48,530 And here you have the dot product of n 229 00:16:48,530 --> 00:16:50,210 with n, which is 1. 230 00:16:50,210 --> 00:16:53,420 So this is 1. 231 00:16:53,420 --> 00:16:56,960 And the cross product of two equal vectors, of course, 232 00:16:56,960 --> 00:17:03,970 is 0 so you get this, which is nice. 233 00:17:03,970 --> 00:17:05,380 Why is this useful? 234 00:17:05,380 --> 00:17:11,760 It's because with this identity, you can understand better 235 00:17:11,760 --> 00:17:15,410 the operator S hat n that we introduced 236 00:17:15,410 --> 00:17:22,950 last time, which was n dot the spin triplet. 237 00:17:22,950 --> 00:17:28,990 So nx, sx, ny, sy, nz, sc. 238 00:17:28,990 --> 00:17:30,370 So what is this? 239 00:17:30,370 --> 00:17:35,260 This is h bar over 2 and dot sigma. 240 00:17:40,320 --> 00:17:42,310 And let's square this. 241 00:17:42,310 --> 00:17:46,510 So Sn vector squared. 242 00:17:46,510 --> 00:17:52,420 This matrix squared would be h bar over 2 squared times 243 00:17:52,420 --> 00:17:55,085 n dot sigma squared, which is 1. 244 00:18:04,600 --> 00:18:06,330 And sigma squared is 1. 245 00:18:06,330 --> 00:18:13,540 Therefore, this spin operator along the n direction squares 246 00:18:13,540 --> 00:18:16,620 to h bar r squared over 2 times 1. 247 00:18:16,620 --> 00:18:25,500 Now, the trace of this Sn operator is also 0. 248 00:18:25,500 --> 00:18:26,860 Why? 249 00:18:26,860 --> 00:18:29,390 Because the trace means that you're 250 00:18:29,390 --> 00:18:32,280 going to sum the elements in the diagonal. 251 00:18:32,280 --> 00:18:36,260 Well, you have a sum of matrices here. 252 00:18:36,260 --> 00:18:40,310 And therefore, you will have to sum the diagonals of each. 253 00:18:40,310 --> 00:18:43,690 But each of the sigmas has 0 trace. 254 00:18:43,690 --> 00:18:46,380 We wrote it there. 255 00:18:46,380 --> 00:18:48,310 Trace of sigma 1 is 0. 256 00:18:48,310 --> 00:18:53,660 All the Pauli matrices have 0 trace, so this has 0 trace. 257 00:18:53,660 --> 00:18:57,930 So you have these two relations. 258 00:18:57,930 --> 00:19:03,250 And again, this tells you that the eigenvalues of this matrix 259 00:19:03,250 --> 00:19:07,210 can be plus minus h bar over 2. 260 00:19:07,210 --> 00:19:10,010 Because the eigenvalues satisfy the same equation 261 00:19:10,010 --> 00:19:11,250 as the matrix. 262 00:19:11,250 --> 00:19:14,460 Therefor,e plus minus h bar over 2. 263 00:19:14,460 --> 00:19:19,090 And this one says that the eigenvalues add up to 0. 264 00:19:19,090 --> 00:19:29,070 So the eigenvalues of S hat n vector are plus h bar over 2 265 00:19:29,070 --> 00:19:31,410 and minus h bar over 2. 266 00:19:31,410 --> 00:19:36,170 We did that last time, but we do that by just taking that matrix 267 00:19:36,170 --> 00:19:37,820 and finding the eigenvalues. 268 00:19:37,820 --> 00:19:44,340 But this shows that its property is almost manifest. 269 00:19:44,340 --> 00:19:47,180 And this is fundamental for the interpretation 270 00:19:47,180 --> 00:19:49,310 of this operator. 271 00:19:49,310 --> 00:19:50,310 Why? 272 00:19:50,310 --> 00:19:54,110 Well, we saw that if n points along the z-direction, 273 00:19:54,110 --> 00:19:56,340 it becomes the operator sz. 274 00:19:56,340 --> 00:19:58,490 If it points about the x-direction, 275 00:19:58,490 --> 00:20:00,810 it becomes the operator sx. 276 00:20:00,810 --> 00:20:03,890 If it points along y, it becomes sy. 277 00:20:03,890 --> 00:20:07,460 But in an arbitrary direction, it's a funny thing. 278 00:20:07,460 --> 00:20:10,410 But it still has the key property. 279 00:20:10,410 --> 00:20:14,200 If you measured the spin along an arbitrary direction, 280 00:20:14,200 --> 00:20:19,740 you should find only plus h bar over 2 or minus h bar over 2. 281 00:20:19,740 --> 00:20:23,320 Because after all, the universe is isotopic. 282 00:20:23,320 --> 00:20:25,560 It doesn't depend on direction. 283 00:20:25,560 --> 00:20:27,500 So a spin one-half particle. 284 00:20:27,500 --> 00:20:30,660 If you find out that whenever you measure the z component, 285 00:20:30,660 --> 00:20:33,250 it's either plus minus h bar over 2. 286 00:20:33,250 --> 00:20:35,355 Well, when you measure any direction, 287 00:20:35,355 --> 00:20:39,120 it should be plus minus h bar over 2. 288 00:20:39,120 --> 00:20:42,940 And this shows that this operator has those eigenvalues. 289 00:20:42,940 --> 00:20:47,570 And therefore, it makes sense that this is the operator 290 00:20:47,570 --> 00:20:52,180 that measures spins in an arbitrary direction. 291 00:20:52,180 --> 00:20:56,595 There's a little more of an aside in there, 292 00:20:56,595 --> 00:20:58,610 in the notes about something that 293 00:20:58,610 --> 00:21:02,590 will be useful and fun to do. 294 00:21:02,590 --> 00:21:04,830 And it corresponds to the case in which you 295 00:21:04,830 --> 00:21:10,150 have two triplets of operators-- x1, x2, x3. 296 00:21:10,150 --> 00:21:12,700 These are operators now. 297 00:21:12,700 --> 00:21:18,945 And y equal y1, y2, y3. 298 00:21:22,120 --> 00:21:24,160 Two triplets of operators. 299 00:21:24,160 --> 00:21:29,585 So you define the dot product of these two triplets 300 00:21:29,585 --> 00:21:36,149 as xi yi summed. 301 00:21:36,149 --> 00:21:37,065 That's the definition. 302 00:21:41,920 --> 00:21:45,190 Now, the dot product of two triplets of operators 303 00:21:45,190 --> 00:21:48,680 defined that way may not commute. 304 00:21:48,680 --> 00:21:52,190 Because the operators x and y may not commute. 305 00:21:52,190 --> 00:21:58,150 So this new dot product of both phase operators 306 00:21:58,150 --> 00:22:01,380 is not commutative-- probably. 307 00:22:01,380 --> 00:22:03,800 It may happen that these operators commute, 308 00:22:03,800 --> 00:22:07,630 in which case x dot y is equal to y dot x. 309 00:22:07,630 --> 00:22:10,620 Similarly, you can define the cross product 310 00:22:10,620 --> 00:22:14,000 of these two things. 311 00:22:14,000 --> 00:22:23,610 And the k-th component is epsilon i j k xi yj like this. 312 00:22:26,830 --> 00:22:30,565 Just like you would define it for two number vectors. 313 00:22:33,460 --> 00:22:36,860 Now, what do you know about the cross product in general? 314 00:22:36,860 --> 00:22:37,875 It's anti-symmetric. 315 00:22:37,875 --> 00:22:40,400 A cross B is equal to minus B cross A. 316 00:22:40,400 --> 00:22:48,290 But this one won't be because the operators x and y 317 00:22:48,290 --> 00:22:49,490 may not commute. 318 00:22:49,490 --> 00:22:56,490 Even x cross x may be nonzero. 319 00:22:56,490 --> 00:23:00,180 So one thing I will ask you to compute in the homework 320 00:23:00,180 --> 00:23:03,180 is not a long calculation. 321 00:23:03,180 --> 00:23:04,150 It's three lines. 322 00:23:04,150 --> 00:23:14,300 But what is S cross S equal to? 323 00:23:14,300 --> 00:23:15,040 Question there? 324 00:23:15,040 --> 00:23:15,956 AUDIENCE: [INAUDIBLE]. 325 00:23:19,770 --> 00:23:21,860 PROFESSOR: Yes, it's the sum [INAUDIBLE]. 326 00:23:21,860 --> 00:23:24,710 Just in the same way that here you're 327 00:23:24,710 --> 00:23:29,270 summing over i's and j's to produce the cross product. 328 00:23:29,270 --> 00:23:31,810 So whenever an index is repeated, 329 00:23:31,810 --> 00:23:34,780 we'll assume it's summed. 330 00:23:34,780 --> 00:23:37,920 And when it is not summed, I will put to the right, 331 00:23:37,920 --> 00:23:41,110 not summed explicitly-- the words-. 332 00:23:41,110 --> 00:23:45,040 Because in some occasions, it matters. 333 00:23:45,040 --> 00:23:46,820 So how much is this? 334 00:23:46,820 --> 00:23:50,840 It will involve i, h bar, and something. 335 00:23:50,840 --> 00:23:53,790 And you will try to find out what this is. 336 00:23:53,790 --> 00:23:56,930 It's a cute thing. 337 00:23:56,930 --> 00:23:59,533 All right, any other questions? 338 00:24:07,880 --> 00:24:09,270 More questions? 339 00:24:09,270 --> 00:24:09,770 Nope. 340 00:24:13,120 --> 00:24:13,800 OK. 341 00:24:13,800 --> 00:24:16,950 So now, finally, we get to that part 342 00:24:16,950 --> 00:24:20,980 of the course that has to do with linear algebra. 343 00:24:20,980 --> 00:24:25,050 And I'm going to do an experiment. 344 00:24:25,050 --> 00:24:27,710 I'm going to do it differently than I did it 345 00:24:27,710 --> 00:24:28,815 in the previous years. 346 00:24:32,450 --> 00:24:35,720 There is this nice book. 347 00:24:35,720 --> 00:24:37,710 It's here. 348 00:24:37,710 --> 00:24:40,090 I don't know if you can read from that far, 349 00:24:40,090 --> 00:24:47,430 but it has a pretty-- you might almost say an arrogant title. 350 00:24:47,430 --> 00:24:54,280 It says, Linear Algebra Done Right by Sheldon Axler. 351 00:24:54,280 --> 00:25:00,140 This is the book, actually, MIT's course 18.700 of linear 352 00:25:00,140 --> 00:25:02,560 algebra uses. 353 00:25:02,560 --> 00:25:05,450 And when you first get the book that looks like that, 354 00:25:05,450 --> 00:25:08,340 you read it and open-- I'm going to show you 355 00:25:08,340 --> 00:25:11,760 that this is not that well done. 356 00:25:11,760 --> 00:25:15,510 But actually, I think it's actually true. 357 00:25:15,510 --> 00:25:18,200 The title is not a lie. 358 00:25:18,200 --> 00:25:21,530 It's really done right. 359 00:25:21,530 --> 00:25:26,370 I actually wish I had learned linear algebra this way. 360 00:25:26,370 --> 00:25:29,500 It may be a little difficult if you've never 361 00:25:29,500 --> 00:25:32,550 done any linear algebra. 362 00:25:32,550 --> 00:25:34,780 You don't know what the matrix is-- 363 00:25:34,780 --> 00:25:36,640 I don't think that's the case anybody here. 364 00:25:36,640 --> 00:25:41,580 A determinant, or eigenvalue. 365 00:25:41,580 --> 00:25:43,670 If you never heard any of those words, 366 00:25:43,670 --> 00:25:46,050 this might be a little hard. 367 00:25:46,050 --> 00:25:48,110 But if you've heard those words and you've 368 00:25:48,110 --> 00:25:50,930 had a little linear algebra, this is quite nice. 369 00:25:50,930 --> 00:25:54,125 Now, this book has also a small problem. 370 00:25:57,000 --> 00:25:59,930 Unless you study it seriously, it's 371 00:25:59,930 --> 00:26:03,750 not all that easy to grab results that you need from it. 372 00:26:03,750 --> 00:26:05,860 You have to study it. 373 00:26:05,860 --> 00:26:08,180 So I don't know if it might help you 374 00:26:08,180 --> 00:26:10,420 or not during this semester. 375 00:26:10,420 --> 00:26:12,290 It may. 376 00:26:12,290 --> 00:26:14,060 It's not necessary to get it. 377 00:26:14,060 --> 00:26:15,620 Absolutely not. 378 00:26:15,620 --> 00:26:19,090 But it is quite lovely. 379 00:26:19,090 --> 00:26:21,630 And the emphasis is quite interesting. 380 00:26:21,630 --> 00:26:25,780 It really begins from very basic things 381 00:26:25,780 --> 00:26:28,520 and logically develops everything 382 00:26:28,520 --> 00:26:31,520 and asks at every point the right questions. 383 00:26:31,520 --> 00:26:32,580 It's quite nice. 384 00:26:32,580 --> 00:26:36,860 So what I'm going to do is-- inspired by that, 385 00:26:36,860 --> 00:26:42,220 I want to introduce some of the linear algebra little 386 00:26:42,220 --> 00:26:42,960 by little. 387 00:26:42,960 --> 00:26:45,910 And I don't know very well how this will go. 388 00:26:45,910 --> 00:26:47,630 Maybe there's too much detail. 389 00:26:47,630 --> 00:26:52,120 Maybe it's a lot of detail, but not enough so 390 00:26:52,120 --> 00:26:53,530 it's not all that great. 391 00:26:53,530 --> 00:26:55,750 I don't know, you will have to tell me. 392 00:26:58,490 --> 00:27:01,130 But we'll try to get some ideas clear. 393 00:27:01,130 --> 00:27:03,990 And the reason I want to get some ideas clear 394 00:27:03,990 --> 00:27:08,880 is that good books on this subject 395 00:27:08,880 --> 00:27:12,070 allow you to understand how much structure you 396 00:27:12,070 --> 00:27:16,680 have to put in a vector space to define certain things. 397 00:27:16,680 --> 00:27:20,470 And unless you do this carefully, 398 00:27:20,470 --> 00:27:25,730 you probably miss some of the basic things. 399 00:27:25,730 --> 00:27:30,240 Like many physicists don't quite realize 400 00:27:30,240 --> 00:27:32,980 that talking about the matrix representation, 401 00:27:32,980 --> 00:27:34,970 you don't need brass and [INAUDIBLE] 402 00:27:34,970 --> 00:27:38,040 to talk about the matrix representation of an operator. 403 00:27:38,040 --> 00:27:40,670 At first sight, it seems like you'd need it, 404 00:27:40,670 --> 00:27:42,970 but you actually don't. 405 00:27:42,970 --> 00:27:46,640 Then, the differences between a complex and a vector space-- 406 00:27:46,640 --> 00:27:51,300 complex and a real vector space become much clearer 407 00:27:51,300 --> 00:27:53,570 if you take your time to understand it. 408 00:27:53,570 --> 00:27:55,320 They are very different. 409 00:27:55,320 --> 00:27:57,740 And in a sense, complex vector spaces 410 00:27:57,740 --> 00:28:01,350 are more powerful, more elegant, have stronger results. 411 00:28:04,260 --> 00:28:07,870 So anyway, it's enough of an introduction. 412 00:28:07,870 --> 00:28:10,710 Let's see how we do. 413 00:28:10,710 --> 00:28:14,870 And let's just begin there for our story. 414 00:28:14,870 --> 00:28:21,770 So we begin with vector spaces and dimensionality. 415 00:28:21,770 --> 00:28:22,510 Yes. 416 00:28:22,510 --> 00:28:24,438 AUDIENCE: Quick question. 417 00:28:24,438 --> 00:28:28,776 The length between the trace of matrix 418 00:28:28,776 --> 00:28:33,450 equals 0 and [INAUDIBLE] is proportional to the identity. 419 00:28:33,450 --> 00:28:35,997 One is the product of the eigenvalues is 1 420 00:28:35,997 --> 00:28:39,893 and the other one was the sum is equal to 0. 421 00:28:39,893 --> 00:28:42,328 Are those two statements related causally, 422 00:28:42,328 --> 00:28:44,710 or are they just separate statements [INAUDIBLE]? 423 00:28:44,710 --> 00:28:46,210 PROFESSOR: OK, the question is, what 424 00:28:46,210 --> 00:28:48,560 is the relation between these two statements? 425 00:28:48,560 --> 00:28:50,570 Those are separate observations. 426 00:28:50,570 --> 00:28:53,050 One does not imply the other. 427 00:28:53,050 --> 00:28:56,490 You can have matrices that square to the identity, 428 00:28:56,490 --> 00:29:00,290 like the identity itself, and don't have 0 trace. 429 00:29:00,290 --> 00:29:02,590 So these are separate properties. 430 00:29:02,590 --> 00:29:06,785 This tells us that the eigenvalue squared 431 00:29:06,785 --> 00:29:10,010 are h bar over 2. 432 00:29:10,010 --> 00:29:14,280 And this one tells me that lambda 1 plus lambda 2-- 433 00:29:14,280 --> 00:29:16,780 there are two eigenvalues-- are 0. 434 00:29:16,780 --> 00:29:20,250 So from here, you deduce that the eigenvalues 435 00:29:20,250 --> 00:29:22,350 could be plus minus h bar over 2. 436 00:29:22,350 --> 00:29:25,450 And in fact, have to be plus minus h bar over 2. 437 00:29:28,520 --> 00:29:32,850 All right, so let's talk about vector spaces 438 00:29:32,850 --> 00:29:35,590 and dimensionality. 439 00:29:35,590 --> 00:29:39,273 Spaces and dimensionality. 440 00:29:47,260 --> 00:29:50,480 So why do we care about this? 441 00:29:50,480 --> 00:29:53,060 Because the end result of our discussion 442 00:29:53,060 --> 00:29:56,600 is that the states of a physical system 443 00:29:56,600 --> 00:30:01,050 are vectors in a complex vector space. 444 00:30:01,050 --> 00:30:04,820 That's, in a sense, the result we're going to get. 445 00:30:04,820 --> 00:30:09,520 Observables, moreover, are linear operators 446 00:30:09,520 --> 00:30:11,420 on those vector spaces. 447 00:30:11,420 --> 00:30:15,070 So we need to understand what are complex vector spaces, what 448 00:30:15,070 --> 00:30:18,930 linear operators on them mean. 449 00:30:18,930 --> 00:30:22,650 So as I said, complex vector spaces 450 00:30:22,650 --> 00:30:26,350 have subtle properties that make them different from real vector 451 00:30:26,350 --> 00:30:28,840 spaces and we want to appreciate that. 452 00:30:28,840 --> 00:30:32,110 In a vector space, what do you have? 453 00:30:32,110 --> 00:30:36,830 You have vectors and you have numbers. 454 00:30:36,830 --> 00:30:39,280 So the two things must exist. 455 00:30:39,280 --> 00:30:43,710 The numbers could be the real numbers, in which case 456 00:30:43,710 --> 00:30:46,310 we're talking about the real vector space. 457 00:30:46,310 --> 00:30:49,680 And the numbers could be complex numbers, in which case 458 00:30:49,680 --> 00:30:52,530 we're talking about the complex vector space. 459 00:30:52,530 --> 00:30:58,670 We don't say the vectors are real, or complex, or imaginary. 460 00:30:58,670 --> 00:31:03,620 We just say there are vectors and there are numbers. 461 00:31:03,620 --> 00:31:08,930 Now, the vectors can be added and the numbers 462 00:31:08,930 --> 00:31:12,180 can be multiplied by vectors to give vectors. 463 00:31:12,180 --> 00:31:15,500 That's basically what is happening. 464 00:31:15,500 --> 00:31:20,320 Now, these numbers can be real or complex. 465 00:31:20,320 --> 00:31:26,933 And the numbers-- so there are vectors and numbers. 466 00:31:30,050 --> 00:31:33,550 And we will focus on just either real numbers 467 00:31:33,550 --> 00:31:36,720 or complex numbers, but either one. 468 00:31:36,720 --> 00:31:41,400 So these sets of numbers form what 469 00:31:41,400 --> 00:31:43,770 is called in mathematics a field. 470 00:31:43,770 --> 00:31:46,970 So I will not define the field. 471 00:31:46,970 --> 00:31:52,430 But a field-- use the letter F for field. 472 00:31:52,430 --> 00:31:53,720 And our results. 473 00:31:53,720 --> 00:31:58,220 I will state results whenever-- it doesn't matter whether it's 474 00:31:58,220 --> 00:32:01,460 real or complex, I may use the letter F 475 00:32:01,460 --> 00:32:05,480 to say the numbers are in F. And you say real or complex. 476 00:32:10,260 --> 00:32:12,250 What is a vector space? 477 00:32:12,250 --> 00:32:22,310 So the vector space, V. Vector space, V, 478 00:32:22,310 --> 00:32:36,410 is a set of vectors with an operation called addition-- 479 00:32:36,410 --> 00:32:54,160 and we represent it as plus-- that assigns a vector u plus v 480 00:32:54,160 --> 00:33:05,320 in the vector space when u and v belong to the vector space. 481 00:33:05,320 --> 00:33:08,170 So for any u and v in the vector space, 482 00:33:08,170 --> 00:33:14,170 there's a rule called addition that assigns another vector. 483 00:33:14,170 --> 00:33:18,410 This also means that this space is closed under addition. 484 00:33:18,410 --> 00:33:21,500 That is, you cannot get out of the vector space by adding 485 00:33:21,500 --> 00:33:22,640 vectors. 486 00:33:22,640 --> 00:33:25,930 The vector space must contain a set that 487 00:33:25,930 --> 00:33:28,390 is consistent in that you can add vectors 488 00:33:28,390 --> 00:33:29,896 and you're always there. 489 00:33:29,896 --> 00:33:31,104 And there's a multiplication. 490 00:33:33,732 --> 00:33:49,420 And a scalar multiplication by elements 491 00:33:49,420 --> 00:34:02,130 of the numbers of F such that a, which is a number, 492 00:34:02,130 --> 00:34:07,110 times v belongs to the vector space 493 00:34:07,110 --> 00:34:16,889 when a belongs to the numbers and v belongs to the vectors. 494 00:34:16,889 --> 00:34:19,050 So every time you have a vector, you 495 00:34:19,050 --> 00:34:22,730 can multiply by those numbers and the result 496 00:34:22,730 --> 00:34:25,940 of that multiplication is another vector. 497 00:34:25,940 --> 00:34:31,610 So we say the space is also closed under multiplication. 498 00:34:31,610 --> 00:34:33,800 Now, these properties exist, but they 499 00:34:33,800 --> 00:34:37,710 must-- these operations exist, but they 500 00:34:37,710 --> 00:34:39,870 must satisfy the following properties. 501 00:34:39,870 --> 00:34:41,809 So the definition is not really over. 502 00:34:45,590 --> 00:34:49,753 These operations satisfy-- 503 00:34:54,520 --> 00:34:56,030 1. 504 00:34:56,030 --> 00:34:59,770 u plus v is equal to v plus u. 505 00:34:59,770 --> 00:35:03,150 The order doesn't matter how you sum vectors. 506 00:35:03,150 --> 00:35:07,830 And here, u and v in V. 507 00:35:07,830 --> 00:35:09,490 2. 508 00:35:09,490 --> 00:35:10,480 Associative. 509 00:35:10,480 --> 00:35:21,130 So u plus v plus w is equal to u plus v plus w. 510 00:35:21,130 --> 00:35:30,950 Moreover, two numbers a times b times v is the same as a times 511 00:35:30,950 --> 00:35:33,420 bv. 512 00:35:33,420 --> 00:35:35,624 You can add with the first number on the vector 513 00:35:35,624 --> 00:35:36,790 and you add with the second. 514 00:35:41,920 --> 00:35:42,420 3. 515 00:35:45,400 --> 00:35:52,245 There is an additive identity. 516 00:35:55,930 --> 00:35:57,130 And that is what? 517 00:35:57,130 --> 00:36:01,450 It's a vector 0 belonging to the vector space. 518 00:36:01,450 --> 00:36:03,370 I could write an arrow. 519 00:36:03,370 --> 00:36:07,570 But actually, for some reason they just 520 00:36:07,570 --> 00:36:09,280 don't like to write it because they say 521 00:36:09,280 --> 00:36:11,780 it's always ambiguous whether you're 522 00:36:11,780 --> 00:36:15,910 talking about the 0 number or the 0 vector. 523 00:36:15,910 --> 00:36:18,050 We do have that problem also in the notation 524 00:36:18,050 --> 00:36:19,480 in quantum mechanics. 525 00:36:19,480 --> 00:36:28,950 But here it is, here is a 0 vector such that 0 526 00:36:28,950 --> 00:36:37,812 plus any vector v is equal to v. 527 00:36:37,812 --> 00:36:40,200 4. 528 00:36:40,200 --> 00:36:42,850 Well, in the field, in the set of numbers, 529 00:36:42,850 --> 00:36:47,420 there's the number 1, which multiplied by any other number 530 00:36:47,420 --> 00:36:49,790 keeps that number. 531 00:36:49,790 --> 00:36:58,210 So the number 1 that belongs to the field 532 00:36:58,210 --> 00:37:08,050 satisfies that 1 times any vector is equal to the vector. 533 00:37:08,050 --> 00:37:13,540 So we declare that that number multiplied by other numbers 534 00:37:13,540 --> 00:37:14,780 is an identity. 535 00:37:14,780 --> 00:37:17,302 [INAUDIBLE] identity also multiplying vectors. 536 00:37:17,302 --> 00:37:18,385 Yes, there was a question. 537 00:37:18,385 --> 00:37:21,120 AUDIENCE: [INAUDIBLE]. 538 00:37:21,120 --> 00:37:24,945 PROFESSOR: There is an additive identity. 539 00:37:24,945 --> 00:37:31,130 Additive identity, the 0 vector. 540 00:37:31,130 --> 00:37:35,532 Finally, distributive laws. 541 00:37:35,532 --> 00:37:37,850 No. 542 00:37:37,850 --> 00:37:38,690 One second. 543 00:37:38,690 --> 00:37:46,290 One, two, three-- the zero vector. 544 00:37:46,290 --> 00:37:50,670 Oh, actually in my list I put them in different orders 545 00:37:50,670 --> 00:37:53,141 in the notes, but never mind. 546 00:37:53,141 --> 00:37:53,640 5. 547 00:37:56,560 --> 00:38:00,540 There's an additive inverse in the vector space. 548 00:38:00,540 --> 00:38:07,230 So for each v belonging to the vector space, 549 00:38:07,230 --> 00:38:17,310 there is a u belonging to the vector space such 550 00:38:17,310 --> 00:38:25,000 that v plus u is equal to 0. 551 00:38:25,000 --> 00:38:31,250 So additive identity you can find 552 00:38:31,250 --> 00:38:34,780 for every element its opposite vector. 553 00:38:34,780 --> 00:38:36,150 It always can be found. 554 00:38:39,670 --> 00:38:44,240 And last is this [INAUDIBLE] which 555 00:38:44,240 --> 00:38:52,820 says that a times u plus v is equal to au plus av, 556 00:38:52,820 --> 00:39:01,660 and a plus b on v is equal to av plus bv. 557 00:39:01,660 --> 00:39:06,220 And a's and b's belong to the numbers. 558 00:39:06,220 --> 00:39:08,760 a and b's belong to the field. 559 00:39:08,760 --> 00:39:14,675 And u and v belong to the vector space. 560 00:39:14,675 --> 00:39:15,175 OK. 561 00:39:18,180 --> 00:39:20,450 It's a little disconcerting. 562 00:39:20,450 --> 00:39:21,820 There's a lot of things. 563 00:39:21,820 --> 00:39:26,630 But actually, they are quite minimal. 564 00:39:26,630 --> 00:39:28,422 It's well done, this definition. 565 00:39:28,422 --> 00:39:29,880 They're all kind of things that you 566 00:39:29,880 --> 00:39:36,930 know that follow quite immediately by little proofs. 567 00:39:36,930 --> 00:39:38,800 You will see more in the notes, but let 568 00:39:38,800 --> 00:39:42,030 me just say briefly a few of them. 569 00:39:42,030 --> 00:39:48,740 So here is the additive identity, the vector 0. 570 00:39:48,740 --> 00:39:53,880 It's easy to prove that this vector 0 is unique. 571 00:39:53,880 --> 00:39:58,890 If you find another 0 prime that also satisfies this property, 572 00:39:58,890 --> 00:40:00,650 0 is equal to 0 prime. 573 00:40:00,650 --> 00:40:03,690 So it's unique. 574 00:40:03,690 --> 00:40:16,570 You can also show that 0 times any vector is equal to 0. 575 00:40:16,570 --> 00:40:20,320 And here, this 0 belongs to the field 576 00:40:20,320 --> 00:40:23,250 and this 0 belongs to the vector space. 577 00:40:23,250 --> 00:40:27,340 So the 0-- you had to postulate that the 1 in the field 578 00:40:27,340 --> 00:40:29,710 does the right thing, but you don't need to postulate 579 00:40:29,710 --> 00:40:33,730 that 0, the number 0, multiplied by a vector is 0. 580 00:40:33,730 --> 00:40:35,480 You can prove that. 581 00:40:35,480 --> 00:40:37,930 And these are not difficult to prove. 582 00:40:37,930 --> 00:40:41,210 All of them are one-line exercises. 583 00:40:41,210 --> 00:40:43,110 They're done in that book. 584 00:40:43,110 --> 00:40:46,060 You can look at them. 585 00:40:46,060 --> 00:40:48,520 Moreover, another one. 586 00:40:48,520 --> 00:40:56,940 a any number times the 0 vector is equal to the 0 vector. 587 00:40:56,940 --> 00:40:59,810 So in this case, those both are vectors. 588 00:40:59,810 --> 00:41:03,690 That's also another property. 589 00:41:03,690 --> 00:41:08,740 So the 0 vector and the 0 number really do the right thing. 590 00:41:08,740 --> 00:41:12,960 Then, another property, the additive inverse. 591 00:41:12,960 --> 00:41:14,560 This is sort of interesting. 592 00:41:14,560 --> 00:41:21,240 So the additive inverse, you can prove it's unique. 593 00:41:21,240 --> 00:41:22,960 So the additive inverse is unique. 594 00:41:31,420 --> 00:41:44,040 And it's called-- for v, it's called minus v, just a name. 595 00:41:44,040 --> 00:41:49,420 And actually, you can prove it's equal to the number minus 1 596 00:41:49,420 --> 00:41:50,185 times the vector. 597 00:41:54,390 --> 00:41:59,380 Might sound totally trivial but try to prove them. 598 00:41:59,380 --> 00:42:02,950 They're all simple, but they're not trivial, all these things. 599 00:42:02,950 --> 00:42:08,200 So you call it minus v, but it's actually-- this is a proof. 600 00:42:12,420 --> 00:42:14,170 OK. 601 00:42:14,170 --> 00:42:16,910 So examples. 602 00:42:16,910 --> 00:42:21,010 Let's do a few examples. 603 00:42:21,010 --> 00:42:24,125 I'll have five examples that we're going to use. 604 00:42:28,330 --> 00:42:35,180 So I think the main thing for a physicist that I remember 605 00:42:35,180 --> 00:42:37,930 being confused about is the statement 606 00:42:37,930 --> 00:42:41,500 that there's no characterization that the vectors are 607 00:42:41,500 --> 00:42:42,930 real or complex. 608 00:42:42,930 --> 00:42:45,240 The vectors are the vectors and you 609 00:42:45,240 --> 00:42:47,690 multiply by a real or complex numbers. 610 00:42:47,690 --> 00:42:51,293 So I will have one example that makes that very dramatic. 611 00:42:54,520 --> 00:42:57,320 As dramatic as it can be. 612 00:42:57,320 --> 00:43:10,280 So one example of vector spaces, the set of N component vectors. 613 00:43:10,280 --> 00:43:16,055 So here it is, a1, a2, up to a n. 614 00:43:16,055 --> 00:43:23,030 For example, with capital N. With a i belongs to the real 615 00:43:23,030 --> 00:43:35,140 and i going from 1 up to N is a vector space 616 00:43:35,140 --> 00:43:41,330 over r, the real numbers. 617 00:43:41,330 --> 00:43:45,930 So people use that terminology, a vector space 618 00:43:45,930 --> 00:43:49,240 over the kind of numbers. 619 00:43:49,240 --> 00:43:51,270 You could call it also a real vector 620 00:43:51,270 --> 00:43:52,920 space, that would be the same. 621 00:43:52,920 --> 00:43:55,490 You see, these components are real. 622 00:43:55,490 --> 00:43:58,430 And you have to think for a second 623 00:43:58,430 --> 00:44:02,020 if you believe all of them are true or how would you do it. 624 00:44:02,020 --> 00:44:05,330 Well, if I would be really precise, 625 00:44:05,330 --> 00:44:07,060 I would have to tell you a lot of things 626 00:44:07,060 --> 00:44:08,370 that you would find boring. 627 00:44:08,370 --> 00:44:12,430 That, for example, you have this vector and you add a set 628 00:44:12,430 --> 00:44:13,120 of b's. 629 00:44:13,120 --> 00:44:14,990 Well, you add the components. 630 00:44:14,990 --> 00:44:17,140 That's the definition of plus. 631 00:44:17,140 --> 00:44:19,930 And what's the definition of multiplying by a number? 632 00:44:19,930 --> 00:44:22,860 Well, if a number is multiplied by this vector, 633 00:44:22,860 --> 00:44:25,950 it goes in and multiplies everybody. 634 00:44:25,950 --> 00:44:29,160 Those are implicit, or you can fill-in the details. 635 00:44:29,160 --> 00:44:31,110 But if you define them that way, it 636 00:44:31,110 --> 00:44:33,340 will satisfy all the properties. 637 00:44:33,340 --> 00:44:35,040 What is the 0 vector? 638 00:44:35,040 --> 00:44:39,220 It must be the one with all entries 0. 639 00:44:39,220 --> 00:44:41,360 What is the additive inverse? 640 00:44:41,360 --> 00:44:43,920 Well, change the sign of all these things. 641 00:44:43,920 --> 00:44:48,380 So it's kind of obvious that this satisfies everything, 642 00:44:48,380 --> 00:44:52,250 if you understand how the sum and the multiplication goes. 643 00:44:54,970 --> 00:44:57,970 Another one, it's kind of similar. 644 00:44:57,970 --> 00:44:59,800 2. 645 00:44:59,800 --> 00:45:10,625 The set of M cross N matrices with complex entries. 646 00:45:13,966 --> 00:45:18,120 Complex entries. 647 00:45:18,120 --> 00:45:24,400 So here you have it, a1 1, a1 2, a1 N. 648 00:45:24,400 --> 00:45:30,330 And here it goes up to aM1, aM2, aMN. 649 00:45:35,560 --> 00:45:45,630 With all the a i j's belonging to the complex numbers, 650 00:45:45,630 --> 00:45:52,540 then-- I'll erase here. 651 00:45:52,540 --> 00:45:56,300 Then you have that this is a complex vector space. 652 00:46:00,590 --> 00:46:09,415 Is a complex vector space. 653 00:46:12,760 --> 00:46:14,710 How do you multiply by a number? 654 00:46:14,710 --> 00:46:18,070 You multiply a number times every entry of the matrices. 655 00:46:18,070 --> 00:46:20,590 How do sum two matrices? 656 00:46:20,590 --> 00:46:25,000 They have the same size, so you sum each element the way 657 00:46:25,000 --> 00:46:25,740 it should be. 658 00:46:25,740 --> 00:46:28,860 And that should be a vector space. 659 00:46:31,630 --> 00:46:34,100 Here is an example that is, perhaps, 660 00:46:34,100 --> 00:46:37,070 a little more surprising. 661 00:46:37,070 --> 00:46:54,150 So the space of 2 by 2 Hermitian matrices 662 00:46:54,150 --> 00:46:58,415 is a real vector space. 663 00:47:06,450 --> 00:47:11,050 You see, this can be easily thought [INAUDIBLE] naturally 664 00:47:11,050 --> 00:47:12,600 thought as a real vector space. 665 00:47:12,600 --> 00:47:16,750 This is a little surprising because Hermitian matrices have 666 00:47:16,750 --> 00:47:17,640 i's. 667 00:47:17,640 --> 00:47:21,370 You remember the most general Hermitian matrix 668 00:47:21,370 --> 00:47:30,470 was of the form-- well, a plus-- no, 669 00:47:30,470 --> 00:47:38,790 c plus d, c minus d, a plus ib, a minus ib, 670 00:47:38,790 --> 00:47:44,480 with all these numbers c, d, b in real. 671 00:47:44,480 --> 00:47:47,870 But they're complex numbers. 672 00:47:47,870 --> 00:47:51,960 Why is this naturally a real vector space? 673 00:47:51,960 --> 00:47:56,590 The problem is that if you multiply by a number, 674 00:47:56,590 --> 00:47:59,530 it should still be a Hermitian matrix in order 675 00:47:59,530 --> 00:48:01,825 for it to be a vector space. 676 00:48:01,825 --> 00:48:03,250 It should be in the vector. 677 00:48:03,250 --> 00:48:06,430 But if you multiply by a real number, there's no problem. 678 00:48:06,430 --> 00:48:08,670 The matrix remains Hermitian. 679 00:48:08,670 --> 00:48:10,640 You multiplied by a complex number, 680 00:48:10,640 --> 00:48:12,550 you use the Hermiticity. 681 00:48:12,550 --> 00:48:16,440 But an i somewhere here for all the factors and it 682 00:48:16,440 --> 00:48:18,790 will not be Hermitian. 683 00:48:18,790 --> 00:48:22,620 So this is why it's a real vector space. 684 00:48:22,620 --> 00:48:31,180 Multiplication by real numbers preserves Hermiticity. 685 00:48:38,400 --> 00:48:41,620 So that's surprising. 686 00:48:41,620 --> 00:48:44,520 So again, illustrates that nobody 687 00:48:44,520 --> 00:48:47,170 would say this is a real vector. 688 00:48:47,170 --> 00:48:52,130 But it really should be thought as a vector over real numbers. 689 00:48:52,130 --> 00:48:56,130 Vector space over real numbers. 690 00:48:56,130 --> 00:48:58,640 Two more examples. 691 00:48:58,640 --> 00:49:04,250 And they are kind of interesting. 692 00:49:16,210 --> 00:49:22,710 So the next example is the set of polynomials as vector space. 693 00:49:22,710 --> 00:49:26,210 So that, again, is sort of a very imaginative thing. 694 00:49:26,210 --> 00:49:33,600 The set of polynomials p of z. 695 00:49:37,380 --> 00:49:45,230 Here, z belongs to some field and p of z, which 696 00:49:45,230 --> 00:49:50,120 is a function of z, also belongs to the same field. 697 00:49:50,120 --> 00:49:52,700 And each polynomial has coefficient. 698 00:49:52,700 --> 00:50:01,790 So any p of z is a0 plus a1 z plus a2 z 699 00:50:01,790 --> 00:50:07,170 squared plus-- up to some an zn. 700 00:50:07,170 --> 00:50:10,580 A polynomial is supposed to end That's 701 00:50:10,580 --> 00:50:12,160 pretty important about polynomials. 702 00:50:12,160 --> 00:50:16,330 So the dots don't go up forever. 703 00:50:16,330 --> 00:50:21,770 So here it is, the a i's also belong to the field. 704 00:50:21,770 --> 00:50:23,030 So looked at this polynomials. 705 00:50:26,310 --> 00:50:29,630 We have the letter z and they have these coefficients 706 00:50:29,630 --> 00:50:30,520 which are numbers. 707 00:50:30,520 --> 00:50:38,230 So a real polynomial-- you know 2 plus x plus x squared. 708 00:50:38,230 --> 00:50:42,380 So you have your real numbers times this general variable 709 00:50:42,380 --> 00:50:44,910 that it's also supposed to be real. 710 00:50:44,910 --> 00:50:47,250 So you could have it real. 711 00:50:47,250 --> 00:50:48,370 You could have it complex. 712 00:50:48,370 --> 00:50:50,540 So that's a polynomial. 713 00:50:50,540 --> 00:50:53,170 How is that a vector space? 714 00:50:53,170 --> 00:51:00,190 Well, it's a vector space-- the space 715 00:51:00,190 --> 00:51:16,490 p of F of those polynomials-- of all polynomials 716 00:51:16,490 --> 00:51:25,410 is a vector space over F. And why is that? 717 00:51:25,410 --> 00:51:27,910 Well, you can take-- again, there's 718 00:51:27,910 --> 00:51:29,620 some implicit definitions. 719 00:51:29,620 --> 00:51:31,410 How do you sum polynomials? 720 00:51:31,410 --> 00:51:34,540 Well, you sum the independent coefficients. 721 00:51:34,540 --> 00:51:37,440 You just sum them and factor out. 722 00:51:37,440 --> 00:51:40,910 So there's an obvious definition of sum. 723 00:51:40,910 --> 00:51:44,120 How do you multiply a polynomial by a number? 724 00:51:44,120 --> 00:51:47,550 Obvious definition, you multiply everything by a number. 725 00:51:47,550 --> 00:51:50,150 If you sum polynomials, you get polynomials. 726 00:51:50,150 --> 00:51:53,690 Given a polynomial, there is a negative polynomial 727 00:51:53,690 --> 00:51:56,410 that adds up to 0. 728 00:51:56,410 --> 00:52:00,480 There's a 0 when all the coefficients is 0. 729 00:52:00,480 --> 00:52:02,760 And it has all the nice properties. 730 00:52:02,760 --> 00:52:06,710 Now, this example is more nontrivial 731 00:52:06,710 --> 00:52:10,710 because you would think, as opposed 732 00:52:10,710 --> 00:52:13,830 to the previous examples, that this is probably 733 00:52:13,830 --> 00:52:17,960 infinite dimensional because it has the linear polynomial, 734 00:52:17,960 --> 00:52:21,360 the quadratic, the cubic, the quartic, the quintic, all 735 00:52:21,360 --> 00:52:23,500 of them together. 736 00:52:23,500 --> 00:52:28,090 And yes, we'll see that in a second. 737 00:52:28,090 --> 00:52:31,900 So set of polynomials. 738 00:52:31,900 --> 00:52:32,740 5. 739 00:52:32,740 --> 00:52:34,210 Another example, 5. 740 00:52:37,200 --> 00:52:43,660 The set F infinity of infinite sequences. 741 00:52:49,680 --> 00:52:55,800 Sequences x1, x2, infinite sequences where the x i's 742 00:52:55,800 --> 00:52:59,170 are in the field. 743 00:52:59,170 --> 00:53:01,210 So you've got an infinite sequence 744 00:53:01,210 --> 00:53:03,610 and you want to add another infinite sequence. 745 00:53:03,610 --> 00:53:05,890 Well, you add the first element, the second elements. 746 00:53:05,890 --> 00:53:08,433 It's like an infinite column vector. 747 00:53:08,433 --> 00:53:12,900 Sometimes mathematicians like to write column vectors like that 748 00:53:12,900 --> 00:53:14,460 because it's practical. 749 00:53:14,460 --> 00:53:16,680 It saves space on a page. 750 00:53:16,680 --> 00:53:21,010 The vertical one, you start writing and the pages 751 00:53:21,010 --> 00:53:22,020 grow very fast. 752 00:53:22,020 --> 00:53:24,840 So here's an infinite sequence. 753 00:53:24,840 --> 00:53:28,330 And think of it as a vertical one if you wish. 754 00:53:28,330 --> 00:53:30,350 And all elements are here, but there 755 00:53:30,350 --> 00:53:34,310 are infinitely many in every sequence. 756 00:53:34,310 --> 00:53:40,140 And of course, the set of all infinite sequences is infinite. 757 00:53:40,140 --> 00:53:43,380 So this is a vector space over F. 758 00:53:43,380 --> 00:53:45,580 Again, because all the numbers are here, 759 00:53:45,580 --> 00:53:55,820 so it's a vector space over F. 760 00:53:55,820 --> 00:53:57,400 And last example. 761 00:54:07,340 --> 00:54:12,470 Our last example is a familiar one in physics, 762 00:54:12,470 --> 00:54:18,010 is the set of complex functions in an interval. 763 00:54:18,010 --> 00:54:36,110 Set of complex functions on an interval x from 0 to L. 764 00:54:36,110 --> 00:54:39,490 So a set of complex functions f of x 765 00:54:39,490 --> 00:54:41,930 I could put here on an interval [INAUDIBLE]. 766 00:54:41,930 --> 00:54:47,810 So this is a complex vector space. 767 00:54:47,810 --> 00:54:49,060 Vector space. 768 00:54:54,290 --> 00:54:58,080 The last three examples, probably you 769 00:54:58,080 --> 00:55:01,570 would agree that there are infinite dimensional, 770 00:55:01,570 --> 00:55:06,920 even though I've not defined what that means very precisely. 771 00:55:06,920 --> 00:55:09,340 But that's what we're going to try to understand now. 772 00:55:09,340 --> 00:55:12,490 We're supposed to understand the concept of dimensionality. 773 00:55:12,490 --> 00:55:16,580 So let's get to that concept now. 774 00:55:16,580 --> 00:55:23,010 So in terms of dimensionality, to build this idea 775 00:55:23,010 --> 00:55:24,743 you need a definition. 776 00:55:27,830 --> 00:55:32,670 You need to know the term subspace of a vector space. 777 00:55:32,670 --> 00:55:36,160 What is a subspace of a vector space? 778 00:55:36,160 --> 00:55:41,320 A subspace of a vector space is a subset of the vector space 779 00:55:41,320 --> 00:55:43,400 that is still a vector space. 780 00:55:43,400 --> 00:55:45,920 So that's why it's called subspace. 781 00:55:45,920 --> 00:55:47,560 It's different from subset. 782 00:55:47,560 --> 00:56:08,870 So a subspace of V is a subset of V that is a vector space. 783 00:56:14,690 --> 00:56:18,800 So in particular, it must contain the vector 0 784 00:56:18,800 --> 00:56:23,535 because any vector space contains the vector 0. 785 00:56:27,410 --> 00:56:31,060 One of the ways you sometimes want to understand the vector 786 00:56:31,060 --> 00:56:37,500 space is by representing it as a sum of smaller vector spaces. 787 00:56:37,500 --> 00:56:40,180 And we will do that when we consider, for example, 788 00:56:40,180 --> 00:56:42,160 angular momentum in detail. 789 00:56:42,160 --> 00:56:51,770 So you want to write a vector space as a sum of subspaces. 790 00:56:51,770 --> 00:56:53,780 So what is that called? 791 00:56:53,780 --> 00:56:56,220 It's called a direct sum. 792 00:56:56,220 --> 00:57:02,400 So if you can write-- here is the equation. 793 00:57:02,400 --> 00:57:09,570 You say V is equal to u1 direct sum with u2 direct sum 794 00:57:09,570 --> 00:57:16,680 with u3 direct sum with u m. 795 00:57:16,680 --> 00:57:23,970 When we say this, we mean the following. 796 00:57:23,970 --> 00:57:33,276 That the ui's are subspaces of V. 797 00:57:33,276 --> 00:57:42,760 And any V in the vector space can 798 00:57:42,760 --> 00:58:01,550 be written uniquely as a1 u1 plus a2 u2 plus a n u 799 00:58:01,550 --> 00:58:10,130 n with ui [INAUDIBLE] capital Ui. 800 00:58:10,130 --> 00:58:14,500 So let me review what we just said. 801 00:58:14,500 --> 00:58:17,190 So you have a vector space and you 802 00:58:17,190 --> 00:58:21,750 want to decompose it in sort of basic ingredients. 803 00:58:21,750 --> 00:58:23,680 This is called a direct sum. 804 00:58:26,460 --> 00:58:30,270 V is a direct sum of subspaces. 805 00:58:30,270 --> 00:58:31,237 Direct sum. 806 00:58:35,140 --> 00:58:39,300 And the Ui's are subspaces of V. But what 807 00:58:39,300 --> 00:58:42,090 must happen for this to be true is 808 00:58:42,090 --> 00:58:45,310 that once you take any vector here, 809 00:58:45,310 --> 00:58:47,800 you can write it as a sum of a vector 810 00:58:47,800 --> 00:58:51,530 here, a vector here, a vector here, a vector everywhere. 811 00:58:51,530 --> 00:58:54,040 And it must be done uniquely. 812 00:58:54,040 --> 00:58:56,810 If you can do this in more than one way, 813 00:58:56,810 --> 00:58:59,160 this is not a direct sum. 814 00:58:59,160 --> 00:59:02,270 These subspaces kind of overlap. 815 00:59:02,270 --> 00:59:06,056 They're not doing the decomposition in a minimal way. 816 00:59:06,056 --> 00:59:06,760 Yes. 817 00:59:06,760 --> 00:59:09,010 AUDIENCE: Does the expression of V have to be a linear 818 00:59:09,010 --> 00:59:10,600 combination of the vectors of the U, 819 00:59:10,600 --> 00:59:15,367 or just sums of the U sub i's? 820 00:59:15,367 --> 00:59:17,033 PROFESSOR: It's some linear combination. 821 00:59:21,430 --> 00:59:23,900 Look, the interpretation, for example, R2. 822 00:59:26,670 --> 00:59:29,880 The normal vector space R2. 823 00:59:29,880 --> 00:59:34,100 You have an intuition quite clearly that any vector here 824 00:59:34,100 --> 00:59:41,150 is a unique sum of this component along this subspace 825 00:59:41,150 --> 00:59:45,220 and this component along this subspace. 826 00:59:45,220 --> 00:59:52,730 So it's a trivial example, but the vector space R2 827 00:59:52,730 --> 00:59:58,120 has a vector subspace R1 here and a vector subspace R1. 828 00:59:58,120 --> 01:00:01,250 Any vector in R2 is uniquely written 829 01:00:01,250 --> 01:00:03,780 as a sum of these two vectors. 830 01:00:03,780 --> 01:00:09,345 That means that R2 is really R1 plus R1. 831 01:00:12,020 --> 01:00:13,093 Yes. 832 01:00:13,093 --> 01:00:14,905 AUDIENCE: [INAUDIBLE]. 833 01:00:14,905 --> 01:00:19,520 Is it redundant to say that that-- because a1 u1 is also 834 01:00:19,520 --> 01:00:22,030 in big U sub 1. 835 01:00:22,030 --> 01:00:22,900 PROFESSOR: Oh. 836 01:00:22,900 --> 01:00:23,590 Oh, yes. 837 01:00:23,590 --> 01:00:24,430 You're right. 838 01:00:24,430 --> 01:00:25,460 No, I'm sorry. 839 01:00:25,460 --> 01:00:26,520 I shouldn't write those. 840 01:00:26,520 --> 01:00:28,720 I'm sorry. 841 01:00:28,720 --> 01:00:31,040 That's absolutely right. 842 01:00:31,040 --> 01:00:34,770 If I had that in my notes, it was a mistake. 843 01:00:34,770 --> 01:00:35,370 Thank you. 844 01:00:35,370 --> 01:00:36,380 That was very good. 845 01:00:36,380 --> 01:00:38,470 Did I have that in my notes? 846 01:00:38,470 --> 01:00:42,030 No, I had it as you said it. 847 01:00:42,030 --> 01:00:42,560 True. 848 01:00:42,560 --> 01:00:46,770 So can be written uniquely as a vector in 849 01:00:46,770 --> 01:00:48,310 first, a vector in the second. 850 01:00:48,310 --> 01:00:52,220 And the a's are absolutely not necessary. 851 01:00:52,220 --> 01:00:53,200 OK. 852 01:00:53,200 --> 01:01:06,070 So let's go ahead then and say the following things. 853 01:01:06,070 --> 01:01:07,870 So here we're going to try to get 854 01:01:07,870 --> 01:01:14,210 to the concept of dimensionality in a precise way. 855 01:01:14,210 --> 01:01:16,269 Yes. 856 01:01:16,269 --> 01:01:18,684 AUDIENCE: [INAUDIBLE]. 857 01:01:18,684 --> 01:01:21,082 PROFESSOR: Right, the last one is m. 858 01:01:21,082 --> 01:01:21,582 Thank you. 859 01:01:38,600 --> 01:01:39,290 All right. 860 01:01:46,220 --> 01:01:49,070 The concept of dimensionality of a vector space 861 01:01:49,070 --> 01:01:51,350 is something that you intuitively understand. 862 01:01:51,350 --> 01:01:56,900 It's sort of how many linearly independent vectors 863 01:01:56,900 --> 01:02:01,610 you need to describe the whole set of vectors. 864 01:02:01,610 --> 01:02:05,970 So that is the number you're trying to get to. 865 01:02:05,970 --> 01:02:10,320 I'll follow it up in a slightly rigorous way 866 01:02:10,320 --> 01:02:13,150 to be able to do infinite dimensional space as well. 867 01:02:13,150 --> 01:02:18,470 So we will consider something called a list of vectors. 868 01:02:18,470 --> 01:02:22,210 List of vectors. 869 01:02:22,210 --> 01:02:27,020 And that will be something like v1, v2 vectors in a vector 870 01:02:27,020 --> 01:02:28,050 space up to vn. 871 01:02:30,640 --> 01:02:38,650 Any list of vectors has finite length. 872 01:02:38,650 --> 01:02:44,390 So we don't accept infinite lists by definition. 873 01:02:48,480 --> 01:02:51,420 You can ask, once you have a list of vectors, 874 01:02:51,420 --> 01:02:57,580 what is the vector subspace spanned by this list? 875 01:02:57,580 --> 01:03:00,420 How much do you reach with that list? 876 01:03:00,420 --> 01:03:04,440 So we call it the span of the list. 877 01:03:04,440 --> 01:03:10,680 The span of the list, vn. 878 01:03:10,680 --> 01:03:23,570 And it's the set of all linear combinations a1 v1 plus a2 v2 879 01:03:23,570 --> 01:03:32,500 plus a n vn for ai in the field. 880 01:03:32,500 --> 01:03:36,200 So the span of the list is all possible products 881 01:03:36,200 --> 01:03:42,915 of your vectors on the list are-- and put like that. 882 01:03:42,915 --> 01:03:51,620 So if we say that the list spans a vector space, 883 01:03:51,620 --> 01:03:55,170 if the span of the list is the vector space. 884 01:03:55,170 --> 01:03:57,650 So that's natural language. 885 01:03:57,650 --> 01:04:01,010 We say, OK, this list spans the vector space. 886 01:04:01,010 --> 01:04:01,520 Why? 887 01:04:01,520 --> 01:04:04,170 Because if you produce the span of the list, 888 01:04:04,170 --> 01:04:06,820 it fills a vector space. 889 01:04:06,820 --> 01:04:12,050 OK, so I could say it that way. 890 01:04:12,050 --> 01:04:26,710 So here is the definition, V is finite dimensional 891 01:04:26,710 --> 01:04:29,180 if it's spanned by some list. 892 01:04:29,180 --> 01:04:35,480 If V is spanned by some list. 893 01:04:40,180 --> 01:04:41,620 So why is that? 894 01:04:41,620 --> 01:04:47,200 Because if the list is-- a definition, finite dimensional. 895 01:04:47,200 --> 01:04:48,800 If it's spanned by some list. 896 01:04:48,800 --> 01:04:52,090 If you got your list, by definition it's finite length. 897 01:04:52,090 --> 01:04:54,790 And with some set of vectors, you span everything. 898 01:04:58,380 --> 01:05:02,990 And moreover, it's infinite dimensional 899 01:05:02,990 --> 01:05:05,840 if it's not finite dimensional. 900 01:05:05,840 --> 01:05:10,775 It's kind of silly, but infinite-- a space 901 01:05:10,775 --> 01:05:24,280 V is infinite dimensional if it is not finite dimensional. 902 01:05:24,280 --> 01:05:32,370 Which is to say that there is no list that spans the space. 903 01:05:32,370 --> 01:05:36,210 So for example, this definition is tailored in a nice way. 904 01:05:36,210 --> 01:05:38,116 Like let's think of the polynomials. 905 01:05:42,010 --> 01:05:45,270 And we want to see if it's finite dimensional or infinite 906 01:05:45,270 --> 01:05:45,980 dimensional. 907 01:05:45,980 --> 01:05:51,160 So you claim it's finite dimensional. 908 01:05:51,160 --> 01:05:53,020 Let's see if it's finite dimensional. 909 01:05:53,020 --> 01:05:55,800 So we make a list of polynomials. 910 01:05:55,800 --> 01:06:00,630 The list must have some length, at least, that spans it. 911 01:06:00,630 --> 01:06:03,800 You put all these 730 polynomials 912 01:06:03,800 --> 01:06:09,830 that you think span the list, span the space, in this list. 913 01:06:09,830 --> 01:06:12,900 Now, if you look at the list, it's 720. 914 01:06:12,900 --> 01:06:14,800 You can check one by one until you 915 01:06:14,800 --> 01:06:17,385 find what is the one of highest order, 916 01:06:17,385 --> 01:06:20,700 the polynomial of highest degree. 917 01:06:20,700 --> 01:06:26,560 But if the highest degree is say, z to the 1 million, 918 01:06:26,560 --> 01:06:30,540 then any polynomial that has a z to the 2 million cannot be 919 01:06:30,540 --> 01:06:32,110 spanned by this one. 920 01:06:32,110 --> 01:06:36,240 So there's no finite list that can span this, 921 01:06:36,240 --> 01:06:40,820 so this set-- the example in 4 is 922 01:06:40,820 --> 01:06:42,720 infinite dimensional for sure. 923 01:06:45,240 --> 01:06:49,590 Example 4 is infinite dimensional. 924 01:06:57,060 --> 01:07:04,940 Well, example one is finite dimensional. 925 01:07:07,560 --> 01:07:09,900 You can see that because we can produce 926 01:07:09,900 --> 01:07:13,250 a list that spans the space. 927 01:07:13,250 --> 01:07:14,965 So look at the example 1. 928 01:07:17,620 --> 01:07:18,120 It's there. 929 01:07:22,710 --> 01:07:24,555 Well, what would be the list? 930 01:07:24,555 --> 01:07:28,090 The list would be-- list. 931 01:07:28,090 --> 01:07:33,310 You would put a vector e1, e2, up to en. 932 01:07:33,310 --> 01:07:40,500 And the vector e1 would be 1, 0, 0, 0, 0. 933 01:07:40,500 --> 01:07:45,680 The vector e2 would be 0, 1, 0, 0, 0. 934 01:07:45,680 --> 01:07:47,300 And go on like that. 935 01:07:47,300 --> 01:07:50,190 So you put 1's and 0's. 936 01:07:50,190 --> 01:07:52,000 And you have n of them. 937 01:07:52,000 --> 01:07:58,540 And certainly, the most general one is a1 times e1 a2 times e2. 938 01:07:58,540 --> 01:08:00,550 And you got the list. 939 01:08:00,550 --> 01:08:06,160 So example 1 is finite dimensional. 940 01:08:06,160 --> 01:08:10,320 A list of vectors is linearly independent. 941 01:08:10,320 --> 01:08:23,250 A list is linearly independent if a list v1 up to vn 942 01:08:23,250 --> 01:08:32,649 is linearly independent, If a1 v1 plus a2 v2 plus a n vn 943 01:08:32,649 --> 01:08:44,710 is equal to 0 has the unique solution a1 equal a2 equal 944 01:08:44,710 --> 01:08:48,649 all of them equal 0. 945 01:08:48,649 --> 01:08:57,990 So that is to mean that whenever this list satisfies 946 01:08:57,990 --> 01:09:02,060 this property-- if you want to represent the vector 947 01:09:02,060 --> 01:09:05,859 0 with this list, you must set all of them 948 01:09:05,859 --> 01:09:08,260 equal to 0, all the coefficients. 949 01:09:08,260 --> 01:09:10,850 That's clear as well in this example. 950 01:09:10,850 --> 01:09:13,170 If you want to represent the 0 vector, 951 01:09:13,170 --> 01:09:18,370 you must have 0 component against the basis vector x 952 01:09:18,370 --> 01:09:19,920 and basis vector y. 953 01:09:19,920 --> 01:09:23,550 So the list of this vector and this vector 954 01:09:23,550 --> 01:09:27,069 is linearly independent because the 0 vector 955 01:09:27,069 --> 01:09:31,180 must have 0 numbers multiplying each of them. 956 01:09:31,180 --> 01:09:36,830 So finally, we define what is a basis. 957 01:09:36,830 --> 01:09:58,266 A basis of V is a list of vectors in V 958 01:09:58,266 --> 01:10:07,255 that spans V and is linearly independent. 959 01:10:13,500 --> 01:10:16,200 So what is a basis? 960 01:10:16,200 --> 01:10:18,420 Well, you should have enough vectors 961 01:10:18,420 --> 01:10:21,970 to represent every vector. 962 01:10:21,970 --> 01:10:25,970 So it must span V. And what else should it have? 963 01:10:25,970 --> 01:10:29,650 It shouldn't have extra vectors that you don't need. 964 01:10:29,650 --> 01:10:31,110 It should be minimal. 965 01:10:31,110 --> 01:10:33,150 It should be all linearly independent. 966 01:10:33,150 --> 01:10:36,610 You shouldn't have added more stuff to it. 967 01:10:36,610 --> 01:10:43,040 So any finite dimensional vector space has a basis. 968 01:10:45,740 --> 01:10:50,200 It's easy to do it. 969 01:10:50,200 --> 01:10:53,170 There's another thing that one can prove. 970 01:10:53,170 --> 01:10:58,110 It may look kind of obvious, but it requires a small proof 971 01:10:58,110 --> 01:11:00,940 that if you have-- the bases are not unique. 972 01:11:00,940 --> 01:11:03,070 It's something we're going to exploit all the time. 973 01:11:03,070 --> 01:11:05,800 One basis, another basis, a third basis. 974 01:11:05,800 --> 01:11:08,650 We're going to change basis all the time. 975 01:11:08,650 --> 01:11:13,070 Well, the bases are not unique, but the length 976 01:11:13,070 --> 01:11:17,130 of the bases of a vector space is always the same. 977 01:11:17,130 --> 01:11:20,850 So the length of the list is-- a number 978 01:11:20,850 --> 01:11:23,810 is the same whatever base you choose. 979 01:11:23,810 --> 01:11:25,920 And that length is what is called 980 01:11:25,920 --> 01:11:28,610 the dimension of the vector space. 981 01:11:28,610 --> 01:11:41,700 So the dimension of a vector space 982 01:11:41,700 --> 01:12:00,230 is the length of any bases of V. And therefore, 983 01:12:00,230 --> 01:12:01,850 it's a well-defined concept. 984 01:12:01,850 --> 01:12:06,060 Any base of a finite vector space has the same length, 985 01:12:06,060 --> 01:12:08,860 and the dimension is that number. 986 01:12:08,860 --> 01:12:11,070 So there was a question. 987 01:12:11,070 --> 01:12:12,658 Yes? 988 01:12:12,658 --> 01:12:14,514 AUDIENCE: Is there any difference 989 01:12:14,514 --> 01:12:16,370 between bases [INAUDIBLE]? 990 01:12:20,900 --> 01:12:22,900 PROFESSOR: No, absolutely not. 991 01:12:22,900 --> 01:12:25,600 You could have a basis, for example, 992 01:12:25,600 --> 01:12:28,570 of R2, which is this vector. 993 01:12:28,570 --> 01:12:33,010 The first and the second is this vector. 994 01:12:33,010 --> 01:12:37,020 And any vector is a linear superposition 995 01:12:37,020 --> 01:12:40,680 of these two vectors with some coefficients and it's unique. 996 01:12:40,680 --> 01:12:43,580 You can find the coefficients. 997 01:12:43,580 --> 01:12:46,426 AUDIENCE: [INAUDIBLE]. 998 01:12:46,426 --> 01:12:47,050 PROFESSOR: Yes. 999 01:12:47,050 --> 01:12:52,140 But you see, here is exactly what I wanted to make clear. 1000 01:12:52,140 --> 01:12:54,480 We're putting the vector space and we're 1001 01:12:54,480 --> 01:12:56,630 putting the least possible structure. 1002 01:12:56,630 --> 01:13:00,650 I didn't say how to take the inner product of two vectors. 1003 01:13:00,650 --> 01:13:02,660 It's not a definition of a vector space. 1004 01:13:02,660 --> 01:13:04,610 It's something we'll put later. 1005 01:13:04,610 --> 01:13:08,190 And then, we will be able to ask whether the basis is 1006 01:13:08,190 --> 01:13:09,790 orthonormal or not. 1007 01:13:09,790 --> 01:13:12,240 But the basis exists. 1008 01:13:12,240 --> 01:13:15,520 Even though you have no definition of an inner product, 1009 01:13:15,520 --> 01:13:19,610 you can talk about basis without any confusion. 1010 01:13:19,610 --> 01:13:23,420 You can also talk about the matrix representation 1011 01:13:23,420 --> 01:13:24,530 of an operator. 1012 01:13:24,530 --> 01:13:27,230 And you don't need an inner product, 1013 01:13:27,230 --> 01:13:30,370 which is sometimes very unclear. 1014 01:13:30,370 --> 01:13:34,490 You can talk about the trace of an operator 1015 01:13:34,490 --> 01:13:37,390 and you don't need an inner product. 1016 01:13:37,390 --> 01:13:40,610 You can talk about eigenvectors and eigenvalues 1017 01:13:40,610 --> 01:13:43,470 and you don't need an inner product. 1018 01:13:43,470 --> 01:13:45,185 The only thing you need the inner product 1019 01:13:45,185 --> 01:13:46,820 is to get numbers. 1020 01:13:46,820 --> 01:13:51,330 And we'll use them to use [INAUDIBLE] to get numbers. 1021 01:13:51,330 --> 01:13:53,380 But it can wait. 1022 01:13:53,380 --> 01:13:55,490 It's better than you see all that you 1023 01:13:55,490 --> 01:13:58,770 can do without introducing more things, 1024 01:13:58,770 --> 01:14:00,980 and then introduce them. 1025 01:14:00,980 --> 01:14:07,020 So let me explain a little more this concept. 1026 01:14:07,020 --> 01:14:12,900 We were talking about this base, this vector space 1, 1027 01:14:12,900 --> 01:14:13,590 for example. 1028 01:14:13,590 --> 01:14:20,645 And we produced a list that spans e1, e2, up to en. 1029 01:14:20,645 --> 01:14:23,000 And those were these vectors. 1030 01:14:23,000 --> 01:14:26,340 Now, this list not only spans, but they 1031 01:14:26,340 --> 01:14:28,010 are linearly independent. 1032 01:14:28,010 --> 01:14:31,650 If you put a1 times this plus a2 times 1033 01:14:31,650 --> 01:14:33,520 this and you set it all equal to 0. 1034 01:14:33,520 --> 01:14:37,990 Well, each entry will be 0, and all the a's are 0. 1035 01:14:37,990 --> 01:14:42,880 So these e's that you put here on that list is actually 1036 01:14:42,880 --> 01:14:44,520 a basis. 1037 01:14:44,520 --> 01:14:47,810 Therefore, the length of that basis is the dimensionality. 1038 01:14:47,810 --> 01:14:54,700 And this space has dimension N. You 1039 01:14:54,700 --> 01:14:58,960 should be able to prove that this space has been dimension 1040 01:14:58,960 --> 01:15:12,550 m times N. Now, let me do the Hermitian-- these matrices. 1041 01:15:12,550 --> 01:15:15,670 And try to figure out the dimensionality 1042 01:15:15,670 --> 01:15:19,360 of the space of Hermitian matrices. 1043 01:15:19,360 --> 01:15:20,870 So here they are. 1044 01:15:20,870 --> 01:15:24,390 This is the most general Hermitian matrix. 1045 01:15:24,390 --> 01:15:31,190 And I'm going to produce for you a list of four vectors. 1046 01:15:31,190 --> 01:15:34,570 Vectors-- yes, they're matrices, but we call them vectors. 1047 01:15:34,570 --> 01:15:35,900 So here is the list. 1048 01:15:40,340 --> 01:15:45,110 The unit matrix, the first Pauli matrix, the second Pauli 1049 01:15:45,110 --> 01:15:49,270 matrix, and the third Pauli matrix. 1050 01:15:49,270 --> 01:15:53,330 All right, let's see how far do we get from there. 1051 01:15:53,330 --> 01:15:56,710 OK, this is a list of vectors in the vector space 1052 01:15:56,710 --> 01:15:59,090 because all of them are Hermitian. 1053 01:15:59,090 --> 01:15:59,590 Good. 1054 01:16:02,690 --> 01:16:04,250 Do they span? 1055 01:16:04,250 --> 01:16:08,570 Well, you calculate the most general Hermitian matrix 1056 01:16:08,570 --> 01:16:09,420 of this form. 1057 01:16:09,420 --> 01:16:12,750 You just put arbitrary complex numbers 1058 01:16:12,750 --> 01:16:17,980 and require that the matrix be equal to its matrix complex 1059 01:16:17,980 --> 01:16:19,420 conjugate and transpose. 1060 01:16:19,420 --> 01:16:21,245 So this is the most general one. 1061 01:16:21,245 --> 01:16:25,210 Do I obtain this matrix from this one's? 1062 01:16:25,210 --> 01:16:34,440 Yes I just have to put 1 times c plus a times sigma 1 plus b 1063 01:16:34,440 --> 01:16:38,990 times sigma 2 plus d times sigma 3. 1064 01:16:38,990 --> 01:16:44,840 So any Hermitian matrix can be obtained 1065 01:16:44,840 --> 01:16:47,150 as the span of this list. 1066 01:16:50,160 --> 01:16:53,630 Is this list linearly independent? 1067 01:16:53,630 --> 01:16:58,350 So I have to go here and set this equal to 0 1068 01:16:58,350 --> 01:17:03,890 and see if it sets to 0 all these coefficients. 1069 01:17:03,890 --> 01:17:08,510 Well, it's the same thing as setting to 0 all this matrix. 1070 01:17:08,510 --> 01:17:15,160 Well, if c plus d and c minus d are 0, then c and d are 0. 1071 01:17:15,160 --> 01:17:20,030 If this is 0, it must be a 0 and b 0, so all of them are 0. 1072 01:17:20,030 --> 01:17:22,970 So yes, it's linearly independent. 1073 01:17:22,970 --> 01:17:24,440 It spans. 1074 01:17:24,440 --> 01:17:27,890 Therefore, you've proven completely rigorously 1075 01:17:27,890 --> 01:17:32,525 that this vector space is dimension 4. 1076 01:17:41,940 --> 01:17:45,270 This vector space-- I will actually 1077 01:17:45,270 --> 01:17:49,890 leave it as an exercise for you to show that this vector 1078 01:17:49,890 --> 01:17:51,214 space is infinite dimensional. 1079 01:17:51,214 --> 01:17:53,130 You say, of course, it's infinite dimensional. 1080 01:17:53,130 --> 01:17:55,530 It has infinite sequences. 1081 01:17:55,530 --> 01:17:58,000 Well, you have to show that if you 1082 01:17:58,000 --> 01:18:01,740 have a finite list of those infinite sequences, 1083 01:18:01,740 --> 01:18:07,240 like 300 sequences, they span that. 1084 01:18:07,240 --> 01:18:08,650 They cannot span that. 1085 01:18:08,650 --> 01:18:12,620 So it takes a little work. 1086 01:18:12,620 --> 01:18:14,235 It's interesting to think about it. 1087 01:18:14,235 --> 01:18:18,510 I think you will enjoy trying to think about this stuff. 1088 01:18:18,510 --> 01:18:24,400 So that's our discussion of dimensionality. 1089 01:18:24,400 --> 01:18:29,620 So this one is a little harder to make sure 1090 01:18:29,620 --> 01:18:31,100 it's infinite dimensional. 1091 01:18:31,100 --> 01:18:34,310 And this one is, yet, a bit harder than that one 1092 01:18:34,310 --> 01:18:36,210 but it can also be done. 1093 01:18:36,210 --> 01:18:37,535 This is infinite dimensional. 1094 01:18:40,356 --> 01:18:41,730 And this is infinite dimensional. 1095 01:18:44,600 --> 01:18:49,300 In the last two minute, I want to tell you a little bit-- one 1096 01:18:49,300 --> 01:18:53,130 definition and let you go with that, 1097 01:18:53,130 --> 01:18:56,413 is the definition of a linear operator. 1098 01:19:01,280 --> 01:19:03,130 So here is one thing. 1099 01:19:03,130 --> 01:19:09,270 So you can be more general, and we won't be that general. 1100 01:19:09,270 --> 01:19:13,350 But when you talk about linear maps, 1101 01:19:13,350 --> 01:19:20,980 you have one vector space and another vector space, v and w. 1102 01:19:20,980 --> 01:19:26,930 This is a vector space and this is a vector space. 1103 01:19:26,930 --> 01:19:31,640 And in general, a map from here is sometimes called, 1104 01:19:31,640 --> 01:19:34,610 if it satisfies the property, a linear map. 1105 01:19:37,120 --> 01:19:39,970 And the key thing is that in all generality, 1106 01:19:39,970 --> 01:19:43,695 these two vector spaces may not have the same dimension. 1107 01:19:43,695 --> 01:19:46,901 It might be one vector space and another very different vector 1108 01:19:46,901 --> 01:19:47,400 space. 1109 01:19:47,400 --> 01:19:50,230 You go from one to the other. 1110 01:19:50,230 --> 01:19:54,530 Now, when you have a vector space v 1111 01:19:54,530 --> 01:19:57,770 and you map to the same vector space, 1112 01:19:57,770 --> 01:20:00,060 this is also a linear map, but this 1113 01:20:00,060 --> 01:20:04,860 is called an operator or a linear operator. 1114 01:20:07,600 --> 01:20:10,980 And what is a linear operator therefore? 1115 01:20:10,980 --> 01:20:20,310 A linear operator is a function T. Let's 1116 01:20:20,310 --> 01:20:26,480 call the linear operator T. It takes v to v. In which way? 1117 01:20:26,480 --> 01:20:35,110 Well, T acting u plus v, on the sum of vectors, 1118 01:20:35,110 --> 01:20:44,790 is Tu plus T v. And T acting on a times a vector 1119 01:20:44,790 --> 01:20:49,380 is a times T of the vector. 1120 01:20:49,380 --> 01:20:53,160 These two things make it into something 1121 01:20:53,160 --> 01:20:55,710 we call a linear operator. 1122 01:20:55,710 --> 01:21:00,400 It acts on the sum of vectors linearly 1123 01:21:00,400 --> 01:21:02,900 and on a number times a vector. 1124 01:21:02,900 --> 01:21:06,236 The number goes out and you act on the vector. 1125 01:21:06,236 --> 01:21:11,300 So all you need to know for what a linear operator is, 1126 01:21:11,300 --> 01:21:14,890 is how it acts on basis vectors. 1127 01:21:14,890 --> 01:21:17,620 Because any vector on the vector space 1128 01:21:17,620 --> 01:21:19,800 is a superposition of basis vectors. 1129 01:21:19,800 --> 01:21:23,410 So if you tell me how it acts on the basis vectors, 1130 01:21:23,410 --> 01:21:24,960 you know everything. 1131 01:21:24,960 --> 01:21:29,000 So we will figure out how the matrix representation 1132 01:21:29,000 --> 01:21:34,530 of the operators arises from how it acts on the basis vectors. 1133 01:21:34,530 --> 01:21:36,940 And you don't need an inner product. 1134 01:21:36,940 --> 01:21:39,460 The reason people think of this is they say, 1135 01:21:39,460 --> 01:21:44,970 oh, the T i j matrix element of T 1136 01:21:44,970 --> 01:21:49,310 is the inner product of the operator between i and j. 1137 01:21:49,310 --> 01:21:51,960 And this is true. 1138 01:21:51,960 --> 01:21:54,340 But for that you need [? brass ?] 1139 01:21:54,340 --> 01:21:56,690 and inner product, all these things. 1140 01:21:56,690 --> 01:21:58,460 And they're not necessary. 1141 01:21:58,460 --> 01:22:00,620 We'll define this without that. 1142 01:22:00,620 --> 01:22:02,100 We don't need it. 1143 01:22:02,100 --> 01:22:06,640 So see you next time, and we'll continue that. 1144 01:22:06,640 --> 01:22:09,940 [APPLAUSE] 1145 01:22:09,940 --> 01:22:11,790 Thank you.