1 00:00:00,100 --> 00:00:01,680 The following content is provided 2 00:00:01,680 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,140 to offer high quality educational resources for free. 5 00:00:10,140 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:16,498 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,498 --> 00:00:17,390 at ocw.mit.edu. 8 00:00:28,700 --> 00:00:31,780 ARAM HARROW: So let's get started. 9 00:00:31,780 --> 00:00:35,990 This week Professor Zwiebach is away, 10 00:00:35,990 --> 00:00:37,770 and I'll be doing today's lecture. 11 00:00:37,770 --> 00:00:42,200 And Will Detmold will do the one on Wednesday. 12 00:00:42,200 --> 00:00:45,010 The normal office hours, unfortunately, 13 00:00:45,010 --> 00:00:46,440 will not be held today. 14 00:00:46,440 --> 00:00:49,820 One of us will cover his hours on Wednesday though. 15 00:00:49,820 --> 00:00:53,950 And you should also just email either me or Professor Detmold 16 00:00:53,950 --> 00:00:55,450 if you want to set up an appointment 17 00:00:55,450 --> 00:00:58,720 to talk to in the next few days. 18 00:00:58,720 --> 00:01:00,720 What I'm going to talk about today 19 00:01:00,720 --> 00:01:04,629 will be more about the linear algebra that's 20 00:01:04,629 --> 00:01:07,430 behind all of quantum mechanics. 21 00:01:07,430 --> 00:01:10,980 And, at the end of last time-- last lecture 22 00:01:10,980 --> 00:01:14,740 you heard about vector spaces from a more abstract 23 00:01:14,740 --> 00:01:17,030 perspective than the usual vectors 24 00:01:17,030 --> 00:01:19,450 are columns of numbers perspective. 25 00:01:19,450 --> 00:01:21,955 Today we're going to look at operators, 26 00:01:21,955 --> 00:01:23,330 which act on vector spaces, which 27 00:01:23,330 --> 00:01:27,390 are linear maps from a vector space to itself. 28 00:01:27,390 --> 00:01:31,660 And they're, in a sense, equivalent 29 00:01:31,660 --> 00:01:34,430 to the familiar idea of matrices, 30 00:01:34,430 --> 00:01:37,940 which are squares or rectangles of numbers. 31 00:01:37,940 --> 00:01:42,930 But are work in this more abstract setting 32 00:01:42,930 --> 00:01:46,300 of vector spaces, which has a number of advantages. 33 00:01:46,300 --> 00:01:47,830 For example, of being able to deal 34 00:01:47,830 --> 00:01:51,110 with infinite dimensional vector spaces and also 35 00:01:51,110 --> 00:01:54,340 of being able to talk about basis independent properties. 36 00:01:54,340 --> 00:01:56,940 And so I'll tell you all about that today. 37 00:01:56,940 --> 00:01:59,329 So we'll talk about how to define operators, 38 00:01:59,329 --> 00:02:01,370 some examples, some of their properties, and then 39 00:02:01,370 --> 00:02:07,050 finally how to relate them to the familiar idea of matrices. 40 00:02:07,050 --> 00:02:09,169 I'll then talk about eigenvectors and eigenvalues 41 00:02:09,169 --> 00:02:11,250 from this operator prospective. 42 00:02:11,250 --> 00:02:13,670 And, depending on time today, a little bit 43 00:02:13,670 --> 00:02:15,300 about inner products, which you'll 44 00:02:15,300 --> 00:02:16,960 hear more about the future. 45 00:02:16,960 --> 00:02:20,280 These numbers here correspond to the sections of the notes 46 00:02:20,280 --> 00:02:23,020 that these refer to. 47 00:02:23,020 --> 00:02:27,890 So let me first-- this is a little bit mathematical 48 00:02:27,890 --> 00:02:31,120 and perhaps dry at first. 49 00:02:31,120 --> 00:02:34,010 The payoff is more distant than usual 50 00:02:34,010 --> 00:02:35,844 for things you'll hear in quantum mechanics. 51 00:02:35,844 --> 00:02:37,301 I just want to mention a little bit 52 00:02:37,301 --> 00:02:38,570 about the motivation for it. 53 00:02:43,930 --> 00:02:49,025 So operators, of course, are how we define observables. 54 00:02:52,550 --> 00:02:56,370 And so if we want to know what the properties of observables, 55 00:02:56,370 --> 00:03:05,440 of which a key example are of Hamiltonians, 56 00:03:05,440 --> 00:03:08,430 then we need to know about operators. 57 00:03:08,430 --> 00:03:11,766 They also, as you will see in the future, 58 00:03:11,766 --> 00:03:13,265 are useful for talking about states. 59 00:03:17,090 --> 00:03:21,140 Right now, states are described as elements of a vector space, 60 00:03:21,140 --> 00:03:23,460 but in the future you'll learn a different formalism 61 00:03:23,460 --> 00:03:27,810 in which states are also described as dense operators. 62 00:03:27,810 --> 00:03:30,555 What are called density operators or density matrices. 63 00:03:33,150 --> 00:03:36,560 And finally, operators are also useful in describing 64 00:03:36,560 --> 00:03:38,050 symmetries of quantum systems. 65 00:03:41,000 --> 00:03:42,990 So already in classical mechanics, 66 00:03:42,990 --> 00:03:45,480 symmetries have been very important for understanding 67 00:03:45,480 --> 00:03:48,560 things like momentum conservation and energy 68 00:03:48,560 --> 00:03:50,000 conservation so on. 69 00:03:50,000 --> 00:03:52,240 They'll be even more important in quantum mechanics 70 00:03:52,240 --> 00:03:56,540 and will be understood through the formalism of operators. 71 00:03:56,540 --> 00:03:59,160 So these are not things that I will talk about today 72 00:03:59,160 --> 00:04:01,890 but are sort of the motivation for understanding 73 00:04:01,890 --> 00:04:05,690 very well the structure of operators now. 74 00:04:05,690 --> 00:04:15,900 So at the end of the last lecture, 75 00:04:15,900 --> 00:04:18,810 Professor Zwiebach defined linear maps. 76 00:04:18,810 --> 00:04:29,600 So this is the set of linear maps from a vector space, v, 77 00:04:29,600 --> 00:04:31,600 to a vector space w. 78 00:04:31,600 --> 00:04:33,630 And just to remind you what it means for a map 79 00:04:33,630 --> 00:04:53,050 to be linear, so T is linear if for all pairs of vectors in v, 80 00:04:53,050 --> 00:04:56,260 the way T acts on their sum is given by just T of u 81 00:04:56,260 --> 00:05:07,840 plus T of v. That's the first property. 82 00:05:07,840 --> 00:05:19,680 And second, for all vectors u and for all scalars a-- 83 00:05:19,680 --> 00:05:22,680 so f is the field that we're working over, 84 00:05:22,680 --> 00:05:28,751 it could be reals are complexes-- we have that 85 00:05:28,751 --> 00:05:41,110 if T acts on a times u, that's equal to a times t acting on u. 86 00:05:41,110 --> 00:05:44,840 So if you put these together what this means 87 00:05:44,840 --> 00:05:48,260 is that t essentially looks like multiplication. 88 00:05:48,260 --> 00:05:52,410 The way T acts on vectors is precisely what you would expect 89 00:05:52,410 --> 00:05:55,020 from the multiplication map, right? 90 00:05:55,020 --> 00:06:01,390 It has the distributive property and it commutes with scalars. 91 00:06:01,390 --> 00:06:03,879 So this is sort of informal-- I mean, 92 00:06:03,879 --> 00:06:06,045 the formal definition is here, but the informal idea 93 00:06:06,045 --> 00:06:13,045 is that T acts like multiplication. 94 00:06:16,820 --> 00:06:22,440 So if the map that squares every entry of a vector 95 00:06:22,440 --> 00:06:26,080 does not act like this, but linear operators do. 96 00:06:26,080 --> 00:06:29,220 And for this reason we often neglect the parentheses. 97 00:06:29,220 --> 00:06:38,310 So we just write TU to mean T of u, which is justified because 98 00:06:38,310 --> 00:06:40,240 of this analogy with multiplication. 99 00:06:43,720 --> 00:06:51,710 So an important special case of this is when v is equal to w. 100 00:06:54,280 --> 00:06:57,560 And so we just write l of v to denote the maps from v 101 00:06:57,560 --> 00:06:59,820 to itself. 102 00:06:59,820 --> 00:07:03,240 Which you could also write like this. 103 00:07:03,240 --> 00:07:11,758 And these are called operators on v. 104 00:07:11,758 --> 00:07:14,200 So when we talk about operators on a vector space, 105 00:07:14,200 --> 00:07:18,320 v, we mean linear maps from that vector space to itself. 106 00:07:23,370 --> 00:07:27,680 So let me illustrate this with a few examples. 107 00:07:36,810 --> 00:07:39,280 Starting with some of the examples of vector spaces 108 00:07:39,280 --> 00:07:40,960 that you saw from last time. 109 00:07:40,960 --> 00:07:48,250 So one example of a vector space is an example 110 00:07:48,250 --> 00:07:50,350 you've seen before but a different notation. 111 00:07:50,350 --> 00:07:59,680 This is the vector space of all real polynomials 112 00:07:59,680 --> 00:08:00,650 in one variable. 113 00:08:07,010 --> 00:08:12,550 So real polynomials over some variable, x. 114 00:08:12,550 --> 00:08:15,350 And over-- this is an infinite dimensional vector space-- 115 00:08:15,350 --> 00:08:17,880 and we can define various operators over it. 116 00:08:17,880 --> 00:08:22,770 For example, we can define one operator, T, 117 00:08:22,770 --> 00:08:25,740 to be like differentiation. 118 00:08:25,740 --> 00:08:32,640 So what you might write as ddx hat, 119 00:08:32,640 --> 00:08:38,179 and it's defined for any polynomial, p, to map 120 00:08:38,179 --> 00:08:39,120 p to p prime. 121 00:08:42,390 --> 00:08:45,850 So this is certainly a function from polynomials 122 00:08:45,850 --> 00:08:47,450 to polynomials. 123 00:08:47,450 --> 00:08:50,140 And you can check that it's also linear 124 00:08:50,140 --> 00:08:53,150 if you multiply the polynomial by a scalar, then 125 00:08:53,150 --> 00:08:55,620 the derivative multiplied by the same scale. 126 00:08:55,620 --> 00:08:59,330 If I take the derivative of a sum of two polynomials, 127 00:08:59,330 --> 00:09:01,450 then I get the sum of the derivatives 128 00:09:01,450 --> 00:09:03,100 of those polynomials. 129 00:09:03,100 --> 00:09:04,960 I won't write that down, but you can 130 00:09:04,960 --> 00:09:06,520 check that the properties are true. 131 00:09:06,520 --> 00:09:08,070 And this is indeed a linear operator. 132 00:09:11,680 --> 00:09:16,720 Another operator, which you've seen before, 133 00:09:16,720 --> 00:09:18,740 is multiplication by x. 134 00:09:18,740 --> 00:09:27,190 So this is defined as the map that 135 00:09:27,190 --> 00:09:30,550 simply multiplies the polynomial by x. 136 00:09:30,550 --> 00:09:32,800 Of course, this gives you another polynomial. 137 00:09:32,800 --> 00:09:36,211 And, again, you can check easily that it satisfies these two 138 00:09:36,211 --> 00:09:36,710 conditions. 139 00:09:47,640 --> 00:09:50,990 So this gives you a sense of why things 140 00:09:50,990 --> 00:09:53,630 that don't appear to be matrix-like 141 00:09:53,630 --> 00:09:56,570 can still be viewed in this operator picture. 142 00:09:59,250 --> 00:10:02,960 Another example, which you'll see 143 00:10:02,960 --> 00:10:06,750 later shows some of the slightly paradoxical features 144 00:10:06,750 --> 00:10:12,260 of infinite dimensional vector spaces, 145 00:10:12,260 --> 00:10:16,630 comes from the vector space of infinite sequences. 146 00:10:16,630 --> 00:10:30,100 So these are all the infinite sequences of reals or complexes 147 00:10:30,100 --> 00:10:33,490 or whatever f is. 148 00:10:33,490 --> 00:10:42,190 One operator we can define is the left shift operator, 149 00:10:42,190 --> 00:10:50,930 which is simply defined by shifting this entire infinite 150 00:10:50,930 --> 00:10:54,540 sequence left by one place and throwing away 151 00:10:54,540 --> 00:10:57,250 the first position. 152 00:10:57,250 --> 00:11:01,750 So you start with x2, x3, and so. 153 00:11:01,750 --> 00:11:03,480 Still goes to infinity so it still 154 00:11:03,480 --> 00:11:06,280 gives you an infinite sequence. 155 00:11:06,280 --> 00:11:08,840 So it is indeed a map-- that's the first thing you should 156 00:11:08,840 --> 00:11:12,020 check that this is indeed a map from v to itself-- 157 00:11:12,020 --> 00:11:14,769 and you can also check that it's linear, 158 00:11:14,769 --> 00:11:16,310 that it satisfies these two products. 159 00:11:18,920 --> 00:11:21,530 Another example is right shift. 160 00:11:26,310 --> 00:11:32,818 And here-- Yeah? 161 00:11:32,818 --> 00:11:34,740 AUDIENCE: So left shift was the first one or-- 162 00:11:34,740 --> 00:11:36,980 ARAM HARROW: That's right. 163 00:11:36,980 --> 00:11:41,820 So there's no back, really. 164 00:11:41,820 --> 00:11:42,780 It's a good point. 165 00:11:42,780 --> 00:11:46,710 So you'd like to not throw out the first one, perhaps, 166 00:11:46,710 --> 00:11:50,160 but there's no canonical place to put it in. 167 00:11:50,160 --> 00:11:55,270 This just goes off to infinity and just falls off the edge. 168 00:11:55,270 --> 00:11:58,310 It's a little bit like differentiation. 169 00:11:58,310 --> 00:11:58,811 Right? 170 00:11:58,811 --> 00:11:59,435 AUDIENCE: Yeah. 171 00:11:59,435 --> 00:12:01,166 I guess it loses some information. 172 00:12:01,166 --> 00:12:02,790 ARAM HARROW: It loses some information. 173 00:12:02,790 --> 00:12:04,260 That's right. 174 00:12:04,260 --> 00:12:05,570 It's a little bit weird, right? 175 00:12:05,570 --> 00:12:08,080 Because how many numbers do you have 176 00:12:08,080 --> 00:12:10,830 before you applied the left shift? 177 00:12:10,830 --> 00:12:11,820 Infinity. 178 00:12:11,820 --> 00:12:14,450 How many do you have after you applied the left shift? 179 00:12:14,450 --> 00:12:15,620 Infinity. 180 00:12:15,620 --> 00:12:17,680 But you lost some information. 181 00:12:17,680 --> 00:12:22,311 So you have to be a little careful with the infinities. 182 00:12:22,311 --> 00:12:22,810 OK 183 00:12:22,810 --> 00:12:25,940 The right shift. 184 00:12:25,940 --> 00:12:30,500 Here it's not so obvious what to do. 185 00:12:30,500 --> 00:12:36,140 We've kind of made space for another number, 186 00:12:36,140 --> 00:12:38,450 and so we have to put something in that first position. 187 00:12:38,450 --> 00:12:47,480 So this will be question mark x1, x2, dot, dot, dot. 188 00:12:47,480 --> 00:12:50,420 Any guesses what should go in the question mark? 189 00:12:50,420 --> 00:12:51,380 AUDIENCE: 0? 190 00:12:51,380 --> 00:12:52,180 ARAM HARROW: 0. 191 00:12:52,180 --> 00:12:52,680 Right. 192 00:12:56,410 --> 00:12:58,060 And why should that be 0? 193 00:12:58,060 --> 00:12:59,000 AUDIENCE: [INAUDIBLE]. 194 00:12:59,000 --> 00:12:59,490 ARAM HARROW: What's that? 195 00:12:59,490 --> 00:13:00,532 AUDIENCE: So it's linear. 196 00:13:00,532 --> 00:13:02,406 ARAM HARROW: Otherwise it wouldn't be linear. 197 00:13:02,406 --> 00:13:03,100 Right. 198 00:13:03,100 --> 00:13:04,600 So imagine what happens if you apply 199 00:13:04,600 --> 00:13:07,920 the right shift to the all 0 string. 200 00:13:07,920 --> 00:13:10,390 If you were to get something non-zero here, 201 00:13:10,390 --> 00:13:14,420 then you would map to the 0 vector to a non-zero vector. 202 00:13:14,420 --> 00:13:18,130 But, by linearity, that's impossible. 203 00:13:18,130 --> 00:13:22,340 Because I could take any vector and multiply it by the scalar 204 00:13:22,340 --> 00:13:25,180 0 and I get the vector 0. 205 00:13:25,180 --> 00:13:28,260 And that should be equal to the scalar 206 00:13:28,260 --> 00:13:32,360 0 multiplied by the output of it. 207 00:13:32,360 --> 00:13:39,260 And so that means that T should always map 0 to 0. 208 00:13:39,260 --> 00:13:41,900 T should always map the vector 0 to the vector 0. 209 00:13:41,900 --> 00:13:45,400 And so if we want right shift to be a linear operator, 210 00:13:45,400 --> 00:13:46,820 we have to put a 0 in there. 211 00:13:50,720 --> 00:13:56,030 And this one is strange also because it creates more space 212 00:13:56,030 --> 00:13:58,330 but still preserves all of the information. 213 00:14:01,330 --> 00:14:08,500 So two other small examples of linear operators 214 00:14:08,500 --> 00:14:10,090 that come up very often. 215 00:14:10,090 --> 00:14:16,930 There's, of course, the 0 operator, 216 00:14:16,930 --> 00:14:20,180 which takes any vector to the 0 vector. 217 00:14:20,180 --> 00:14:22,960 Here I'm not distinguishing between-- here 218 00:14:22,960 --> 00:14:25,130 the 0 means an operator, here it means a vector. 219 00:14:25,130 --> 00:14:33,060 I guess I can clarify it that way. 220 00:14:33,060 --> 00:14:36,380 And this is, of course, linear and sends any vector space 221 00:14:36,380 --> 00:14:37,500 to itself. 222 00:14:37,500 --> 00:14:39,850 One important thing is that the output 223 00:14:39,850 --> 00:14:42,370 doesn't have to be the entire vector space. 224 00:14:42,370 --> 00:14:44,510 The fact that it sends a vector space to itself 225 00:14:44,510 --> 00:14:47,040 only means that the output is contained within the vector 226 00:14:47,040 --> 00:14:47,820 space. 227 00:14:47,820 --> 00:14:49,580 It could be something as boring is 228 00:14:49,580 --> 00:14:54,630 0 that just sends all the vectors to a single point. 229 00:14:54,630 --> 00:14:57,000 And finally, one other important operator 230 00:14:57,000 --> 00:15:04,470 is the identity operator that sends-- actually I 231 00:15:04,470 --> 00:15:08,587 won't use the arrows here. 232 00:15:08,587 --> 00:15:10,170 We'll get used to the mathematical way 233 00:15:10,170 --> 00:15:13,220 of writing it-- that sends any vector to itself. 234 00:15:32,980 --> 00:15:35,600 Those are a few examples of operators. 235 00:15:35,600 --> 00:15:39,900 I guess you've seen already kind of the more familiar 236 00:15:39,900 --> 00:15:44,660 matrix-type of operators, but these show you 237 00:15:44,660 --> 00:15:46,670 also the range of what is possible. 238 00:15:50,680 --> 00:15:58,760 So the space l of v of all operators-- 239 00:15:58,760 --> 00:16:01,950 I want to talk now about its properties. 240 00:16:01,950 --> 00:16:08,100 So l of v is the space of all linear maps from v to itself. 241 00:16:08,100 --> 00:16:11,100 So this is the space of maps on a vector space, 242 00:16:11,100 --> 00:16:13,930 but itself is also a vector space. 243 00:16:21,910 --> 00:16:26,550 So the set of operators satisfies all the axioms 244 00:16:26,550 --> 00:16:27,600 of a vector space. 245 00:16:27,600 --> 00:16:31,010 It contains a 0 operator. 246 00:16:31,010 --> 00:16:33,680 That's this one right here. 247 00:16:33,680 --> 00:16:35,750 It's closed under a linear combination. 248 00:16:35,750 --> 00:16:38,040 If I add together two linear operators, 249 00:16:38,040 --> 00:16:39,990 I get another linear operator. 250 00:16:39,990 --> 00:16:41,830 It's closed under a scalar multiplication. 251 00:16:41,830 --> 00:16:44,200 If I multiply a linear operator by a scalar, 252 00:16:44,200 --> 00:16:48,500 I get another linear operator, et cetera. 253 00:16:48,500 --> 00:16:51,150 And so everything we can do on a vector space, 254 00:16:51,150 --> 00:16:53,280 like finding a basis and so on, we 255 00:16:53,280 --> 00:16:58,280 can do for the space of linear operators. 256 00:16:58,280 --> 00:17:02,080 However, in addition to having the vector space structure, 257 00:17:02,080 --> 00:17:05,745 it has an additional structure, which is multiplication. 258 00:17:17,349 --> 00:17:21,690 And here we're finally making use of the fact 259 00:17:21,690 --> 00:17:26,240 that we're talking about linear maps from a vector space 260 00:17:26,240 --> 00:17:27,589 to itself. 261 00:17:27,589 --> 00:17:31,140 If we were talking about maps from v to w, 262 00:17:31,140 --> 00:17:35,060 we couldn't necessarily multiply them by other maps from v to w, 263 00:17:35,060 --> 00:17:40,401 we could only multiply them by maps from w to something else. 264 00:17:40,401 --> 00:17:41,900 Just like how, if you're multiplying 265 00:17:41,900 --> 00:17:45,050 rectangular matrices, the multiplication is not 266 00:17:45,050 --> 00:17:48,430 always defined if the dimensions don't match up, 267 00:17:48,430 --> 00:17:52,730 But since these operators are like square matrices, 268 00:17:52,730 --> 00:17:54,880 multiplication is always defined, 269 00:17:54,880 --> 00:18:01,330 and this can be used to prove many nice things about them. 270 00:18:01,330 --> 00:18:03,500 So this type of structure being a vector 271 00:18:03,500 --> 00:18:06,970 space of multiplication makes it, in many ways, 272 00:18:06,970 --> 00:18:11,910 like a field-- like real numbers or complexes-- 273 00:18:11,910 --> 00:18:14,760 but without all of the properties. 274 00:18:14,760 --> 00:18:29,610 So the properties that the multiplication does have first 275 00:18:29,610 --> 00:18:32,960 is that it's associative. 276 00:18:32,960 --> 00:18:35,640 So let's see what this looks like. 277 00:18:35,640 --> 00:18:51,270 So if we have a times bc is equal to ab times c. 278 00:18:51,270 --> 00:19:00,830 And the way we can check this is just 279 00:19:00,830 --> 00:19:04,180 by verifying the action of this on any vector. 280 00:19:04,180 --> 00:19:10,440 So an operator is defined by its action and all 281 00:19:10,440 --> 00:19:13,340 of the vectors in a vector space. 282 00:19:13,340 --> 00:19:23,430 So the definition of ab can be thought 283 00:19:23,430 --> 00:19:31,150 of as asking how does it act on all the possible vectors? 284 00:19:31,150 --> 00:19:39,600 And this is defined just in terms of the action of a and b 285 00:19:39,600 --> 00:19:44,940 as you first apply b and then you apply A. 286 00:19:44,940 --> 00:19:47,170 So this can be thought of as the definition 287 00:19:47,170 --> 00:19:49,500 of how to multiply operators. 288 00:19:49,500 --> 00:19:51,970 And then from this, you can easily 289 00:19:51,970 --> 00:19:56,100 check the associativity property that in both cases, however 290 00:19:56,100 --> 00:20:06,880 you write it out, you obtain A of B of C of v. 291 00:20:06,880 --> 00:20:09,440 I'm writing out all the parentheses 292 00:20:09,440 --> 00:20:12,275 just to sort of emphasize this is C acting on v, 293 00:20:12,275 --> 00:20:16,910 and then B acting on C of v, and then A acting on all of this. 294 00:20:16,910 --> 00:20:20,820 The fact that this is equal-- that this is the same no matter 295 00:20:20,820 --> 00:20:23,260 how A, B, and C are grouped is again 296 00:20:23,260 --> 00:20:27,500 part of what let's us justify this right here, 297 00:20:27,500 --> 00:20:30,310 where we drop-- we just don't use parentheses 298 00:20:30,310 --> 00:20:34,600 when we have operators acting. 299 00:20:34,600 --> 00:20:38,560 So, yes, we have the associative property. 300 00:20:38,560 --> 00:20:41,460 Another property of multiplication that operators 301 00:20:41,460 --> 00:20:47,050 satisfy is the existence of an identity. 302 00:20:47,050 --> 00:20:50,400 That's just the identity operator, here, 303 00:20:50,400 --> 00:20:52,870 which for any vector space can always be defined. 304 00:20:54,951 --> 00:20:56,950 But there are other properties of multiplication 305 00:20:56,950 --> 00:20:59,370 that it doesn't have. 306 00:20:59,370 --> 00:21:08,890 So inverses are not always defined. 307 00:21:08,890 --> 00:21:10,730 They sometimes are. 308 00:21:10,730 --> 00:21:12,980 I can't say that a matrix is never invertible, 309 00:21:12,980 --> 00:21:16,410 but for things like the reals and the complexes, 310 00:21:16,410 --> 00:21:19,840 every nonzero element has an inverse. 311 00:21:19,840 --> 00:21:21,380 And for matrices, that's not true. 312 00:21:24,250 --> 00:21:27,850 And another property-- a more interesting one that 313 00:21:27,850 --> 00:21:36,691 these lack-- is that the multiplication is not 314 00:21:36,691 --> 00:21:37,190 commutative. 315 00:21:41,420 --> 00:21:44,340 So this is something that you've seen for matrices. 316 00:21:44,340 --> 00:21:47,150 If you multiply two matrices, the order matters, 317 00:21:47,150 --> 00:21:50,635 and so it's not surprising that same is true for operators. 318 00:21:59,260 --> 00:22:03,110 And just to give a quick example of that, 319 00:22:03,110 --> 00:22:08,280 let's look at this example one here with polynomials. 320 00:22:08,280 --> 00:22:17,610 And let's consider S times T acting on the monomial x 321 00:22:17,610 --> 00:22:20,500 to the n. 322 00:22:20,500 --> 00:22:24,400 So T is differentiation so it sends 323 00:22:24,400 --> 00:22:28,410 this to n times x to the n minus 1. 324 00:22:28,410 --> 00:22:33,790 So we get S times n, x to the n minus 1. 325 00:22:33,790 --> 00:22:38,570 Linearity means we can move the n past the S. S acting here 326 00:22:38,570 --> 00:22:46,250 multiplies by x, and so we get n times x to the n. 327 00:22:46,250 --> 00:22:52,040 Whereas if we did the other order, 328 00:22:52,040 --> 00:22:57,240 we get T times S acting on x to the n, which 329 00:22:57,240 --> 00:22:59,740 is x to the n plus 1. 330 00:22:59,740 --> 00:23:10,910 When you differentiate this you get n plus 1 times x to the n. 331 00:23:10,910 --> 00:23:12,600 So these numbers are different meaning 332 00:23:12,600 --> 00:23:16,550 that S and T do not commute. 333 00:23:16,550 --> 00:23:20,320 And it's kind of cute to measure to what extent do they 334 00:23:20,320 --> 00:23:22,740 not commute. 335 00:23:22,740 --> 00:23:25,320 This is done by the commutator. 336 00:23:25,320 --> 00:23:28,750 And what these equations say is that if the commutator acts 337 00:23:28,750 --> 00:23:33,650 on x to the n, then you get n plus 1 times 338 00:23:33,650 --> 00:23:39,130 x to the n minus n times x to the n, 339 00:23:39,130 --> 00:23:40,280 which is just x to the n. 340 00:23:43,600 --> 00:23:49,160 And we can write this another way as identity 341 00:23:49,160 --> 00:23:50,400 times x to the n. 342 00:23:54,280 --> 00:23:59,990 And since this is true for any choice of n, 343 00:23:59,990 --> 00:24:03,420 it's true for what turns out to be 344 00:24:03,420 --> 00:24:05,790 a basis for the space of polynomials. 345 00:24:05,790 --> 00:24:11,540 So 1x, x squared, x cubed, et cetera, 346 00:24:11,540 --> 00:24:13,960 these span the space of polynomials. 347 00:24:13,960 --> 00:24:15,904 So if you know what an operator does and all 348 00:24:15,904 --> 00:24:17,320 of the x to the n's, you know what 349 00:24:17,320 --> 00:24:20,030 it does on all the polynomials. 350 00:24:20,030 --> 00:24:26,540 And so this means, actually, that the commutator 351 00:24:26,540 --> 00:24:28,610 of these two is the identity. 352 00:24:34,670 --> 00:24:42,545 And so the significance of this is-- well, 353 00:24:42,545 --> 00:24:44,670 I won't dwell on the physical significance of this, 354 00:24:44,670 --> 00:24:48,860 but it's related to what you've seen for position and momentum. 355 00:24:48,860 --> 00:24:51,040 And essentially the fact that these don't commute 356 00:24:51,040 --> 00:24:54,520 is actually an important feature of the theory. 357 00:25:00,370 --> 00:25:06,180 So these are some of the key properties 358 00:25:06,180 --> 00:25:09,300 of the space of operators. 359 00:25:09,300 --> 00:25:10,970 I want to also now tell you about some 360 00:25:10,970 --> 00:25:15,630 of the key properties of individual operators. 361 00:25:15,630 --> 00:25:18,340 And basically, if you're given an operator 362 00:25:18,340 --> 00:25:21,090 and want to know the gross features of it, 363 00:25:21,090 --> 00:25:22,490 what should you look at? 364 00:25:25,070 --> 00:25:33,710 So one of these things is the null space of an operator. 365 00:25:33,710 --> 00:25:45,440 So this is the set of all v, of all vectors, 366 00:25:45,440 --> 00:25:46,820 that are killed by the operator. 367 00:25:46,820 --> 00:25:47,790 They're sent to 0. 368 00:25:52,740 --> 00:25:56,990 In some case-- so this will always include the vector 0. 369 00:26:00,350 --> 00:26:06,950 So this always at least includes the vector 0, but in some cases 370 00:26:06,950 --> 00:26:09,360 it will be a lot bigger. 371 00:26:09,360 --> 00:26:11,740 So for the identity operator, the null space 372 00:26:11,740 --> 00:26:12,985 is only the vector 0. 373 00:26:12,985 --> 00:26:16,270 The only thing that gets sent to 0 is 0 itself. 374 00:26:16,270 --> 00:26:20,450 Whereas, for the 0 operator, everything gets sent to 0. 375 00:26:20,450 --> 00:26:24,595 So the null space is the entire vector space. 376 00:26:24,595 --> 00:26:32,220 For left shift, the null space is only 0 itself-- sorry, 377 00:26:32,220 --> 00:26:37,840 for right shift the null space is only 0 itself. 378 00:26:37,840 --> 00:26:39,730 And what about for left shift? 379 00:26:39,730 --> 00:26:41,496 What's the null space here? 380 00:26:41,496 --> 00:26:41,995 Yeah? 381 00:26:41,995 --> 00:26:44,310 AUDIENCE: Some numer with a string of 0s following it. 382 00:26:44,310 --> 00:26:45,101 ARAM HARROW: Right. 383 00:26:45,101 --> 00:26:47,310 Any sequence where the first number 384 00:26:47,310 --> 00:26:51,920 is arbitrary, but everything after the first number is 0. 385 00:26:51,920 --> 00:26:55,500 And so from all of these examples 386 00:26:55,500 --> 00:26:58,510 you might guess that this is a linear subspace, 387 00:26:58,510 --> 00:27:02,620 because in every case it's been a vector space, and, in fact, 388 00:27:02,620 --> 00:27:04,220 this is correct. 389 00:27:04,220 --> 00:27:13,050 So this is a subspace of v because, if there's 390 00:27:13,050 --> 00:27:15,820 a vector that gets sent to 0, any multiple of it 391 00:27:15,820 --> 00:27:17,800 also will be sent to 0. 392 00:27:17,800 --> 00:27:19,760 And of the two vectors that get sent to 0, 393 00:27:19,760 --> 00:27:22,830 their sum will also be sent to 0. 394 00:27:22,830 --> 00:27:31,010 So the fact that it's a linear subspace 395 00:27:31,010 --> 00:27:35,490 can be a helpful way of understanding this set. 396 00:27:35,490 --> 00:27:40,090 And it's related to the properties of T as a function. 397 00:27:40,090 --> 00:27:43,290 So for a function we often want to know whether it's 1 to 1, 398 00:27:43,290 --> 00:27:47,620 or injective, or whether it's [? onto ?] or surjective. 399 00:27:47,620 --> 00:27:59,940 And you can check that if T is injective, 400 00:27:59,940 --> 00:28:11,010 meaning that if u is not equal to v, then T of u 401 00:28:11,010 --> 00:28:14,340 is not equal to T of v. So this property, 402 00:28:14,340 --> 00:28:19,790 that T maps distinct vectors two distinct vectors, 403 00:28:19,790 --> 00:28:22,290 turns out to be equivalent to the null space 404 00:28:22,290 --> 00:28:23,660 being only the 0 vector. 405 00:28:27,040 --> 00:28:27,870 So why is that? 406 00:28:34,900 --> 00:28:37,925 This statement here, that whenever u is not equal to v, 407 00:28:37,925 --> 00:28:41,070 T of u is not equal to T of v, another way 408 00:28:41,070 --> 00:28:50,166 to write that is whenever u is not equal to v, T of u minus v 409 00:28:50,166 --> 00:28:51,450 is not equal to 0. 410 00:28:54,600 --> 00:28:58,030 And if you look at this statement 411 00:28:58,030 --> 00:29:00,260 a little more carefully, you'll realize 412 00:29:00,260 --> 00:29:04,580 that all we cared about on both sides was u minus v. Here, 413 00:29:04,580 --> 00:29:07,010 obviously, we care about u minus v. Here 414 00:29:07,010 --> 00:29:11,090 we only care if u is not equal to v. 415 00:29:11,090 --> 00:29:18,910 So that's the same as saying if u minus v is non-zero, 416 00:29:18,910 --> 00:29:24,630 then T of u minus v is non-zero. 417 00:29:24,630 --> 00:29:30,180 And this in turn is equivalent to saying 418 00:29:30,180 --> 00:29:36,010 that the null space of T is only 0. 419 00:29:36,010 --> 00:29:40,640 In other words, the set of vectors that get sent to 0 420 00:29:40,640 --> 00:29:43,090 consists only of the 0 vector itself. 421 00:29:46,710 --> 00:29:50,470 So the null space for linear operators 422 00:29:50,470 --> 00:29:53,290 is how we can characterize whether they're 1 to 1, 423 00:29:53,290 --> 00:29:55,160 whether they destroy any information. 424 00:30:14,780 --> 00:30:19,060 The other subspace that will be important that we will use 425 00:30:19,060 --> 00:30:20,585 is the range of an operator. 426 00:30:28,800 --> 00:30:34,970 So the range of an operator, which we can also just 427 00:30:34,970 --> 00:30:44,260 write as T of v, is the set of all points that vectors in v 428 00:30:44,260 --> 00:30:45,190 get mapped to. 429 00:30:45,190 --> 00:30:55,290 So the set of all Tv for some vector, v. 430 00:30:55,290 --> 00:31:00,120 So this, too, can be shown to be a subspace. 431 00:31:13,100 --> 00:31:19,310 And that's because-- it takes a little more work to show it, 432 00:31:19,310 --> 00:31:23,850 but not very much-- if there's something in the output of T, 433 00:31:23,850 --> 00:31:26,870 then whatever the corresponding input is 434 00:31:26,870 --> 00:31:30,220 we could have multiplied that by a scalar. 435 00:31:30,220 --> 00:31:32,790 And then the corresponding output also 436 00:31:32,790 --> 00:31:35,010 would get multiplied by a scalar, 437 00:31:35,010 --> 00:31:38,910 and so that, too, would be in the range. 438 00:31:38,910 --> 00:31:41,140 And so that means that for anything in the range, 439 00:31:41,140 --> 00:31:43,450 we can multiply it by any scalar and again get 440 00:31:43,450 --> 00:31:45,110 something in the range. 441 00:31:45,110 --> 00:31:46,190 Similarly for addition. 442 00:31:46,190 --> 00:31:48,050 A similar argument shows that the range 443 00:31:48,050 --> 00:31:49,150 is closed under addition. 444 00:31:49,150 --> 00:31:53,290 So indeed, it's a linear subspace. 445 00:31:53,290 --> 00:31:58,180 Again, since it's a linear subspace, it always contains 0. 446 00:31:58,180 --> 00:32:05,010 And depending on the operator, may contain a lot more. 447 00:32:05,010 --> 00:32:08,550 So whereas the null space determined 448 00:32:08,550 --> 00:32:10,870 whether T was injective, the range 449 00:32:10,870 --> 00:32:13,680 determines whether T is surjective. 450 00:32:13,680 --> 00:32:29,420 So the range of T equals v if and only if T is surjective. 451 00:32:34,040 --> 00:32:38,620 And here this is simply the definition of being surjective. 452 00:32:38,620 --> 00:32:41,590 It's not really a theorem like it was in the case of T 453 00:32:41,590 --> 00:32:43,180 being injective. 454 00:32:43,180 --> 00:32:45,230 Here that's really what it means to be surjective 455 00:32:45,230 --> 00:32:46,990 is that your output is the entire space. 456 00:32:50,280 --> 00:32:53,330 So one important property of the range 457 00:32:53,330 --> 00:32:57,160 of the null space whenever v is finite dimensional 458 00:32:57,160 --> 00:33:08,200 is that the dimension of v is equal to the dimension 459 00:33:08,200 --> 00:33:19,275 of the null space plus the dimension of the range. 460 00:33:23,970 --> 00:33:29,390 And this is actually not trivial to prove. 461 00:33:29,390 --> 00:33:33,118 And I'm actually not going to prove it right now. 462 00:33:33,118 --> 00:33:37,020 But the intuition of it is as follows. 463 00:33:37,020 --> 00:33:40,510 Imagine that v is some n dimensional space 464 00:33:40,510 --> 00:33:43,430 and the null space has dimension k. 465 00:33:43,430 --> 00:33:47,370 So that means you have input of n degrees of freedom, 466 00:33:47,370 --> 00:33:50,320 but T kills k of n. 467 00:33:50,320 --> 00:33:52,320 And so k at different degrees of freedom, 468 00:33:52,320 --> 00:33:54,960 no matter how you vary them, have no effect on the output. 469 00:33:54,960 --> 00:33:56,860 They just get mapped to 0. 470 00:33:56,860 --> 00:34:00,480 And so what's left are n minus k degrees of freedom 471 00:34:00,480 --> 00:34:01,950 that do affect the outcome. 472 00:34:01,950 --> 00:34:05,820 Where, if you vary them, it does change the output in some way. 473 00:34:05,820 --> 00:34:09,350 And those correspond to the n minus k dimensions 474 00:34:09,350 --> 00:34:10,924 of the range. 475 00:34:10,924 --> 00:34:12,340 And if you want to get formal, you 476 00:34:12,340 --> 00:34:13,798 have to formalize what I was saying 477 00:34:13,798 --> 00:34:16,991 about what's left is n minus k. 478 00:34:16,991 --> 00:34:18,949 If you talk about something like the orthogonal 479 00:34:18,949 --> 00:34:23,179 complement or completing a basis or in some way 480 00:34:23,179 --> 00:34:26,290 formalize that intuition. 481 00:34:26,290 --> 00:34:28,040 And, in fact, you can do a little further, 482 00:34:28,040 --> 00:34:30,180 and you can decompose the space. 483 00:34:30,180 --> 00:34:31,679 So this is just dimensions counting. 484 00:34:31,679 --> 00:34:35,679 You can even decompose the space into the null space 485 00:34:35,679 --> 00:34:40,760 and the complement of that and show that T is 1 to 1 486 00:34:40,760 --> 00:34:44,929 on the complement of the null space. 487 00:34:44,929 --> 00:34:48,330 But for now, I think this is all that we'll need for now. 488 00:34:50,989 --> 00:34:52,861 Any questions so far? 489 00:34:52,861 --> 00:34:53,360 Yeah? 490 00:34:53,360 --> 00:34:57,380 AUDIENCE: Why isn't the null space part of the range? 491 00:34:57,380 --> 00:34:59,320 ARAM HARROW: Why isn't it part of the range? 492 00:34:59,320 --> 00:35:01,280 AUDIENCE: So you're taking T of v 493 00:35:01,280 --> 00:35:05,004 and null space is just the special case when T of v 494 00:35:05,004 --> 00:35:06,400 is equal to 0. 495 00:35:06,400 --> 00:35:07,720 ARAM HARROW: Right. 496 00:35:07,720 --> 00:35:19,480 So the null space are all of the-- This theorem, 497 00:35:19,480 --> 00:35:23,799 I guess, would be a little bit more surprising if you realized 498 00:35:23,799 --> 00:35:25,340 that it works not only for operators, 499 00:35:25,340 --> 00:35:27,740 but for general linear maps. 500 00:35:27,740 --> 00:35:32,300 And in that case, the range is a subset space of w. 501 00:35:32,300 --> 00:35:34,680 Because the range is about the output. 502 00:35:34,680 --> 00:35:36,740 And the null space is a [? subset ?] space 503 00:35:36,740 --> 00:35:39,570 of v, which is part of the input. 504 00:35:39,570 --> 00:35:42,310 And so in that case, they're not even comparable. 505 00:35:42,310 --> 00:35:44,860 The vectors might just have different lengths. 506 00:35:44,860 --> 00:35:51,212 And so it can never-- like the null space in a range, 507 00:35:51,212 --> 00:35:53,420 in that case, would live in totally different spaces. 508 00:35:58,690 --> 00:36:01,600 So let me give you a very simple example. 509 00:36:01,600 --> 00:36:10,680 Let's suppose that T is equal to 3, 0, minus 1, 4. 510 00:36:10,680 --> 00:36:14,690 So just a diagonal 4 by 4 matrix. 511 00:36:14,690 --> 00:36:22,840 Then the null space would be the span 512 00:36:22,840 --> 00:36:28,440 of e2, that's the vector with a 1 in the second position. 513 00:36:31,770 --> 00:36:42,680 And the range would be the span of e1, e3, in e4. 514 00:36:42,680 --> 00:36:44,930 So in fact, usually it's the opposite that happens. 515 00:36:44,930 --> 00:36:48,260 The null space in the range are-- 516 00:36:48,260 --> 00:36:51,105 in this case they're actually orthogonal subspaces. 517 00:36:58,100 --> 00:37:03,040 But this picture is actually a little bit 518 00:37:03,040 --> 00:37:05,180 deceptive in how nice it is. 519 00:37:05,180 --> 00:37:06,830 So if you look at this, total space 520 00:37:06,830 --> 00:37:09,350 is 4, four dimensions, it divides up 521 00:37:09,350 --> 00:37:11,360 into one dimension that gets killed, 522 00:37:11,360 --> 00:37:15,674 and three dimensions where the output still tells you 523 00:37:15,674 --> 00:37:17,340 something about the input, where there's 524 00:37:17,340 --> 00:37:20,920 some variation of the output. 525 00:37:20,920 --> 00:37:29,850 But this picture makes it seem-- the simplicity of this picture 526 00:37:29,850 --> 00:37:30,860 does not always exist. 527 00:37:35,910 --> 00:37:47,340 A much more horrible example is this matrix. 528 00:37:55,840 --> 00:37:57,735 So what's the null space of this matrix? 529 00:38:03,940 --> 00:38:04,702 Yeah? 530 00:38:04,702 --> 00:38:07,118 AUDIENCE: You just don't care about the upper [INAUDIBLE]. 531 00:38:07,118 --> 00:38:16,902 ARAM HARROW: You don't care about the-- informally, 532 00:38:16,902 --> 00:38:18,110 it's everything of this form. 533 00:38:18,110 --> 00:38:20,470 Everything with something in the first position, 534 00:38:20,470 --> 00:38:21,705 0 in the second position. 535 00:38:21,705 --> 00:38:27,350 In other words, it's the span of e1. 536 00:38:32,476 --> 00:38:33,350 What about the range? 537 00:38:38,853 --> 00:38:40,280 AUDIENCE: [INAUDIBLE]. 538 00:38:40,280 --> 00:38:41,321 ARAM HARROW: What's that? 539 00:38:45,769 --> 00:38:46,269 Yeah? 540 00:38:46,269 --> 00:38:48,764 AUDIENCE: [INAUDIBLE]. 541 00:38:48,764 --> 00:38:50,261 ARAM HARROW: It's actually-- 542 00:38:50,261 --> 00:38:51,270 AUDIENCE: Isn't it e1? 543 00:38:51,270 --> 00:38:52,353 ARAM HARROW: It's also e1. 544 00:38:52,353 --> 00:38:53,446 It's the same thing. 545 00:38:59,410 --> 00:39:02,570 So you have this intuition that some degrees of freedom 546 00:39:02,570 --> 00:39:06,010 are preserved and some are killed. 547 00:39:06,010 --> 00:39:08,940 And here they look totally different. 548 00:39:08,940 --> 00:39:11,600 And there they look the same. 549 00:39:11,600 --> 00:39:14,240 So you should be a little bit nervous 550 00:39:14,240 --> 00:39:16,730 about trying to apply that intuition. 551 00:39:16,730 --> 00:39:19,400 You should be reassured that at least the theorem is still 552 00:39:19,400 --> 00:39:20,610 true. 553 00:39:20,610 --> 00:39:23,465 At least 1 plus 1 is equal to 2. 554 00:39:23,465 --> 00:39:27,120 We still have that. 555 00:39:27,120 --> 00:39:31,630 But the null space and the range are the same thing here. 556 00:39:31,630 --> 00:39:36,530 And the way around that paradox-- Yeah? 557 00:39:36,530 --> 00:39:38,363 AUDIENCE: So can you just change the basis-- 558 00:39:38,363 --> 00:39:39,766 is there always a way of changing 559 00:39:39,766 --> 00:39:40,765 the basis of the matrix? 560 00:39:40,765 --> 00:39:43,582 In this case it becomes [INAUDIBLE]? 561 00:39:43,582 --> 00:39:44,580 Or not necessarily? 562 00:39:44,580 --> 00:39:45,246 ARAM HARROW: No. 563 00:39:45,246 --> 00:39:47,460 It turns out that, even with the change of basis, 564 00:39:47,460 --> 00:39:51,150 you cannot guarantee that the null space and the range will 565 00:39:51,150 --> 00:39:52,850 be perpendicular. 566 00:39:52,850 --> 00:39:54,320 Yeah? 567 00:39:54,320 --> 00:40:01,250 AUDIENCE: What if you reduce it to only measures on the-- 568 00:40:01,250 --> 00:40:04,220 or what if you reduce the matrix of-- [? usability ?] 569 00:40:04,220 --> 00:40:07,390 on only [INAUDIBLE] on the diagonal? 570 00:40:07,390 --> 00:40:08,390 ARAM HARROW: Right Good. 571 00:40:08,390 --> 00:40:16,620 So if you do that, then-- if you do row [? eduction ?] with two 572 00:40:16,620 --> 00:40:19,170 different row and column operations, 573 00:40:19,170 --> 00:40:21,320 then what you've done is you have a different input 574 00:40:21,320 --> 00:40:23,540 and output basis. 575 00:40:23,540 --> 00:40:31,160 And so that would-- then once you kind of unpack 576 00:40:31,160 --> 00:40:33,770 what's going on in terms of the basis, 577 00:40:33,770 --> 00:40:37,360 then it would turn out that you could still 578 00:40:37,360 --> 00:40:39,294 have strange behavior like this. 579 00:40:39,294 --> 00:40:40,710 What your intuition is based on is 580 00:40:40,710 --> 00:40:43,330 that if the matrix is diagonal in some basis, 581 00:40:43,330 --> 00:40:44,770 then you don't have this trouble. 582 00:40:44,770 --> 00:40:47,890 But the problem is that not all matrices can be diagonalized. 583 00:40:47,890 --> 00:40:49,354 Yeah? 584 00:40:49,354 --> 00:40:50,818 AUDIENCE: So is it just the trouble 585 00:40:50,818 --> 00:40:53,258 that the null is what you're acting on 586 00:40:53,258 --> 00:40:56,136 and the range is what results from it? 587 00:40:56,136 --> 00:40:57,010 ARAM HARROW: Exactly. 588 00:40:57,010 --> 00:41:00,470 And they could even live in different space. 589 00:41:00,470 --> 00:41:06,710 And so they really just don't-- to compare them is dangerous. 590 00:41:06,710 --> 00:41:12,010 So it turns out that the degrees of freedom corresponding 591 00:41:12,010 --> 00:41:14,820 to the range-- what you should think about 592 00:41:14,820 --> 00:41:18,620 are the degrees of freedom that get sent to the range. 593 00:41:18,620 --> 00:41:21,490 And in this case, that would be e2. 594 00:41:21,490 --> 00:41:26,720 And so then you can say that e1 gets sent to 0 and e2 595 00:41:26,720 --> 00:41:28,130 gets sent to the range. 596 00:41:28,130 --> 00:41:31,520 And now you really have decomposed the input space 597 00:41:31,520 --> 00:41:34,730 into two orthogonal parts. 598 00:41:34,730 --> 00:41:37,800 And because we're talking about a single space, the input 599 00:41:37,800 --> 00:41:39,990 space, it actually makes sense to break it up 600 00:41:39,990 --> 00:41:41,710 into these parts. 601 00:41:41,710 --> 00:41:45,790 Whereas here, they look like they're the same, but really 602 00:41:45,790 --> 00:41:47,400 input and output spaces you should 603 00:41:47,400 --> 00:41:51,260 think of as potentially different. 604 00:41:51,260 --> 00:41:57,400 So this is just a mild warning about reading too much 605 00:41:57,400 --> 00:42:02,100 into this formula, even though it's the rough idea 606 00:42:02,100 --> 00:42:05,170 it counting degrees of freedom is still roughly accurate. 607 00:42:09,430 --> 00:42:18,300 So I want to say one more thing about properties of operators, 608 00:42:18,300 --> 00:42:20,600 which is about invertibility. 609 00:42:20,600 --> 00:42:24,430 And maybe I'll leave this up for now. 610 00:42:41,090 --> 00:42:45,780 So we say that a linear operator, T, 611 00:42:45,780 --> 00:42:59,770 has a left inverse, S, if multiplying T on the left by s 612 00:42:59,770 --> 00:43:02,750 will give you the identity. 613 00:43:02,750 --> 00:43:12,706 And T has a right inverse, S prime, 614 00:43:12,706 --> 00:43:16,160 you can guess what will happen here 615 00:43:16,160 --> 00:43:22,140 if multiplying T on the right by S prime gives you identity. 616 00:43:22,140 --> 00:43:25,728 And what if T has both? 617 00:43:25,728 --> 00:43:31,620 Then in that next case, it turns out 618 00:43:31,620 --> 00:43:35,447 that S and S prime have to be the same. 619 00:43:35,447 --> 00:43:36,280 So here's the proof. 620 00:43:36,280 --> 00:43:55,140 So if both exist, then S is equal to s times identity-- 621 00:43:55,140 --> 00:43:57,090 by the definition of the identity. 622 00:43:57,090 --> 00:44:00,225 And then we can replace identity with TS prime. 623 00:44:05,370 --> 00:44:13,810 Then we can group these and cancel them and get S prime. 624 00:44:13,810 --> 00:44:18,150 So if a matrix has both a left and a right inverse, 625 00:44:18,150 --> 00:44:21,860 then it turns out that the left and right inverse are the same. 626 00:44:21,860 --> 00:44:31,740 And in this case, we say that T is invertible, 627 00:44:31,740 --> 00:44:50,330 and we define T inverse to be S. 628 00:44:50,330 --> 00:44:52,680 One question that you often want to ask 629 00:44:52,680 --> 00:44:55,650 is when do left to right inverses exist? 630 00:45:03,335 --> 00:45:05,080 Actually, maybe I'll write it here. 631 00:45:15,300 --> 00:45:17,620 Intuitively, there should exist a left inverse 632 00:45:17,620 --> 00:45:22,200 when, after we've applied T, we haven't 633 00:45:22,200 --> 00:45:24,610 done irreparable damage. 634 00:45:24,610 --> 00:45:26,640 So whatever we're left with, there's 635 00:45:26,640 --> 00:45:30,390 still enough information that some linear operator 636 00:45:30,390 --> 00:45:35,530 can restore our original vector and give us back the identity. 637 00:45:35,530 --> 00:45:39,550 And so that condition is when-- of not doing 638 00:45:39,550 --> 00:45:42,140 irreparable damage, of not losing information, 639 00:45:42,140 --> 00:45:45,250 is asking essentially whether T is injective. 640 00:45:45,250 --> 00:45:54,350 So there exists a left inverse if and only if T is injective. 641 00:46:01,540 --> 00:46:07,370 Now for a right inverse the situation 642 00:46:07,370 --> 00:46:10,260 is sort of dual to this. 643 00:46:10,260 --> 00:46:14,470 And here what we want-- we can multiply 644 00:46:14,470 --> 00:46:16,520 on the right by whatever we like, 645 00:46:16,520 --> 00:46:18,890 but there won't be anything on the left. 646 00:46:18,890 --> 00:46:22,920 So after the action of T, there won't be any further room 647 00:46:22,920 --> 00:46:25,760 to explore the whole vector space. 648 00:46:25,760 --> 00:46:29,920 So the output of T had better cover all of the possibilities 649 00:46:29,920 --> 00:46:35,320 if we want to be able to achieve identity by multiplying T 650 00:46:35,320 --> 00:46:37,860 by something on the right. 651 00:46:37,860 --> 00:46:41,145 So any guesses for what the condition 652 00:46:41,145 --> 00:46:42,455 is for having a right inverse? 653 00:46:42,455 --> 00:46:43,330 AUDIENCE: Surjective. 654 00:46:43,330 --> 00:46:43,780 ARAM HARROW: Surjective. 655 00:46:43,780 --> 00:46:44,360 Right. 656 00:46:44,360 --> 00:47:00,170 So there exists a right inverse if and only if T is surjective. 657 00:47:00,170 --> 00:47:05,100 Technically, I've only proved one direction. 658 00:47:05,100 --> 00:47:09,410 My hand waving just now proved that, if T is not injective, 659 00:47:09,410 --> 00:47:11,380 there's no way it will have a left inverse. 660 00:47:11,380 --> 00:47:13,310 If it's not surjective, there's no way 661 00:47:13,310 --> 00:47:14,640 it'll have a right inverse. 662 00:47:14,640 --> 00:47:17,660 I haven't actually proved that, if it is injective, 663 00:47:17,660 --> 00:47:19,260 there is such a left inverse. 664 00:47:19,260 --> 00:47:22,480 And if it is surjective, there is such a right universe. 665 00:47:22,480 --> 00:47:24,364 But those I think are good exercises 666 00:47:24,364 --> 00:47:26,780 for you to do to make sure you understand what's going on. 667 00:47:31,220 --> 00:47:34,110 This takes us part of the way there. 668 00:47:34,110 --> 00:47:38,230 In some cases our lives become much easier. 669 00:47:38,230 --> 00:47:45,490 In particular, if v is finite dimensional, 670 00:47:45,490 --> 00:47:51,930 it turns out that all of these are equivalent. 671 00:47:51,930 --> 00:48:12,870 So T is injective if and only if T is surjective if 672 00:48:12,870 --> 00:48:16,950 and only if T is invertible. 673 00:48:31,340 --> 00:48:32,020 And why is this? 674 00:48:32,020 --> 00:48:34,760 Why should it be true that T is surjective if 675 00:48:34,760 --> 00:48:37,456 and only if T is injective? 676 00:48:37,456 --> 00:48:39,205 Why should those be equivalent statements? 677 00:48:44,022 --> 00:48:45,516 Yeah? 678 00:48:45,516 --> 00:48:48,504 AUDIENCE: This isn't really a rigorous statement, 679 00:48:48,504 --> 00:48:53,484 but if the intuition of it is a little bit that you're taking 680 00:48:53,484 --> 00:48:54,726 vectors in v to vectors in v. 681 00:48:54,726 --> 00:48:55,476 ARAM HARROW: Yeah. 682 00:48:55,476 --> 00:48:59,520 AUDIENCE: And so your mapping is 1 to 1 683 00:48:59,520 --> 00:49:04,700 if and only if every vector is mapped to, because then 684 00:49:04,700 --> 00:49:06,310 you're not leaving anything out. 685 00:49:06,310 --> 00:49:07,393 ARAM HARROW: That's right. 686 00:49:07,393 --> 00:49:10,680 In failing to be injective and failing to be surjective 687 00:49:10,680 --> 00:49:12,896 both look like losing information. 688 00:49:12,896 --> 00:49:14,270 Failing to be injective means I'm 689 00:49:14,270 --> 00:49:17,380 sending a whole non-zero vector and its multiples 690 00:49:17,380 --> 00:49:20,230 to 0, that's a degree of freedom lost. 691 00:49:20,230 --> 00:49:22,470 Failing to be surjective means once I look 692 00:49:22,470 --> 00:49:24,270 at all the degrees of freedom I reach, 693 00:49:24,270 --> 00:49:25,720 I haven't reached everything. 694 00:49:25,720 --> 00:49:29,710 So they intuitively look the same. 695 00:49:29,710 --> 00:49:31,720 So that's the right intuition. 696 00:49:31,720 --> 00:49:33,380 There's a proof, actually, that makes 697 00:49:33,380 --> 00:49:36,800 use of something on a current blackboard though. 698 00:49:36,800 --> 00:49:37,487 Yeah? 699 00:49:37,487 --> 00:49:39,112 AUDIENCE: Well, you need the dimensions 700 00:49:39,112 --> 00:49:42,458 of-- so if the [INAUDIBLE] space is 0, 701 00:49:42,458 --> 00:49:44,370 you need dimensions of [? the range to p. ?] 702 00:49:44,370 --> 00:49:45,461 ARAM HARROW: Right. 703 00:49:45,461 --> 00:49:45,960 Right. 704 00:49:45,960 --> 00:49:48,600 So from this dimensions formula you immediately get 705 00:49:48,600 --> 00:49:55,010 because if this is 0, then this is the whole vector space. 706 00:49:55,010 --> 00:49:58,930 And if this is non-zero, this is not the whole vector space. 707 00:49:58,930 --> 00:50:02,470 And this proof is sort of non-illuminating 708 00:50:02,470 --> 00:50:04,600 if you don't know the proof of that thing-- which 709 00:50:04,600 --> 00:50:05,900 I apologize for. 710 00:50:05,900 --> 00:50:08,275 But also, you can see immediately from that 711 00:50:08,275 --> 00:50:13,050 that we've used the fact that v is finite dimensional. 712 00:50:13,050 --> 00:50:17,080 And it turns out this equivalence breaks down 713 00:50:17,080 --> 00:50:21,210 if the vector space is infinite dimensional. 714 00:50:21,210 --> 00:50:23,050 Which is pretty weird. 715 00:50:23,050 --> 00:50:26,020 There's a lot of subtleties of infinite dimensional vector 716 00:50:26,020 --> 00:50:30,890 spaces that it's easy to overlook if you build up 717 00:50:30,890 --> 00:50:32,670 your intuition from matrices. 718 00:50:35,470 --> 00:50:40,580 So does anyone have an idea of a-- so let's think 719 00:50:40,580 --> 00:50:43,580 of an example of a vector of something 720 00:50:43,580 --> 00:50:46,060 that is on an infinite dimensional space that's 721 00:50:46,060 --> 00:50:48,495 surjective but not injective. 722 00:50:51,090 --> 00:50:55,000 Any guesses for such an operation? 723 00:50:55,000 --> 00:50:55,840 Yeah? 724 00:50:55,840 --> 00:50:57,070 AUDIENCE: The left shift. 725 00:50:57,070 --> 00:50:58,267 ARAM HARROW: Yes. 726 00:50:58,267 --> 00:51:00,725 You'll notice I didn't erase this blackboard strategically. 727 00:51:00,725 --> 00:51:01,225 Yes. 728 00:51:01,225 --> 00:51:04,270 The left shift operator is surjective. 729 00:51:04,270 --> 00:51:07,990 I can prepare any vector here I like just 730 00:51:07,990 --> 00:51:11,750 by putting it into the x2, x3, dot, dot, dot parts. 731 00:51:11,750 --> 00:51:15,190 So the range is everything, but it's not injective 732 00:51:15,190 --> 00:51:17,360 because it throws away the first register. 733 00:51:17,360 --> 00:51:21,690 It's maps things with it a non-zero element 734 00:51:21,690 --> 00:51:24,920 in the first position and 0's everywhere else to 0. 735 00:51:24,920 --> 00:51:33,820 So this is surjective not injective. 736 00:51:36,510 --> 00:51:39,190 On the other hand, if you want something 737 00:51:39,190 --> 00:51:42,000 that's injective and not surjective, 738 00:51:42,000 --> 00:51:48,480 you don't have to look very far, the right shift 739 00:51:48,480 --> 00:51:54,940 is injective and not surjective. 740 00:51:54,940 --> 00:51:57,380 It's pretty obvious it's not surjective. 741 00:51:57,380 --> 00:52:01,180 There's that 0 there which definitely means it cannot 742 00:52:01,180 --> 00:52:02,462 achieve any vector. 743 00:52:02,462 --> 00:52:04,295 And it's not too hard to see it's injective. 744 00:52:04,295 --> 00:52:06,520 It hasn't lost any information. 745 00:52:06,520 --> 00:52:09,260 It's like you're in the hotel that's infinitely long 746 00:52:09,260 --> 00:52:14,692 and all the rooms are full and the person at the front desk 747 00:52:14,692 --> 00:52:15,400 says, no problem. 748 00:52:15,400 --> 00:52:18,325 I'll just move everyone down one room to the right, 749 00:52:18,325 --> 00:52:19,820 and you can take the first room. 750 00:52:19,820 --> 00:52:24,370 So that policy is injective-- you'll 751 00:52:24,370 --> 00:52:28,370 always get a room to yourself-- and made 752 00:52:28,370 --> 00:52:33,170 possible by having an infinite dimensional vector space. 753 00:52:33,170 --> 00:52:37,230 So in infinite dimensions we cannot say this. 754 00:52:37,230 --> 00:52:48,050 Instead, we can say that T is invertible if 755 00:52:48,050 --> 00:52:56,720 and only if T is injective and surjective. 756 00:52:59,990 --> 00:53:03,040 So this statement is true in general 757 00:53:03,040 --> 00:53:06,390 for infinite dimensional, whatever, vector spaces. 758 00:53:06,390 --> 00:53:10,460 And only in the nice special case of finite dimensions 759 00:53:10,460 --> 00:53:11,690 do we get this equivalence. 760 00:53:16,012 --> 00:53:17,509 Yeah? 761 00:53:17,509 --> 00:53:23,996 AUDIENCE: Can the range and null space of T a [INAUDIBLE] of T 762 00:53:23,996 --> 00:53:27,084 the operator again use a vector space [INAUDIBLE]? 763 00:53:27,084 --> 00:53:28,500 ARAM HARROW: Yes. the question was 764 00:53:28,500 --> 00:53:32,630 do the null space in a range are they properties just of T 765 00:53:32,630 --> 00:53:34,200 or also of v? 766 00:53:34,200 --> 00:53:35,904 And definitely you also need to know 767 00:53:35,904 --> 00:53:40,350 v. The way I've been writing it, T 768 00:53:40,350 --> 00:53:43,940 is implicitly defined in terms of v, 769 00:53:43,940 --> 00:53:45,550 which in turn is implicitly defined 770 00:53:45,550 --> 00:53:47,820 in terms of the field, f. 771 00:53:47,820 --> 00:53:51,641 And all these things can make a difference. 772 00:53:51,641 --> 00:53:52,140 Yes? 773 00:53:52,140 --> 00:53:55,767 AUDIENCE: So do you have to be a bijection for it to be-- 774 00:53:55,767 --> 00:53:56,850 ARAM HARROW: That's right. 775 00:53:56,850 --> 00:53:57,442 That's right. 776 00:53:57,442 --> 00:53:58,900 Invertible is the same a bijection. 777 00:54:01,800 --> 00:54:08,105 So let me now try and relate this to matrices. 778 00:54:12,732 --> 00:54:14,190 I've been saying that operators are 779 00:54:14,190 --> 00:54:18,075 like the fancy mathematician's form of matrices. 780 00:54:22,490 --> 00:54:24,410 If you're Arrested Development fans, 781 00:54:24,410 --> 00:54:28,810 it's like magic trick versus an illusion. 782 00:54:28,810 --> 00:54:32,165 But are they different or not depends on your perspective. 783 00:54:34,940 --> 00:54:37,160 There are advantages to seeing it both ways, I think. 784 00:54:37,160 --> 00:54:40,560 So let me tell you how you can view 785 00:54:40,560 --> 00:54:44,620 an operator in a matrix form. 786 00:54:44,620 --> 00:54:47,180 The way to do this-- and the reason 787 00:54:47,180 --> 00:54:52,540 why matrices are not universally loved by mathematicians 788 00:54:52,540 --> 00:54:56,730 is I haven't specified a basis this whole time. 789 00:54:56,730 --> 00:54:59,390 But if I want a matrix, all I needed 790 00:54:59,390 --> 00:55:01,330 was a vector space and a function-- 791 00:55:01,330 --> 00:55:03,710 a linear function between two vector spaces-- 792 00:55:03,710 --> 00:55:05,600 or, sorry, from a vector space to itself. 793 00:55:05,600 --> 00:55:09,290 But if I want a matrix, I need additional structure. 794 00:55:09,290 --> 00:55:13,540 And mathematicians try to avoid that whenever possible. 795 00:55:13,540 --> 00:55:16,460 But if you're willing to take this additional structure-- so 796 00:55:16,460 --> 00:55:26,400 if you choose a basis v1 through vn-- 797 00:55:26,400 --> 00:55:28,680 it turns out you can get a simpler 798 00:55:28,680 --> 00:55:31,640 form of the operator that's useful to compute with. 799 00:55:31,640 --> 00:55:32,930 So why is that? 800 00:55:32,930 --> 00:55:35,170 Well, the fact that it's a basis that 801 00:55:35,170 --> 00:55:50,630 means that any v can be written as linear combinations 802 00:55:50,630 --> 00:55:56,250 of these basis elements where a1 though an belong to the field. 803 00:55:56,250 --> 00:56:09,480 And since T is linear, if T acts on v, 804 00:56:09,480 --> 00:56:12,820 we can rewrite it in this way, and you 805 00:56:12,820 --> 00:56:17,570 see that the entire action is determined 806 00:56:17,570 --> 00:56:22,350 by T acting on v1 through vn. 807 00:56:22,350 --> 00:56:24,490 So think about-- if you wanted to represent 808 00:56:24,490 --> 00:56:27,330 an operator in a computer, you'd say, 809 00:56:27,330 --> 00:56:29,620 well, there's an infinite number of input vectors. 810 00:56:29,620 --> 00:56:32,160 And for each input vector I have to write down 811 00:56:32,160 --> 00:56:33,330 the output vector. 812 00:56:33,330 --> 00:56:34,950 And this says, no, you don't. 813 00:56:34,950 --> 00:56:37,390 You only need to restore on your computer what 814 00:56:37,390 --> 00:56:43,415 does T do to v1, what does T do to v2, et cetera. 815 00:56:43,415 --> 00:56:44,040 So that's good. 816 00:56:44,040 --> 00:56:46,850 Now you only have to write down n vectors, 817 00:56:46,850 --> 00:56:50,430 and since these factors in turn can 818 00:56:50,430 --> 00:56:52,310 be expressed in terms of the basis, 819 00:56:52,310 --> 00:56:55,590 you can express this just in terms of a bunch of numbers. 820 00:56:55,590 --> 00:57:03,400 So let's further expand Tvj in this basis. 821 00:57:03,400 --> 00:57:04,820 And so there's some coefficient. 822 00:57:04,820 --> 00:57:12,930 So it's something times v1 plus something times v2 something 823 00:57:12,930 --> 00:57:14,370 times vn. 824 00:57:14,370 --> 00:57:19,100 And I'm going to-- these somethings are a function of T 825 00:57:19,100 --> 00:57:30,120 so I'm just going to call this T sub 1j, T sub 2j, T sub nj. 826 00:57:33,460 --> 00:57:44,980 And this whole thing I can write more succinctly in this way. 827 00:57:44,980 --> 00:57:51,040 And now all I need are these T's of ij, 828 00:57:51,040 --> 00:57:54,120 and that can completely determine for me the action 829 00:57:54,120 --> 00:58:21,630 of T because this Tv here-- so Tv 830 00:58:21,630 --> 00:58:27,685 we can write as a sum over j of T times ajvj. 831 00:58:27,685 --> 00:58:30,420 And we can move the aj past the T. 832 00:58:30,420 --> 00:58:34,550 And then if we expand this out, we 833 00:58:34,550 --> 00:58:45,080 get that it's a sum over i from 1 to n, sum over j from 1 to n, 834 00:58:45,080 --> 00:58:45,755 of Tijajvi. 835 00:58:56,110 --> 00:58:59,640 And so if we act on in general vector, v, 836 00:58:59,640 --> 00:59:04,640 and we know the coefficients of v in some basis, 837 00:59:04,640 --> 00:59:09,690 then we can re-express it in that basis as follows. 838 00:59:09,690 --> 00:59:16,257 And this output in general can always 839 00:59:16,257 --> 00:59:18,215 be written in the basis with some coefficients. 840 00:59:24,187 --> 00:59:25,770 So we could always write it like this. 841 00:59:28,840 --> 00:59:33,630 And this formula tells you what those coefficients should be. 842 00:59:33,630 --> 00:59:40,570 They say, if your input vector has coefficients a1 through an, 843 00:59:40,570 --> 00:59:46,210 then your output vector has coefficients b1 through bn, 844 00:59:46,210 --> 00:59:51,970 where the b sub i are defined by this sum. 845 01:00:03,430 --> 01:00:11,300 And of course there's a more-- this formula is one 846 01:00:11,300 --> 01:00:16,530 that you've seen before, and it's often 847 01:00:16,530 --> 01:00:20,590 written in this more familiar form. 848 01:00:33,380 --> 01:00:36,475 So this is now the familiar matrix-vector multiplication. 849 01:00:36,475 --> 01:00:42,540 And it says that the b vector is obtained from the a vector 850 01:00:42,540 --> 01:00:46,495 by multiplying it by the matrix of these Tij. 851 01:00:49,790 --> 01:00:53,560 And so this T is the matrix form-- 852 01:00:53,560 --> 01:00:59,010 this is a matrix form of the operator T. 853 01:00:59,010 --> 01:01:03,304 And you might find this not very impressive. 854 01:01:03,304 --> 01:01:04,720 You say, well, look I already knew 855 01:01:04,720 --> 01:01:08,440 how to multiply a matrix by vector. 856 01:01:08,440 --> 01:01:13,110 But what I think is nice about this is that the usual way 857 01:01:13,110 --> 01:01:15,790 you learn linear algebra if someone says, 858 01:01:15,790 --> 01:01:18,130 a vector is a list of numbers. 859 01:01:18,130 --> 01:01:20,350 A matrix is a rectangle of numbers. 860 01:01:20,350 --> 01:01:24,200 Here's are the rules for what you do with them. 861 01:01:24,200 --> 01:01:25,660 If you want to put them together, 862 01:01:25,660 --> 01:01:27,170 you do it in this way. 863 01:01:27,170 --> 01:01:30,860 Here this was not an axiom of the theory at all. 864 01:01:30,860 --> 01:01:34,000 We just started with linear maps from one vector space 865 01:01:34,000 --> 01:01:37,780 to another one and the idea of a basis 866 01:01:37,780 --> 01:01:40,390 as something that you can prove has to exist 867 01:01:40,390 --> 01:01:43,140 and you can derive matrix multiplication. 868 01:01:43,140 --> 01:01:45,250 So matrix multiplication emerges-- 869 01:01:45,250 --> 01:01:47,970 or matrix-vector multiplication emerges 870 01:01:47,970 --> 01:01:50,570 as a consequence of the theory rather than as something 871 01:01:50,570 --> 01:01:52,710 that you have to put in. 872 01:01:52,710 --> 01:01:55,940 So that, I think, is what's kind of cute about this 873 01:01:55,940 --> 01:01:58,120 even if it comes back on the end to something 874 01:01:58,120 --> 01:02:02,939 that you had been taught before. 875 01:02:02,939 --> 01:02:03,980 Any questions about that? 876 01:02:09,420 --> 01:02:13,750 So this is matrix-vector multiplication. 877 01:02:13,750 --> 01:02:17,385 You can similarly derive matrix-matrix multiplication. 878 01:02:30,960 --> 01:02:43,230 So if we have two operators, T and S, 879 01:02:43,230 --> 01:02:48,400 and we act on a vector, v sub k-- 880 01:02:48,400 --> 01:02:50,390 and by what I argued before, it's 881 01:02:50,390 --> 01:02:53,120 enough just to know how they act on the basis vectors. 882 01:02:53,120 --> 01:02:55,350 You don't need to know-- and once you 883 01:02:55,350 --> 01:03:00,120 do that, you can figure out how they act on any vector. 884 01:03:00,120 --> 01:03:04,400 So if we just expand out what we wrote before, 885 01:03:04,400 --> 01:03:09,656 this is equal to T times the sum over j of Sjkvj. 886 01:03:18,050 --> 01:03:21,660 So Svk can be re-expressed in terms 887 01:03:21,660 --> 01:03:24,644 of the basis with some coefficients. 888 01:03:24,644 --> 01:03:26,060 And those coefficients will depend 889 01:03:26,060 --> 01:03:32,620 on the vector you start with, k, and the part 890 01:03:32,620 --> 01:03:36,762 of the basis that you're using to express it with j. 891 01:03:36,762 --> 01:03:39,810 Then we apply the same thing again with T. 892 01:03:39,810 --> 01:03:47,940 We get-- this is sum over i, sum over j TijSjkvi. 893 01:03:58,050 --> 01:04:01,690 And now, what have we done? 894 01:04:01,690 --> 01:04:06,800 TS is an operator and when you act of vk 895 01:04:06,800 --> 01:04:10,290 it spat out something that's a linear combination of all 896 01:04:10,290 --> 01:04:19,190 the basis states, v sub i, and the coefficient of v sub i 897 01:04:19,190 --> 01:04:22,280 is this part in the parentheses. 898 01:04:22,280 --> 01:04:27,210 And so this is the matrix element of TS. 899 01:04:30,120 --> 01:04:45,117 So the ik matrix element of ts is the sum over j of Tijsjk. 900 01:04:48,680 --> 01:04:51,530 And so just like we derived matrix-vector multiplication, 901 01:04:51,530 --> 01:04:54,045 here we can derive matrix-matrix multiplication. 902 01:04:58,990 --> 01:05:02,020 And so what was originally just sort of an axiom of the theory 903 01:05:02,020 --> 01:05:04,780 is now the only possible way it could 904 01:05:04,780 --> 01:05:09,490 be if you want to define operator multiplication is 905 01:05:09,490 --> 01:05:12,380 first one operator acts, than the other operator acts. 906 01:05:15,460 --> 01:05:21,200 So in terms of this-- so this, I think, 907 01:05:21,200 --> 01:05:24,400 justifies why you can think of matrices 908 01:05:24,400 --> 01:05:28,750 as a faithful representation of operators. 909 01:05:28,750 --> 01:05:34,170 And once you've chosen a basis, they can-- 910 01:05:34,170 --> 01:05:37,840 the square full of numbers becomes 911 01:05:37,840 --> 01:05:41,300 equivalent to the abstract map between vector spaces. 912 01:05:43,820 --> 01:05:46,320 And the equivalent-- they're so equivalent that I'm just 913 01:05:46,320 --> 01:05:47,950 going to write things like equal signs. 914 01:05:47,950 --> 01:05:52,420 Like I'll write identity equals a bunch of 1's 915 01:05:52,420 --> 01:05:54,020 down the diagonal, right? 916 01:05:54,020 --> 01:05:56,390 And not worry about the fact that technically this 917 01:05:56,390 --> 01:05:59,090 is an operator and this is a matrix. 918 01:05:59,090 --> 01:06:09,640 And similarly, the 0 matrix equals a matrix full of 0's. 919 01:06:09,640 --> 01:06:14,600 Technically, we should write-- if you 920 01:06:14,600 --> 01:06:17,490 want to express the basis dependence, 921 01:06:17,490 --> 01:06:33,425 you can write things like T parentheses-- sorry, 922 01:06:33,425 --> 01:06:35,290 let me write it like this. 923 01:06:41,290 --> 01:06:43,910 If you really want to be very explicit about the basis, 924 01:06:43,910 --> 01:06:46,480 you could use this to refer to the matrix. 925 01:06:46,480 --> 01:06:50,600 Just to really emphasize that the matrix depends 926 01:06:50,600 --> 01:06:54,400 not only on the operator, but also on your choice of basis. 927 01:06:54,400 --> 01:06:56,930 But we'll almost never bothered to do this. 928 01:06:56,930 --> 01:06:59,440 We usually just sort of say it in words what the basis is. 929 01:07:12,650 --> 01:07:16,020 So matrices are an important calculational tool, 930 01:07:16,020 --> 01:07:20,520 and we ultimately want to compute numbers of physical 931 01:07:20,520 --> 01:07:23,730 quantities so we cannot always spend our lives in abstract 932 01:07:23,730 --> 01:07:25,660 vector spaces. 933 01:07:25,660 --> 01:07:29,570 But the basis dependence is an unfortunate thing. 934 01:07:29,570 --> 01:07:32,170 A basis is like a choice of coordinate systems, 935 01:07:32,170 --> 01:07:35,350 and you really don't want your physics to depend on it, 936 01:07:35,350 --> 01:07:39,050 and you don't want quantity if you compute to be dependent on. 937 01:07:39,050 --> 01:07:42,474 And so we often want to formulate-- 938 01:07:42,474 --> 01:07:44,890 we're interested in quantities that are basis independent. 939 01:07:44,890 --> 01:07:47,140 And in fact, that's a big point of the whole operator 940 01:07:47,140 --> 01:07:50,190 picture is that because the quantities we want 941 01:07:50,190 --> 01:07:52,110 are ultimately basis independent, 942 01:07:52,110 --> 01:07:55,720 it's nice to have language that is itself basis independent. 943 01:07:55,720 --> 01:08:00,930 Terminology and theorems that do not refer to a basis. 944 01:08:00,930 --> 01:08:13,120 I'll mention a few basis independent quantities, 945 01:08:13,120 --> 01:08:14,960 and I won't say too much more about them 946 01:08:14,960 --> 01:08:17,710 because you will prove properties [INAUDIBLE] 947 01:08:17,710 --> 01:08:23,790 on your p set, but one of them is the trace 948 01:08:23,790 --> 01:08:25,595 and another one is the determinant. 949 01:08:28,640 --> 01:08:30,920 And when you first look at them-- OK, 950 01:08:30,920 --> 01:08:35,130 you can check that each one is basis independent, 951 01:08:35,130 --> 01:08:37,630 and it really looks kind of mysterious. 952 01:08:37,630 --> 01:08:43,242 I mean, like, who pulled these out of the hat? 953 01:08:43,242 --> 01:08:44,700 They look totally different, right? 954 01:08:44,700 --> 01:08:48,060 They don't look remotely related to each other. 955 01:08:48,060 --> 01:08:50,510 And are these all there is? 956 01:08:50,510 --> 01:08:53,319 Are there many more? 957 01:08:53,319 --> 01:08:57,740 And it turns out that, at least for matrices with eigenvalues, 958 01:08:57,740 --> 01:09:02,210 these can be seen as members of a much larger family. 959 01:09:02,210 --> 01:09:04,710 And the reason is that the trace turns out 960 01:09:04,710 --> 01:09:06,941 to be the sum of all the eigenvalues 961 01:09:06,941 --> 01:09:08,399 and the determinant turns out to be 962 01:09:08,399 --> 01:09:10,899 the product of all of the eigenvalues. 963 01:09:10,899 --> 01:09:14,529 And in general, we'll see in a minute, that basis 964 01:09:14,529 --> 01:09:16,500 independent things-- actually, not in a minute. 965 01:09:16,500 --> 01:09:20,370 In a future lecture-- that basis independent things 966 01:09:20,370 --> 01:09:22,830 are functions of eigenvalues. 967 01:09:22,830 --> 01:09:25,290 And furthermore, that don't care about the ordering 968 01:09:25,290 --> 01:09:26,560 of the eigenvalues. 969 01:09:26,560 --> 01:09:29,080 So they're symmetric functions of eigenvalues. 970 01:09:29,080 --> 01:09:31,240 And then it starts to make a little bit more sense. 971 01:09:31,240 --> 01:09:33,769 Because if you talk about symmetric polynomials, 972 01:09:33,769 --> 01:09:36,060 those are two of the most important ones where you just 973 01:09:36,060 --> 01:09:39,189 add up all the things and when you multiply all the things. 974 01:09:39,189 --> 01:09:40,950 And then, if you add this perspective 975 01:09:40,950 --> 01:09:43,930 of symmetric polynomial of the eigenvalue, 976 01:09:43,930 --> 01:09:47,830 then you can cook up other basis independent quantities. 977 01:09:47,830 --> 01:09:50,252 So this is actually not the approach 978 01:09:50,252 --> 01:09:51,460 you should take on the p set. 979 01:09:51,460 --> 01:09:52,835 The [? p set ?] asks you to prove 980 01:09:52,835 --> 01:09:56,520 more directly that the trace is basis independent, 981 01:09:56,520 --> 01:09:59,710 but the sort of framework that these fit into 982 01:09:59,710 --> 01:10:01,750 is symmetric functions of eigenvalues. 983 01:10:06,110 --> 01:10:11,340 So I want to say a little bit about eigenvalues. 984 01:10:11,340 --> 01:10:13,120 Any questions about matrices before I do? 985 01:10:29,950 --> 01:10:33,852 So eigenvalues-- I guess, these are 986 01:10:33,852 --> 01:10:35,060 basis independent quantities. 987 01:10:42,800 --> 01:10:45,840 Another important basis independent quantity, 988 01:10:45,840 --> 01:10:49,370 or property of a matrix, is its eigenvalue-eigenvector 989 01:10:49,370 --> 01:10:49,870 structure. 990 01:10:58,320 --> 01:11:01,410 The place where eigenvectors come from 991 01:11:01,410 --> 01:11:05,190 is by considering a slightly more general thing, which 992 01:11:05,190 --> 01:11:07,790 is the idea of an invariant subspace. 993 01:11:07,790 --> 01:11:22,490 So we say that U is a T invariant subspace 994 01:11:22,490 --> 01:11:29,310 if T of U-- this is an operator acting on an entire subspace. 995 01:11:29,310 --> 01:11:32,390 So what do I mean by that? 996 01:11:32,390 --> 01:11:39,450 I mean the set of all TU for vectors in the subspace. 997 01:11:39,450 --> 01:11:47,450 If T of U is contained in U. 998 01:11:47,450 --> 01:11:51,470 So I take a vector in this subspace, act on it with T, 999 01:11:51,470 --> 01:11:52,940 and then I'm still in the subspace 1000 01:11:52,940 --> 01:11:55,680 no matter which vector I had. 1001 01:11:55,680 --> 01:12:03,680 So some examples that always work. 1002 01:12:03,680 --> 01:12:06,310 The 0 subspace is invariant. 1003 01:12:06,310 --> 01:12:10,320 T always maps it to itself. 1004 01:12:10,320 --> 01:12:14,760 And the entire space, v, T is a linear operator on v 1005 01:12:14,760 --> 01:12:17,470 so by definition it maps v to itself. 1006 01:12:20,280 --> 01:12:22,730 These are called the trivial examples. 1007 01:12:22,730 --> 01:12:26,410 And usually when people talk about non-trivial invariant 1008 01:12:26,410 --> 01:12:29,810 subspaces they mean not one of these two. 1009 01:12:29,810 --> 01:12:32,610 The particular type that we will be interested in 1010 01:12:32,610 --> 01:12:34,520 are one dimensional ones. 1011 01:12:43,100 --> 01:12:51,330 So this corresponds to a direction that T fixes. 1012 01:12:51,330 --> 01:12:57,430 So U-- this vector space now can be written just 1013 01:12:57,430 --> 01:13:10,710 as the span of a single vector, U, and U being T invariant 1014 01:13:10,710 --> 01:13:19,280 is equivalent to TU being a mu, because they're just 1015 01:13:19,280 --> 01:13:20,020 a single vector. 1016 01:13:20,020 --> 01:13:22,590 So all I have to do is get that single vector right 1017 01:13:22,590 --> 01:13:25,170 and I'll get the whole subspace right. 1018 01:13:25,170 --> 01:13:41,040 And that, in turn, is equivalent to TU being some multiple of U. 1019 01:13:41,040 --> 01:13:47,360 And this equation you've seen before. 1020 01:13:47,360 --> 01:13:50,060 This is the familiar eigenvector equation. 1021 01:13:50,060 --> 01:13:53,734 And if it's a very, very important equation 1022 01:13:53,734 --> 01:13:55,400 it might be named after a mathematician, 1023 01:13:55,400 --> 01:13:59,420 but this one is so important that two of the pieces of it 1024 01:13:59,420 --> 01:14:02,710 have their own special name. 1025 01:14:02,710 --> 01:14:08,920 So these are called-- lambda is called an eigenvalue 1026 01:14:08,920 --> 01:14:15,680 and U is called an eigenvector. 1027 01:14:21,340 --> 01:14:28,500 And more or less it's true that all of the solutions to this 1028 01:14:28,500 --> 01:14:30,590 are called eigenvalues, and all the solutions 1029 01:14:30,590 --> 01:14:32,550 are called eigenvectors. 1030 01:14:32,550 --> 01:14:38,960 There's one exception, which is there's 1031 01:14:38,960 --> 01:14:41,980 one kind of trivial solution to this equation, which 1032 01:14:41,980 --> 01:14:46,780 is when U is 0 this equation is always true. 1033 01:14:46,780 --> 01:14:49,820 And that's not very interesting, but it's 1034 01:14:49,820 --> 01:14:53,520 true for all values of lambda. 1035 01:14:53,520 --> 01:14:56,910 And so that doesn't count as being an eigenvalue. 1036 01:14:56,910 --> 01:14:59,264 And you can tell a doesn't correspond to 1D invariant 1037 01:14:59,264 --> 01:14:59,930 subspace, right? 1038 01:14:59,930 --> 01:15:02,970 It corresponds to a 0 dimensional subspace, 1039 01:15:02,970 --> 01:15:04,090 which is the trivial case. 1040 01:15:11,120 --> 01:15:23,010 So we say that lambda is an eigenvalue of T 1041 01:15:23,010 --> 01:15:34,270 if Tu equals lambda U for some non-zero vector, U. 1042 01:15:34,270 --> 01:15:36,160 So the non 0 is crucial. 1043 01:15:42,250 --> 01:15:51,250 And then the spectrum of T is the collection 1044 01:15:51,250 --> 01:15:52,115 of all eigenvalues. 1045 01:16:10,060 --> 01:16:15,000 So there's something a little bit asymmetric about this, 1046 01:16:15,000 --> 01:16:17,830 which is we still say that 0 vector is 1047 01:16:17,830 --> 01:16:24,710 an eigenvector with all the various eigenvalues, 1048 01:16:24,710 --> 01:16:27,510 but we had to put this here or everything 1049 01:16:27,510 --> 01:16:31,570 would be an eigenvalue and it wouldn't be very interesting. 1050 01:16:31,570 --> 01:16:34,210 So the-- 1051 01:16:38,710 --> 01:16:42,380 Oh, also I want to say this term spectrum you'll see it other 1052 01:16:42,380 --> 01:16:42,880 [INAUDIBLE]. 1053 01:16:42,880 --> 01:16:46,500 You'll see spectral theory or spectral this or that, 1054 01:16:46,500 --> 01:16:49,480 and that means essentially making use of the eigenvalues. 1055 01:16:49,480 --> 01:16:52,900 So people talk about partitioning a graph using 1056 01:16:52,900 --> 01:16:55,690 eigenvalues of the associated matrix, that's 1057 01:16:55,690 --> 01:16:57,910 called spectral partitioning. 1058 01:16:57,910 --> 01:17:04,260 And so throughout math, this term is used a lot. 1059 01:17:06,960 --> 01:17:12,550 So I have only about three minutes 1060 01:17:12,550 --> 01:17:16,340 left to tell-- so I think I will not 1061 01:17:16,340 --> 01:17:21,920 finish the eigenvalue discussion but will just show you 1062 01:17:21,920 --> 01:17:26,280 a few examples of how it's not always 1063 01:17:26,280 --> 01:17:29,810 as nice as you might expect. 1064 01:17:29,810 --> 01:17:40,910 So one example that I'll consider 1065 01:17:40,910 --> 01:17:50,270 is the vector space will be the reals, 3D real space, 1066 01:17:50,270 --> 01:17:56,620 and the operator, T, will be rotation about the z-axis 1067 01:17:56,620 --> 01:17:59,500 by some small angle. 1068 01:17:59,500 --> 01:18:08,345 Let's call it a theta rotation about the z-axis. 1069 01:18:10,860 --> 01:18:13,899 Turns out, if you write this in matrix form, 1070 01:18:13,899 --> 01:18:14,690 it looks like this. 1071 01:18:14,690 --> 01:18:24,840 Cosine theta minus sine theta 0 sine theta cosine theta 0, 0, 1072 01:18:24,840 --> 01:18:27,640 0, 0, 1. 1073 01:18:27,640 --> 01:18:30,560 That 1 is because it leaves the z-axis alone 1074 01:18:30,560 --> 01:18:33,620 and then x and y get rotated. 1075 01:18:33,620 --> 01:18:35,520 You can tell if theta is 0 it does nothing 1076 01:18:35,520 --> 01:18:36,966 so that's reassuring. 1077 01:18:36,966 --> 01:18:38,590 And if theta does a little bit, then it 1078 01:18:38,590 --> 01:18:42,170 starts mixing the x and y components. 1079 01:18:42,170 --> 01:18:46,120 So that is the rotation matrix. 1080 01:18:46,120 --> 01:18:51,125 So what is an eigenvalue-- and anyone 1081 01:18:51,125 --> 01:18:53,670 say what an eigenvalue is of this matrix? 1082 01:18:53,670 --> 01:18:54,170 AUDIENCE: 1. 1083 01:18:54,170 --> 01:18:54,650 ARAM HARROW: 1. 1084 01:18:54,650 --> 01:18:54,840 Good. 1085 01:18:54,840 --> 01:18:56,070 And what's the eigenvector? 1086 01:18:56,070 --> 01:18:57,450 AUDIENCE: The z basis vector. 1087 01:18:57,450 --> 01:18:58,520 ARAM HARROW: The z basis vector. 1088 01:18:58,520 --> 01:18:59,019 Right. 1089 01:18:59,019 --> 01:19:02,290 So it fixes a z basis vector so this 1090 01:19:02,290 --> 01:19:06,430 is an eigenvector with eigenvalue 1. 1091 01:19:06,430 --> 01:19:09,940 Does it have any other eigenvectors? 1092 01:19:09,940 --> 01:19:10,538 Yeah? 1093 01:19:10,538 --> 01:19:13,190 AUDIENCE: If you go to the complex plane, then yes. 1094 01:19:13,190 --> 01:19:15,880 ARAM HARROW: If you are talking about complex numbers, then 1095 01:19:15,880 --> 01:19:19,130 yes, it has complex eigenvalues. 1096 01:19:19,130 --> 01:19:22,900 But if we're talking about a real vector space, 1097 01:19:22,900 --> 01:19:26,040 then it doesn't. 1098 01:19:26,040 --> 01:19:30,910 And so this just has one eigenvalue and one eigenvector. 1099 01:19:30,910 --> 01:19:36,600 And if we were to get rid of the third dimension-- 1100 01:19:36,600 --> 01:19:39,800 so if we just had T-- and let's be even simpler, 1101 01:19:39,800 --> 01:19:41,595 let's just take theta to be pi over 2. 1102 01:19:46,370 --> 01:19:51,540 So let's just take a 90 degree rotation in the plane. 1103 01:19:54,090 --> 01:19:57,620 Now T has no eigenvalues. 1104 01:19:57,620 --> 01:20:02,800 There are no vectors other than 0 that it sends to itself. 1105 01:20:02,800 --> 01:20:11,020 And so this is a slightly unfortunate note 1106 01:20:11,020 --> 01:20:11,996 to end the lecture on. 1107 01:20:11,996 --> 01:20:13,870 You think, well, these eigenvalues are great, 1108 01:20:13,870 --> 01:20:16,140 but maybe they exist, maybe they don't. 1109 01:20:16,140 --> 01:20:19,250 And you'll see next time part of the reason 1110 01:20:19,250 --> 01:20:21,680 why we use complex numbers, even though it looks 1111 01:20:21,680 --> 01:20:26,200 like real space isn't complex, is because any polynomial can 1112 01:20:26,200 --> 01:20:28,300 be completely factored in complex numbers, 1113 01:20:28,300 --> 01:20:32,660 and every matrix has a complex egeinvalue. 1114 01:20:32,660 --> 01:20:34,880 OK, I'll stop here.