1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:31,565 --> 00:00:32,940 HERBERT GROSS: Hi. 9 00:00:32,940 --> 00:00:35,490 Today, we begin our eighth and final block 10 00:00:35,490 --> 00:00:37,900 of material in our course. 11 00:00:37,900 --> 00:00:41,880 And since we began the course with an emphasis 12 00:00:41,880 --> 00:00:45,120 on structure, in particular vector spaces, 13 00:00:45,120 --> 00:00:49,110 it's only appropriate that we conclude on the same level. 14 00:00:49,110 --> 00:00:51,000 Now, you may recall, when we first 15 00:00:51,000 --> 00:00:53,580 started talking about vectors, we 16 00:00:53,580 --> 00:00:57,570 began with the rather simple interpretation of arrows. 17 00:00:57,570 --> 00:01:00,930 Then we got to 2-tuples and 3-tuples 18 00:01:00,930 --> 00:01:04,500 to revisit what arrows looked like numerically. 19 00:01:04,500 --> 00:01:08,840 Then we generalized this to n-dimensional n-tuples 20 00:01:08,840 --> 00:01:13,140 so that we could talk about functions of n real variables. 21 00:01:13,140 --> 00:01:18,060 And what we did was we defined a vector space to be the n-tuples 22 00:01:18,060 --> 00:01:20,800 together with certain structural properties. 23 00:01:20,800 --> 00:01:23,880 And at the very end, we then did the usual thing 24 00:01:23,880 --> 00:01:25,650 that one does in a structure. 25 00:01:25,650 --> 00:01:27,900 We showed what a vector space would 26 00:01:27,900 --> 00:01:31,530 look like without reference to n-tuples. 27 00:01:31,530 --> 00:01:34,200 Now what I'd like to do in this lecture, 28 00:01:34,200 --> 00:01:37,650 and emphasize this for the remainder of our course, 29 00:01:37,650 --> 00:01:41,070 is to show what vector spaces would have looked like, 30 00:01:41,070 --> 00:01:44,520 what enrichment we could have gained if we had stuck 31 00:01:44,520 --> 00:01:47,970 with our more generalized axiomatic definition 32 00:01:47,970 --> 00:01:48,840 in the first place. 33 00:01:48,840 --> 00:01:52,170 And rather than go on about this philosophically, 34 00:01:52,170 --> 00:01:56,220 let me get to the gist of our lecture today, which, as I say, 35 00:01:56,220 --> 00:01:58,710 is simply called vector spaces, but hopefully 36 00:01:58,710 --> 00:02:02,400 from a perspective different from what was had before, 37 00:02:02,400 --> 00:02:06,150 that the structural axiomatic definition, you'll recall, 38 00:02:06,150 --> 00:02:08,850 was that V was called a vector space with respect 39 00:02:08,850 --> 00:02:10,280 to the real numbers R. 40 00:02:10,280 --> 00:02:12,210 And I should mention here that there 41 00:02:12,210 --> 00:02:14,700 are places where one might talk about a vector 42 00:02:14,700 --> 00:02:18,750 space with respect to the complex numbers C, or something 43 00:02:18,750 --> 00:02:19,280 like this. 44 00:02:19,280 --> 00:02:21,120 But, for our purposes, we are going 45 00:02:21,120 --> 00:02:25,500 to be content to stick with what is called real vector 46 00:02:25,500 --> 00:02:27,660 spaces, namely a vector space with respect 47 00:02:27,660 --> 00:02:29,110 to the real numbers. 48 00:02:29,110 --> 00:02:30,850 And what were the structural properties? 49 00:02:30,850 --> 00:02:34,830 Well, the sum of two vectors had to be a vector, 50 00:02:34,830 --> 00:02:38,570 that the sum had to obey the associative rule. 51 00:02:38,570 --> 00:02:41,490 There had to be an additive identity. 52 00:02:41,490 --> 00:02:44,460 There had to be additive inverses. 53 00:02:44,460 --> 00:02:47,520 And the sum had to be commutative. 54 00:02:47,520 --> 00:02:50,910 We also had a scalar multiplication structure, 55 00:02:50,910 --> 00:02:54,770 if you'll recall, that said that if a scalar multiplied 56 00:02:54,770 --> 00:02:58,170 the sum of two vectors, the distributive rule held, 57 00:02:58,170 --> 00:03:01,320 namely that c times alpha plus beta was c alpha 58 00:03:01,320 --> 00:03:04,560 plus c beta, where c was a real number. 59 00:03:04,560 --> 00:03:08,100 And the distributive rule also held if the sum of two numbers 60 00:03:08,100 --> 00:03:10,490 was multiplying a vector, you see, 61 00:03:10,490 --> 00:03:13,030 where c1 and c2 here are real numbers. 62 00:03:13,030 --> 00:03:18,900 And, finally, if a scalar times a scalar multiple of a vector 63 00:03:18,900 --> 00:03:23,160 were given, this product was also associative, namely 64 00:03:23,160 --> 00:03:26,520 that c1 times the vector c2 alpha 65 00:03:26,520 --> 00:03:31,200 is the same as the scalar multiple c1 c2 times alpha. 66 00:03:31,200 --> 00:03:35,790 And, finally, that the number 1 multiplied by the vector alpha, 67 00:03:35,790 --> 00:03:39,360 the scalar multiple 1 times alpha, was still alpha. 68 00:03:39,360 --> 00:03:41,030 In other words, that the scalar-- 69 00:03:41,030 --> 00:03:43,170 that, with respect to scalar multiplication, 70 00:03:43,170 --> 00:03:46,680 1 still behaves like the multiplicative identity. 71 00:03:46,680 --> 00:03:50,940 And what we noted was that the properties of n-tuples 72 00:03:50,940 --> 00:03:54,930 were now theorems with respect to the structural definition. 73 00:03:54,930 --> 00:03:59,430 You see, we could show that if alpha were an n-tuple, that 0 74 00:03:59,430 --> 00:04:04,230 times alpha was 0 for all n-tuples in V. 75 00:04:04,230 --> 00:04:07,560 But now what we're saying is we also showed at that time 76 00:04:07,560 --> 00:04:11,170 that, axiomatically, one could prove all of these results, 77 00:04:11,170 --> 00:04:15,600 that the 0 scalar times a vector was always 0. 78 00:04:15,600 --> 00:04:19,470 The scalar c times the 0 vector was always 79 00:04:19,470 --> 00:04:22,470 the 0 vector for all scalars. 80 00:04:22,470 --> 00:04:25,830 That if alpha plus beta equals alpha plus gamma, 81 00:04:25,830 --> 00:04:26,970 beta equaled gamma. 82 00:04:26,970 --> 00:04:29,820 The cancellation rule held. 83 00:04:29,820 --> 00:04:32,580 If c times alpha was 0, either c was 84 00:04:32,580 --> 00:04:34,920 0 or alpha was 0, et cetera, meaning, 85 00:04:34,920 --> 00:04:36,320 we had all of these theorems. 86 00:04:36,320 --> 00:04:39,150 And, just again by way of refreshing your memories, 87 00:04:39,150 --> 00:04:43,980 I very quickly jotted down a semi-proof of this result, 88 00:04:43,980 --> 00:04:48,570 namely, if c wasn't 0, knowing that c times alpha was 0, 89 00:04:48,570 --> 00:04:51,480 we multiply both sides by 1 over c. 90 00:04:51,480 --> 00:04:57,210 By the associative property, I can group 1 over c with c. 91 00:04:57,210 --> 00:04:59,640 And 1 over c times c is 1. 92 00:04:59,640 --> 00:05:03,930 I also know that c times 0 is still 0. 93 00:05:03,930 --> 00:05:07,040 And in words, any real number time the 0 vector 94 00:05:07,040 --> 00:05:08,580 is still the 0 vector. 95 00:05:08,580 --> 00:05:12,000 Putting all of this together, you see I have what? 96 00:05:12,000 --> 00:05:14,820 That regrouping this in this form, 97 00:05:14,820 --> 00:05:17,130 this must equal the 0 vector. 98 00:05:17,130 --> 00:05:19,920 But this is 1 times alpha equaling 0. 99 00:05:19,920 --> 00:05:23,280 But by our ninth axiom, 1 times alpha is alpha. 100 00:05:23,280 --> 00:05:25,140 Therefore, alpha equals 0. 101 00:05:25,140 --> 00:05:28,830 And we can go on in this particular way, re-deriving 102 00:05:28,830 --> 00:05:32,390 all of the properties that we previously talked about when 103 00:05:32,390 --> 00:05:33,920 we were dealing with n-tuples. 104 00:05:33,920 --> 00:05:35,840 The question that comes up is, what's 105 00:05:35,840 --> 00:05:39,690 so great about doing the same thing in two different ways? 106 00:05:39,690 --> 00:05:42,950 Why do we have to come back to the structural definition 107 00:05:42,950 --> 00:05:46,100 when the n-tuples were working very nicely for us? 108 00:05:46,100 --> 00:05:48,830 And what I'd like to do is two things, 109 00:05:48,830 --> 00:05:51,410 and I'm going to introduce one new concept, 110 00:05:51,410 --> 00:05:54,260 and let it drop fairly rapidly, and then come back 111 00:05:54,260 --> 00:05:57,680 to a older concept from a new point of view. 112 00:05:57,680 --> 00:06:02,150 In the first place, my claim is that the axiomatic definition 113 00:06:02,150 --> 00:06:07,850 of a vector space opens up a whole new avenue of things 114 00:06:07,850 --> 00:06:11,210 that we can call vector spaces that we couldn't have called 115 00:06:11,210 --> 00:06:14,510 vector spaces before, because they couldn't be represented 116 00:06:14,510 --> 00:06:16,165 as n-tuples, for example. 117 00:06:16,165 --> 00:06:19,220 And let me just say that-- so you see it written down here-- 118 00:06:19,220 --> 00:06:22,730 new definition permits new vector spaces. 119 00:06:22,730 --> 00:06:27,650 By way of example, let me take the set of all functions 120 00:06:27,650 --> 00:06:30,980 subject only to the condition that they have a common domain. 121 00:06:30,980 --> 00:06:33,680 For example, let's take a set of all functions f 122 00:06:33,680 --> 00:06:36,170 whose domain of definitions, say, is the closed 123 00:06:36,170 --> 00:06:37,310 interval from a to b. 124 00:06:37,310 --> 00:06:41,030 In other words, all you need to belong to the set V 125 00:06:41,030 --> 00:06:44,000 is a function defined for all real numbers 126 00:06:44,000 --> 00:06:46,340 on the closed interval from a to b. 127 00:06:46,340 --> 00:06:49,700 Notice that, both in part 1 and part 2 of this course, 128 00:06:49,700 --> 00:06:53,090 we had already defined very special meanings 129 00:06:53,090 --> 00:06:55,430 for what it meant by the sum of two functions 130 00:06:55,430 --> 00:06:59,525 and the scalar multiple of a function, namely, by f plus g, 131 00:06:59,525 --> 00:07:03,440 where f and g were functions with a common domain, 132 00:07:03,440 --> 00:07:05,360 and by c times f. 133 00:07:05,360 --> 00:07:07,370 Recall our definitions were what? 134 00:07:07,370 --> 00:07:13,610 That f plus g of x is simply f of x plus g of x, and cf of x 135 00:07:13,610 --> 00:07:16,430 is simply c times f of x, where x 136 00:07:16,430 --> 00:07:19,940 is any number in the close interval from a to b. 137 00:07:19,940 --> 00:07:22,400 Notice, in terms of a picture, what we're saying 138 00:07:22,400 --> 00:07:26,690 is that, for the f plus g machine, if the input is x, 139 00:07:26,690 --> 00:07:29,390 the output is f of x plus g of x. 140 00:07:29,390 --> 00:07:32,510 For the cf machine, if the input is x, 141 00:07:32,510 --> 00:07:34,970 the output is c times f of x. 142 00:07:34,970 --> 00:07:38,450 Notice that if f is defined on the closed interval from a 143 00:07:38,450 --> 00:07:41,260 to b, c times f is defined on the closed 144 00:07:41,260 --> 00:07:42,560 interval from a to b. 145 00:07:42,560 --> 00:07:46,010 And if f and g have a common domain, 146 00:07:46,010 --> 00:07:47,840 in particular if the domain happens 147 00:07:47,840 --> 00:07:49,880 to be the closed interval from a to b, 148 00:07:49,880 --> 00:07:53,180 notice that x belongs to both the domains of f and g, 149 00:07:53,180 --> 00:07:56,700 and that, consequently, this summation makes sense. 150 00:07:56,700 --> 00:07:58,580 In other words, if we had a number which 151 00:07:58,580 --> 00:08:02,480 belonged to the domain of f but not to the domain of g, 152 00:08:02,480 --> 00:08:04,960 this wouldn't even be defined. 153 00:08:04,960 --> 00:08:05,480 All right. 154 00:08:05,480 --> 00:08:08,090 So here is a new type of set that 155 00:08:08,090 --> 00:08:09,440 doesn't look like n-tuples. 156 00:08:09,440 --> 00:08:12,560 In other words, there are a voluminous number 157 00:08:12,560 --> 00:08:16,580 of independent functions which are defined on some interval 158 00:08:16,580 --> 00:08:18,380 from a to b. 159 00:08:18,380 --> 00:08:21,650 But the key point is that this new vector-- 160 00:08:21,650 --> 00:08:26,000 that this new set V satisfies axioms 1 161 00:08:26,000 --> 00:08:30,080 through 9 of our new definition of our structural definition. 162 00:08:30,080 --> 00:08:32,030 In other words, the set of all functions 163 00:08:32,030 --> 00:08:33,870 which are continuous-- 164 00:08:33,870 --> 00:08:36,799 I'm sorry-- which are defined on the interval from a to b, 165 00:08:36,799 --> 00:08:39,500 according to our structural definition, 166 00:08:39,500 --> 00:08:42,539 would also be a vector space. 167 00:08:42,539 --> 00:08:46,010 In other words, our key point is that the new definition 168 00:08:46,010 --> 00:08:48,590 enlarges the concept of a vector space, 169 00:08:48,590 --> 00:08:50,690 where, by enlarges the concept, I 170 00:08:50,690 --> 00:08:55,640 mean it allows new animals to get in to the definition, 171 00:08:55,640 --> 00:08:59,330 that we have enlarged what can now be called a vector space. 172 00:08:59,330 --> 00:09:01,400 Let me pause here for a moment just 173 00:09:01,400 --> 00:09:03,860 to enforce one little detail for you, 174 00:09:03,860 --> 00:09:08,150 and that is keep in mind that I am appreciative of the fact 175 00:09:08,150 --> 00:09:11,330 that, probably, you are not used to visualizing 176 00:09:11,330 --> 00:09:14,690 a set of functions as being a vector space. 177 00:09:14,690 --> 00:09:17,720 Consequently, it's my feeling that if I continue 178 00:09:17,720 --> 00:09:20,990 harping on this particular example, much of what I say 179 00:09:20,990 --> 00:09:24,680 will be lost on you because you haven't had enough time to have 180 00:09:24,680 --> 00:09:26,700 this new concept sink in. 181 00:09:26,700 --> 00:09:30,320 So what I'm going to do is to leave a number of exercises 182 00:09:30,320 --> 00:09:32,150 for our study guide where you can 183 00:09:32,150 --> 00:09:37,490 get adequate drill on what we mean by this new type of vector 184 00:09:37,490 --> 00:09:38,450 space. 185 00:09:38,450 --> 00:09:42,050 What I'd like to do for the next part of our lecture 186 00:09:42,050 --> 00:09:44,730 is to get away from this new concept, which, by the way, 187 00:09:44,730 --> 00:09:48,320 is very important, which we will emphasize 188 00:09:48,320 --> 00:09:51,080 as the remainder of this block continues. 189 00:09:51,080 --> 00:09:53,480 But for those who say, I'm not that interested 190 00:09:53,480 --> 00:09:55,130 in functions as vector spaces-- 191 00:09:55,130 --> 00:09:56,660 I can't visualize that-- 192 00:09:56,660 --> 00:09:59,600 I would like to point out a second key point. 193 00:09:59,600 --> 00:10:03,290 And that is that even when you deal with n-tuples, 194 00:10:03,290 --> 00:10:06,590 the structural definition has advantages 195 00:10:06,590 --> 00:10:09,710 over the n-tuple definition. 196 00:10:09,710 --> 00:10:11,810 In particular, which I call key point number 197 00:10:11,810 --> 00:10:14,510 2, the new definition-- and by new I 198 00:10:14,510 --> 00:10:17,030 mean the structural definition-- is 199 00:10:17,030 --> 00:10:21,270 free of any dependence of a coordinate system. 200 00:10:21,270 --> 00:10:23,660 Now, what do I mean by dependence 201 00:10:23,660 --> 00:10:25,080 of a coordinate system? 202 00:10:25,080 --> 00:10:27,260 Let's see if I can't make that clear by means 203 00:10:27,260 --> 00:10:30,090 of a particular example. 204 00:10:30,090 --> 00:10:32,280 Let's suppose I'm dealing in the-- 205 00:10:32,280 --> 00:10:36,070 I'm in the xy plane, and we're used to the vectors i and j 206 00:10:36,070 --> 00:10:40,240 as being the basic vectors of the xy plane. 207 00:10:40,240 --> 00:10:42,130 Let's assume for the sake of argument 208 00:10:42,130 --> 00:10:44,020 that, for one reason or another-- 209 00:10:44,020 --> 00:10:46,150 and I'll clarify what one reason or another 210 00:10:46,150 --> 00:10:48,430 means as our course goes on. 211 00:10:48,430 --> 00:10:50,080 But, for one reason or another, let's 212 00:10:50,080 --> 00:10:51,880 suppose that, for the particular problems 213 00:10:51,880 --> 00:10:54,940 I'm interested in, the two vectors alpha 214 00:10:54,940 --> 00:10:59,320 and beta, where alpha is defined to be i plus j, and beta, say, 215 00:10:59,320 --> 00:11:01,990 is defined to be 3i plus 2j, let's 216 00:11:01,990 --> 00:11:05,350 suppose that the two vectors alpha and beta 217 00:11:05,350 --> 00:11:07,960 happen to be the two vectors that we, 218 00:11:07,960 --> 00:11:09,940 for some reason or other, happen to be 219 00:11:09,940 --> 00:11:14,350 more interested in than we are interested in i and j. 220 00:11:14,350 --> 00:11:16,250 Now, the idea is something like this. 221 00:11:16,250 --> 00:11:19,180 First of all, observe that, because of the arithmetic, 222 00:11:19,180 --> 00:11:22,480 the axioms that a vector space satisfies, 223 00:11:22,480 --> 00:11:26,440 I can treat alpha, beta, i and j, the same as I would 224 00:11:26,440 --> 00:11:28,730 real variables, so to speak. 225 00:11:28,730 --> 00:11:31,610 And I can solve algebraically for i 226 00:11:31,610 --> 00:11:34,130 and j in terms of alpha and beta, 227 00:11:34,130 --> 00:11:35,830 which I've just done over here. 228 00:11:35,830 --> 00:11:38,020 And I leave the details for you. 229 00:11:38,020 --> 00:11:41,020 You just solve these as two equations and two unknowns. 230 00:11:41,020 --> 00:11:44,420 Solve for the i and j in terms of alpha and beta. 231 00:11:44,420 --> 00:11:45,920 Now the idea is this. 232 00:11:45,920 --> 00:11:47,560 Let's suppose I'm given some vector 233 00:11:47,560 --> 00:11:53,740 v. For the sake of argument, let v be the vector 4i plus 5j. 234 00:11:53,740 --> 00:11:56,740 Now, notice that the vector v is well defined, 235 00:11:56,740 --> 00:11:59,470 and I'll show you that pictorially in a few moments. 236 00:11:59,470 --> 00:12:02,830 But that vector v exists independently of 237 00:12:02,830 --> 00:12:05,110 whether we talk about i and j components or not. 238 00:12:05,110 --> 00:12:08,770 In other words, once I tell you that v is 4i plus 5j, 239 00:12:08,770 --> 00:12:11,050 it exists as an arrow in the plane. 240 00:12:11,050 --> 00:12:14,320 And once I've drawn that arrow, that arrow 241 00:12:14,320 --> 00:12:19,360 is well defined even if I now erase what this definition is. 242 00:12:19,360 --> 00:12:21,870 The point I wanted to make, though, is this. 243 00:12:21,870 --> 00:12:25,030 You will recall that when we dealt with 2-tuples, 244 00:12:25,030 --> 00:12:28,090 we picked i and j as being the important vectors, 245 00:12:28,090 --> 00:12:31,870 and we abbreviated v as 4 comma 5, 246 00:12:31,870 --> 00:12:35,880 indicating that the vector 4i plus 5j, 247 00:12:35,880 --> 00:12:39,040 if it originated at 0, 0, would terminate 248 00:12:39,040 --> 00:12:41,090 at the point 4 comma 5. 249 00:12:41,090 --> 00:12:43,910 Now suppose, for the sake of argument, 250 00:12:43,910 --> 00:12:48,310 we were interested in v not in terms of i and j 251 00:12:48,310 --> 00:12:51,910 but in terms of the new vectors alpha and beta. 252 00:12:51,910 --> 00:12:55,030 Knowing that i is minus 2 alpha plus beta, 253 00:12:55,030 --> 00:12:59,230 and j is 3 alpha minus beta, by direct substitution, 254 00:12:59,230 --> 00:13:02,560 we can replace i and j by what they're equal to in terms 255 00:13:02,560 --> 00:13:04,180 of alpha and beta. 256 00:13:04,180 --> 00:13:05,680 So just by direct substitution. 257 00:13:05,680 --> 00:13:07,610 And I can conclude-- 258 00:13:07,610 --> 00:13:09,850 again, notice the arithmetic here. 259 00:13:09,850 --> 00:13:10,960 You see 4-- 260 00:13:10,960 --> 00:13:12,940 I can multiply this out-- 261 00:13:12,940 --> 00:13:15,700 minus 8 alpha plus 4 beta, because I 262 00:13:15,700 --> 00:13:19,390 have a rule that tells me that a scalar times the sum of two 263 00:13:19,390 --> 00:13:22,180 vectors is a scalar times the first vector 264 00:13:22,180 --> 00:13:24,700 plus a scalar times the second vector, et cetera. 265 00:13:24,700 --> 00:13:26,840 All the rules of arithmetic apply here. 266 00:13:26,840 --> 00:13:31,210 So I very quickly say 15 alpha minus 8 alpha is 7 alpha. 267 00:13:31,210 --> 00:13:34,640 4 beta minus 5 beta is minus beta. 268 00:13:34,640 --> 00:13:37,570 So relative to alpha and beta, notice 269 00:13:37,570 --> 00:13:41,410 that v has the unique representation 7 alpha 270 00:13:41,410 --> 00:13:42,610 minus beta. 271 00:13:42,610 --> 00:13:46,780 Now, suppose I wanted to abbreviate v with reference 272 00:13:46,780 --> 00:13:51,840 to the basis vectors alpha and beta, where 273 00:13:51,840 --> 00:13:54,390 the convention would be that I would list the alpha component 274 00:13:54,390 --> 00:13:57,240 first and the beta component secondly. 275 00:13:57,240 --> 00:13:59,970 Notice that the 2-tuple that names v 276 00:13:59,970 --> 00:14:02,092 would be 7 comma minus 1. 277 00:14:02,092 --> 00:14:04,410 See, 7 is the coefficient of alpha. 278 00:14:04,410 --> 00:14:06,780 Minus 1 is the coefficient of beta. 279 00:14:06,780 --> 00:14:10,530 7 comma minus 1 would be the 2-tuple 280 00:14:10,530 --> 00:14:13,920 that represents v if I use alpha and beta 281 00:14:13,920 --> 00:14:15,870 as my representative vectors. 282 00:14:15,870 --> 00:14:18,720 Whereas the 2-tuple 4 comma 5 represents 283 00:14:18,720 --> 00:14:21,840 v with respect to the, in quotation marks, 284 00:14:21,840 --> 00:14:25,980 the "usual" representation in terms of i and j. 285 00:14:25,980 --> 00:14:27,760 Notice that v is the same vector. 286 00:14:27,760 --> 00:14:28,980 That hasn't changed. 287 00:14:28,980 --> 00:14:32,265 But, certainly, the 2-tuple 7 comma minus 1 288 00:14:32,265 --> 00:14:36,300 and the 2-tuple 4 comma 5 do not look alike. 289 00:14:36,300 --> 00:14:38,460 In other words, there is a big danger-- 290 00:14:38,460 --> 00:14:40,590 as, for one reason or another, we 291 00:14:40,590 --> 00:14:43,020 switch from different representative vectors 292 00:14:43,020 --> 00:14:46,650 to other, often a given set of representative vectors 293 00:14:46,650 --> 00:14:48,600 to another set of representative vectors-- 294 00:14:48,600 --> 00:14:52,710 that we can become confused as to which 2-tuple-- 295 00:14:52,710 --> 00:14:55,230 in this case, 2-tuple would be n-tuple in general-- 296 00:14:55,230 --> 00:15:01,040 but which representation goes with which 2-tuple. 297 00:15:01,040 --> 00:15:04,620 See, the whole idea is that our new definition, structurally, 298 00:15:04,620 --> 00:15:08,650 never pays any attention to what the particular representation 299 00:15:08,650 --> 00:15:09,150 is. 300 00:15:09,150 --> 00:15:11,790 And from the mathematician's point of view, 301 00:15:11,790 --> 00:15:15,060 what we feel is that what should happen in the study of vector 302 00:15:15,060 --> 00:15:18,690 spaces should be consequences of the vectors, not 303 00:15:18,690 --> 00:15:20,280 the particular coordinate system. 304 00:15:20,280 --> 00:15:22,770 And if that phrase confuses you, I'm 305 00:15:22,770 --> 00:15:25,770 using the coordinate system in the following sense. 306 00:15:25,770 --> 00:15:28,770 When I talk about i and j, I think about i and j 307 00:15:28,770 --> 00:15:30,630 as being my coordinate system. 308 00:15:30,630 --> 00:15:34,080 When I talk about alpha and beta, I think of alpha and beta 309 00:15:34,080 --> 00:15:35,880 as being my coordinate system. 310 00:15:35,880 --> 00:15:40,200 For example, given the vector v, let's suppose this is alpha 311 00:15:40,200 --> 00:15:41,460 and this is beta. 312 00:15:41,460 --> 00:15:44,550 Notice that v certainly is a linear combination 313 00:15:44,550 --> 00:15:45,570 of alpha and beta. 314 00:15:45,570 --> 00:15:50,260 In other words, it's this vector plus this vector. 315 00:15:50,260 --> 00:15:54,420 It's also a linear combination of i and j, certainly 316 00:15:54,420 --> 00:15:57,360 a different linear combination of alpha and beta 317 00:15:57,360 --> 00:16:00,180 than it is a linear combination of i and j. 318 00:16:00,180 --> 00:16:03,270 But also certain is the fact that the representation 319 00:16:03,270 --> 00:16:07,770 of v in terms of alpha and beta as our coordinate system 320 00:16:07,770 --> 00:16:10,320 is unique, just as the representation 321 00:16:10,320 --> 00:16:13,590 of v in terms of an i and j coordinate system is unique. 322 00:16:13,590 --> 00:16:16,890 In other words, v has a unique representation 323 00:16:16,890 --> 00:16:21,120 as some scalar multiple of alpha plus a scalar multiple of beta. 324 00:16:21,120 --> 00:16:23,550 But it also has a unique representation 325 00:16:23,550 --> 00:16:28,080 in the form a scalar multiple of i plus a scalar multiple of j. 326 00:16:28,080 --> 00:16:30,930 You see, we do not mean that the representation 327 00:16:30,930 --> 00:16:34,980 is unique in the sense they can only be written as one 328 00:16:34,980 --> 00:16:36,330 set of linear combinations. 329 00:16:36,330 --> 00:16:39,800 What we mean is that, for each coordinate system, 330 00:16:39,800 --> 00:16:41,790 for each coordinate system, there 331 00:16:41,790 --> 00:16:45,450 is a unique representation as a linear combination 332 00:16:45,450 --> 00:16:48,960 of the coordinates, but different coordinate systems 333 00:16:48,960 --> 00:16:51,390 lead to different linear combinations. 334 00:16:51,390 --> 00:16:54,030 But the key point from the mathematician's point of view 335 00:16:54,030 --> 00:16:57,720 is that v exists independently of the choices of alpha 336 00:16:57,720 --> 00:16:58,700 and beta. 337 00:16:58,700 --> 00:17:00,720 You see, again, we ran into this problem 338 00:17:00,720 --> 00:17:03,420 before when we talked about the distance between two points 339 00:17:03,420 --> 00:17:04,319 in a plane. 340 00:17:04,319 --> 00:17:07,470 Certainly, the distance between two points in a plane 341 00:17:07,470 --> 00:17:10,560 was independent of what coordinate system we used. 342 00:17:10,560 --> 00:17:12,480 But with respect to polar coordinates, 343 00:17:12,480 --> 00:17:15,900 the recipe for finding the distance between the two points 344 00:17:15,900 --> 00:17:17,880 did not look the same as the recipe 345 00:17:17,880 --> 00:17:20,880 that one used for finding the distance between the same two 346 00:17:20,880 --> 00:17:22,710 points when the points were expressed 347 00:17:22,710 --> 00:17:24,390 in Cartesian coordinates. 348 00:17:24,390 --> 00:17:26,069 The same thing we're saying here. 349 00:17:26,069 --> 00:17:30,300 We would like our mathematics to depend only on the vectors, 350 00:17:30,300 --> 00:17:33,090 not on their representation. 351 00:17:33,090 --> 00:17:36,990 Well, enough said about that part of the theory. 352 00:17:36,990 --> 00:17:39,000 What we'd next-- again, the exercises 353 00:17:39,000 --> 00:17:41,850 will hopefully fill in any gaps that 354 00:17:41,850 --> 00:17:44,350 may be missing in the logical structure 355 00:17:44,350 --> 00:17:45,660 from your point of view. 356 00:17:45,660 --> 00:17:47,760 But the next thing we'd like to talk about 357 00:17:47,760 --> 00:17:51,000 is a thing called a substructure, 358 00:17:51,000 --> 00:17:54,690 that whenever one talks about a new type of structure, 359 00:17:54,690 --> 00:17:59,820 the concept of what we used to call a subset becomes too weak, 360 00:17:59,820 --> 00:18:01,650 and what one wants to talk about, 361 00:18:01,650 --> 00:18:04,410 then, is the concept of a substructure. 362 00:18:04,410 --> 00:18:07,380 What I mean by that is the next topic I want to talk about 363 00:18:07,380 --> 00:18:08,970 is called subspaces. 364 00:18:08,970 --> 00:18:11,560 And to give you an idea of what I'm talking about, 365 00:18:11,560 --> 00:18:13,540 let's do the following. 366 00:18:13,540 --> 00:18:15,920 Let's go back into the plane and take just 367 00:18:15,920 --> 00:18:18,120 the two individual vectors i and j. 368 00:18:18,120 --> 00:18:20,770 Not the whole plane, just the two vectors-- 369 00:18:20,770 --> 00:18:23,620 the unit vector in the direction of the positive x-axis, 370 00:18:23,620 --> 00:18:26,490 the unit vector in the direction of the positive y-axis. 371 00:18:26,490 --> 00:18:28,090 Just these two vectors. 372 00:18:28,090 --> 00:18:30,720 Now, certainly, the set consisting of just these two 373 00:18:30,720 --> 00:18:36,370 vectors is a subset of two-dimensional vector space. 374 00:18:36,370 --> 00:18:40,020 In other words, the set of all vectors in the xy plane 375 00:18:40,020 --> 00:18:43,570 certainly includes the i vector and the j vector. 376 00:18:43,570 --> 00:18:45,960 So it would certainly be fair to say 377 00:18:45,960 --> 00:18:51,980 that this set A is a subset of two-dimensional vector spaces. 378 00:18:51,980 --> 00:18:55,310 But the idea is that a vector space has a structure. 379 00:18:55,310 --> 00:19:00,110 We care much more about how you combine vectors 380 00:19:00,110 --> 00:19:02,150 and what you get than we worry about 381 00:19:02,150 --> 00:19:03,420 whether you just have a set. 382 00:19:03,420 --> 00:19:05,600 Remember, mathematically, a structure is what? 383 00:19:05,600 --> 00:19:08,400 A set together with certain rules. 384 00:19:08,400 --> 00:19:11,360 And so what we're saying here is look what goes wrong with this. 385 00:19:11,360 --> 00:19:14,090 If we were looking at this thing structurally, 386 00:19:14,090 --> 00:19:17,840 one would say something like, gee, let's add i and j. 387 00:19:17,840 --> 00:19:21,380 If we add i and j, we get i plus j. 388 00:19:21,380 --> 00:19:25,070 i plus j is neither the vector i nor the vector j. 389 00:19:25,070 --> 00:19:31,550 In other words, in terms of a picture, here's i, here's j. 390 00:19:31,550 --> 00:19:33,920 What is i plus j? 391 00:19:33,920 --> 00:19:36,245 i plus j is just the sum of these two vectors. 392 00:19:40,130 --> 00:19:43,730 What we're saying is i and j belong to the set A, 393 00:19:43,730 --> 00:19:46,940 but i plus j certainly is neither i nor j. 394 00:19:46,940 --> 00:19:49,310 Since they consist solely of i and j, 395 00:19:49,310 --> 00:19:52,160 the fact that i plus j is neither i nor j 396 00:19:52,160 --> 00:19:54,890 means that i plus j does not belong 397 00:19:54,890 --> 00:19:58,940 to A. And, similarly, neither does, for example, twice i. 398 00:19:58,940 --> 00:20:01,280 If I double i, I get the vector which 399 00:20:01,280 --> 00:20:05,240 has the same direction and sense as i but it's twice as long. 400 00:20:05,240 --> 00:20:08,330 Consequently, that vector is neither i nor j. 401 00:20:08,330 --> 00:20:13,100 So, in particular, given the set whose only two elements are 402 00:20:13,100 --> 00:20:16,790 i and j, notice that if I add two vectors in the set A, 403 00:20:16,790 --> 00:20:19,400 I need not get a vector in the set A. 404 00:20:19,400 --> 00:20:23,210 And if I take a scalar multiple of any vector in the set A, 405 00:20:23,210 --> 00:20:25,970 I need not get a vector in the set A. 406 00:20:25,970 --> 00:20:29,520 And so, structurally, this is a bad thing, because notice 407 00:20:29,520 --> 00:20:31,970 that, structurally, we will be adding vectors. 408 00:20:31,970 --> 00:20:34,310 Consequently, if we start with a set 409 00:20:34,310 --> 00:20:38,120 and add two members in that set, and the resulting vector 410 00:20:38,120 --> 00:20:40,610 can then be a vector which isn't in that set, 411 00:20:40,610 --> 00:20:43,970 we don't have a very convenient structure. 412 00:20:43,970 --> 00:20:47,960 So what we do is we isolate a very key point. 413 00:20:47,960 --> 00:20:50,510 And let me give you the definition abstractly first, 414 00:20:50,510 --> 00:20:52,970 and then show you in terms of familiar examples 415 00:20:52,970 --> 00:20:55,500 that we already knew this. 416 00:20:55,500 --> 00:20:58,250 Let's suppose that we have a vector space V 417 00:20:58,250 --> 00:21:01,850 and that W is simply a subset of V. 418 00:21:01,850 --> 00:21:03,980 So this just means subset so far. 419 00:21:03,980 --> 00:21:08,000 Then W is called a subspace of V provided 420 00:21:08,000 --> 00:21:13,100 that, one, the sum of any two elements in the subset W 421 00:21:13,100 --> 00:21:16,610 is also an element in W. In other words, that the subset 422 00:21:16,610 --> 00:21:21,140 W is close with respect to the addition operation that 423 00:21:21,140 --> 00:21:23,270 makes V a vector space. 424 00:21:23,270 --> 00:21:27,560 And, secondly, that, for any element in the subset W, 425 00:21:27,560 --> 00:21:29,630 any scalar multiple of that element 426 00:21:29,630 --> 00:21:35,360 is also in W. In other words, W must also be close with respect 427 00:21:35,360 --> 00:21:38,300 to that same scalar multiplication with respect 428 00:21:38,300 --> 00:21:40,930 to which V is defined. 429 00:21:40,930 --> 00:21:44,510 And, by the way, I think if you think about that for a while, 430 00:21:44,510 --> 00:21:46,580 you will notice that this is true. 431 00:21:46,580 --> 00:21:49,730 For example, we have often talked about a line 432 00:21:49,730 --> 00:21:52,165 being a subset of a plane. 433 00:21:52,165 --> 00:21:53,540 But in terms of vectors, we could 434 00:21:53,540 --> 00:21:55,730 have made a stronger statement, namely 435 00:21:55,730 --> 00:21:59,860 a line is a subspace of the plane. 436 00:21:59,860 --> 00:22:02,180 And, in a similar way, a plane is 437 00:22:02,180 --> 00:22:05,090 a subspace of three-dimensional space, 438 00:22:05,090 --> 00:22:08,570 namely, starting with a plane, if you take two vectors which 439 00:22:08,570 --> 00:22:10,700 lie in the plane and you add them, 440 00:22:10,700 --> 00:22:13,555 the sum of those two vectors is still in the plane. 441 00:22:13,555 --> 00:22:14,930 And if you take a scalar multiple 442 00:22:14,930 --> 00:22:17,330 of any vector in a plane, the resulting vector 443 00:22:17,330 --> 00:22:19,160 still lies in that plane. 444 00:22:19,160 --> 00:22:21,980 So by means of a few examples, then, 445 00:22:21,980 --> 00:22:25,640 if V now is usual three-dimensional Euclidean 446 00:22:25,640 --> 00:22:30,290 space, if I let W be the set of all linear combinations of i 447 00:22:30,290 --> 00:22:31,280 and j-- 448 00:22:31,280 --> 00:22:34,520 and, by the way, contrast that with the set W-- 449 00:22:34,520 --> 00:22:35,780 with the set A over here. 450 00:22:35,780 --> 00:22:38,960 Notice that the set A over here consisted only 451 00:22:38,960 --> 00:22:41,120 of the vectors i and j. 452 00:22:41,120 --> 00:22:46,280 The set W here consists of all linear combinations of i and j. 453 00:22:46,280 --> 00:22:50,180 In other words, it's all vectors of the form c1 i plus c2 454 00:22:50,180 --> 00:22:53,460 j, where c1 and c2 are real numbers. 455 00:22:53,460 --> 00:22:56,930 And notice that W is the entire xy plane. 456 00:22:56,930 --> 00:22:59,570 And, after all, any vector in the xy plane 457 00:22:59,570 --> 00:23:03,200 can be written as a linear combination of i and j. 458 00:23:03,200 --> 00:23:04,550 Why is this a subspace? 459 00:23:04,550 --> 00:23:06,860 Well, leaving the details to you, 460 00:23:06,860 --> 00:23:08,810 notice, first of all, that it's clear 461 00:23:08,810 --> 00:23:10,850 that this should certainly be a subset, 462 00:23:10,850 --> 00:23:13,340 namely every vector in here is a vector 463 00:23:13,340 --> 00:23:15,080 that belongs to three-dimensional space 464 00:23:15,080 --> 00:23:16,550 even though it lies in a plane. 465 00:23:16,550 --> 00:23:19,340 But notice that the sum of any two linear combinations of i 466 00:23:19,340 --> 00:23:23,130 and j is again a linear combination of i and j. 467 00:23:23,130 --> 00:23:27,350 And any scalar multiple of a linear combination of i and j 468 00:23:27,350 --> 00:23:30,020 is again a linear combination of i and j. 469 00:23:30,020 --> 00:23:34,700 So W satisfies the criteria for being a subspace. 470 00:23:34,700 --> 00:23:37,340 In other words, W is not only a subspace in this case. 471 00:23:37,340 --> 00:23:40,040 W is the xy plane. 472 00:23:40,040 --> 00:23:45,500 By the way, using the alpha and beta of the previous example, 473 00:23:45,500 --> 00:23:49,100 notice that if I let V again be three-dimensional space 474 00:23:49,100 --> 00:23:53,780 and pick alpha to be i plus j, and beta to be 3i plus 2j, 475 00:23:53,780 --> 00:23:56,510 notice that if I take the set of all linear combinations 476 00:23:56,510 --> 00:24:01,010 of alpha and beta, then W is also a plane, 477 00:24:01,010 --> 00:24:04,340 and it's called the plane spanned by or determined 478 00:24:04,340 --> 00:24:05,750 by alpha and beta. 479 00:24:05,750 --> 00:24:09,530 In other words, it's the plane that has alpha and beta 480 00:24:09,530 --> 00:24:12,980 as consecutive edges of a parallelogram. 481 00:24:12,980 --> 00:24:15,450 By the way, let me make two comments here. 482 00:24:15,450 --> 00:24:19,250 First of all, if these d's look funny to you, they should, 483 00:24:19,250 --> 00:24:22,820 because just before the lecture began, these were c's. 484 00:24:22,820 --> 00:24:27,230 And I changed these c's to d's because the mathematician would 485 00:24:27,230 --> 00:24:32,060 have no danger in confusing the coefficients here 486 00:24:32,060 --> 00:24:35,360 with the coefficients here, but for the uninitiated I think 487 00:24:35,360 --> 00:24:37,730 it's very important to make this observation. 488 00:24:37,730 --> 00:24:39,920 Notice that alpha and beta, being 489 00:24:39,920 --> 00:24:44,720 linear combinations of i and j, both lie in the xy plane. 490 00:24:44,720 --> 00:24:49,220 Consequently, the plane determined by alpha and beta 491 00:24:49,220 --> 00:24:51,740 is also the xy plane. 492 00:24:51,740 --> 00:24:56,690 This, in turn, means that, in both of these two examples, 493 00:24:56,690 --> 00:24:59,300 W is the xy plane. 494 00:24:59,300 --> 00:25:01,860 But, somehow or other, you wouldn't 495 00:25:01,860 --> 00:25:05,160 expect that W would be the same linear combination of alpha 496 00:25:05,160 --> 00:25:08,820 and beta as it is of i and j. 497 00:25:08,820 --> 00:25:11,940 And, therefore, I prefer to use different letters here 498 00:25:11,940 --> 00:25:15,210 to indicate that a given vector in V 499 00:25:15,210 --> 00:25:18,210 might require two numbers c1 and c2 500 00:25:18,210 --> 00:25:21,030 to be coefficients of i and j, but, with respect 501 00:25:21,030 --> 00:25:23,380 to alpha and beta, two different numbers. 502 00:25:23,380 --> 00:25:24,550 OK? 503 00:25:24,550 --> 00:25:26,550 That's one point I wanted to make about this. 504 00:25:26,550 --> 00:25:29,470 The other point I wanted to make about this, you see, 505 00:25:29,470 --> 00:25:32,380 notice that alpha and beta don't look like i and j, 506 00:25:32,380 --> 00:25:33,750 and there's a danger-- 507 00:25:33,750 --> 00:25:36,630 that you may not even realize when you see alpha and beta-- 508 00:25:36,630 --> 00:25:41,190 that alpha and beta determine the same plane as i and j. 509 00:25:41,190 --> 00:25:43,830 Now, you see, that leads to the problem 510 00:25:43,830 --> 00:25:45,720 that we were talking about before, 511 00:25:45,720 --> 00:25:50,100 namely, given the same plane, the representation of the plane 512 00:25:50,100 --> 00:25:54,620 looks different if you use one set of representative vectors 513 00:25:54,620 --> 00:25:57,510 than if you use another set of representative vectors. 514 00:25:57,510 --> 00:25:59,940 But, more importantly, let me point out 515 00:25:59,940 --> 00:26:02,580 that there was nothing sacred about the xy plane here. 516 00:26:02,580 --> 00:26:06,040 Just to change this problem ever so slightly, 517 00:26:06,040 --> 00:26:10,440 let me add on a plus k, say, to both alpha and beta. 518 00:26:10,440 --> 00:26:12,600 That makes this a new example, you see. 519 00:26:12,600 --> 00:26:15,180 Let alpha be i plus j plus k. 520 00:26:15,180 --> 00:26:18,390 Let beta be 3i plus 2j plus k. 521 00:26:18,390 --> 00:26:21,660 Notice that alpha and beta are-- 522 00:26:21,660 --> 00:26:23,730 neither is in the xy plane. 523 00:26:23,730 --> 00:26:26,880 They are not parallel to one another, because this is not 524 00:26:26,880 --> 00:26:29,400 a constant multiple of this. 525 00:26:29,400 --> 00:26:35,010 Therefore, notice that alpha and beta are still two vectors. 526 00:26:35,010 --> 00:26:38,970 W is still the set of all linear combinations of these two 527 00:26:38,970 --> 00:26:39,970 vectors. 528 00:26:39,970 --> 00:26:42,420 See, this part doesn't change. 529 00:26:42,420 --> 00:26:46,800 And W now is still the plane spanned by alpha and beta, 530 00:26:46,800 --> 00:26:50,220 but now notice that the plane spanned by alpha and beta 531 00:26:50,220 --> 00:26:54,260 is the plane which has alpha and beta as consecutive edges, 532 00:26:54,260 --> 00:26:57,600 the parallelogram determined by the plane that has alpha 533 00:26:57,600 --> 00:26:59,670 and beta as consecutive edges. 534 00:26:59,670 --> 00:27:05,160 And this parallelogram clearly is no longer in the xy plane. 535 00:27:05,160 --> 00:27:05,910 All right. 536 00:27:05,910 --> 00:27:10,890 And now, for a finale for today, what I'd like to do now 537 00:27:10,890 --> 00:27:13,530 is to point out, going back ever so briefly 538 00:27:13,530 --> 00:27:17,010 to our first key point, that there are vector spaces which 539 00:27:17,010 --> 00:27:20,100 aren't viewed as n-tuples, namely, for example, 540 00:27:20,100 --> 00:27:23,730 the set of all functions defined on the closed interval 541 00:27:23,730 --> 00:27:24,780 from a to b. 542 00:27:24,780 --> 00:27:27,630 Let me show you what subspaces mean in terms 543 00:27:27,630 --> 00:27:30,300 of this interpretation. 544 00:27:30,300 --> 00:27:34,270 My next example, again let V be, as earlier in the lecture, 545 00:27:34,270 --> 00:27:38,400 the set of all functions having a common domain, 546 00:27:38,400 --> 00:27:41,070 say the closed interval from a to b. 547 00:27:41,070 --> 00:27:45,570 Let's look now at the set W1 of all functions which 548 00:27:45,570 --> 00:27:48,540 not only are defined on the closed interval from a to b, 549 00:27:48,540 --> 00:27:51,210 but they happen to be continuous on the closed 550 00:27:51,210 --> 00:27:53,070 interval from a to b. 551 00:27:53,070 --> 00:27:55,020 You see, notice that, originally, I never had 552 00:27:55,020 --> 00:27:56,790 to assume continuity here. 553 00:27:56,790 --> 00:27:58,590 As long as the function is defined 554 00:27:58,590 --> 00:28:00,450 on the closed interval from a to b, 555 00:28:00,450 --> 00:28:03,180 I can add any two members of this family. 556 00:28:03,180 --> 00:28:06,275 I can scalar multiply any member of this family. 557 00:28:06,275 --> 00:28:11,040 So I do not need continuity to have this be a vector space. 558 00:28:11,040 --> 00:28:13,650 However, suppose I now look at all functions which 559 00:28:13,650 --> 00:28:16,100 are continuous on the closed interval from a to b. 560 00:28:16,100 --> 00:28:18,090 Well, in particular, it should be 561 00:28:18,090 --> 00:28:24,330 trivial to see that this set W sub 1 is a subset of V, namely 562 00:28:24,330 --> 00:28:26,950 every function which is continuous on the closed 563 00:28:26,950 --> 00:28:30,480 interval from a to b, in particular, must be defined 564 00:28:30,480 --> 00:28:32,400 on the closed interval from a to b. 565 00:28:32,400 --> 00:28:35,790 Consequently, every function in this set 566 00:28:35,790 --> 00:28:38,490 certainly belongs to this set. 567 00:28:38,490 --> 00:28:41,010 On the other hand, notice that we already 568 00:28:41,010 --> 00:28:43,770 know that the sum of continuous functions 569 00:28:43,770 --> 00:28:45,720 is again a continuous function. 570 00:28:45,720 --> 00:28:47,850 A scalar multiple of a continuous function 571 00:28:47,850 --> 00:28:50,070 is again a continuous function. 572 00:28:50,070 --> 00:28:54,120 Consequently, according to our definition of subspace, 573 00:28:54,120 --> 00:28:59,640 W1 is not only a subset of V. It's a subspace of V 574 00:28:59,640 --> 00:29:02,250 because it's closed with respect to-- 575 00:29:02,250 --> 00:29:05,940 excuse me-- addition and scalar multiplication. 576 00:29:05,940 --> 00:29:09,060 As another example, let V be the same 577 00:29:09,060 --> 00:29:10,620 as it was in the previous example, 578 00:29:10,620 --> 00:29:13,890 namely the set of all functions defined on the closed 579 00:29:13,890 --> 00:29:15,420 interval from a to b. 580 00:29:15,420 --> 00:29:18,120 And now let's take that subset consisting 581 00:29:18,120 --> 00:29:21,270 of all functions which are differentiable on the closed 582 00:29:21,270 --> 00:29:23,370 interval from a to b. 583 00:29:23,370 --> 00:29:26,100 In other words, this is an even stronger condition 584 00:29:26,100 --> 00:29:27,388 than being continuous. 585 00:29:27,388 --> 00:29:29,430 We say not only must the functions be continuous, 586 00:29:29,430 --> 00:29:32,580 but now we want it to be differentiable as well. 587 00:29:32,580 --> 00:29:35,520 Remember, differentiability implies continuity, 588 00:29:35,520 --> 00:29:39,600 but continuity does not imply differentiability. 589 00:29:39,600 --> 00:29:43,020 All I'm saying now is that, since the sum of two 590 00:29:43,020 --> 00:29:44,940 differentiable functions is again 591 00:29:44,940 --> 00:29:49,050 a differentiable function, and since a scalar 592 00:29:49,050 --> 00:29:51,900 multiple, a constant times a differentiable function, 593 00:29:51,900 --> 00:29:56,940 is still differentiable, not only is W2 a subset of V. 594 00:29:56,940 --> 00:30:04,210 It's a subspace of V. In fact, W2 is also a subspace of W1. 595 00:30:04,210 --> 00:30:06,390 Recall, W1 was what? 596 00:30:06,390 --> 00:30:09,600 The set of continuous functions on the interval from a to b. 597 00:30:09,600 --> 00:30:11,950 That was also a vector space. 598 00:30:11,950 --> 00:30:13,380 And now what we're saying is that 599 00:30:13,380 --> 00:30:16,890 the differentiable functions are a subspace 600 00:30:16,890 --> 00:30:18,540 of the continuous functions. 601 00:30:18,540 --> 00:30:21,910 Again, I'm going to emphasize all of this in the exercises, 602 00:30:21,910 --> 00:30:24,660 and we'll pick this up in more detail next time, 603 00:30:24,660 --> 00:30:28,410 but what I wanted to close with is simply the following remark. 604 00:30:28,410 --> 00:30:30,900 If you have studied calculus in the past 605 00:30:30,900 --> 00:30:33,090 and have been away from it for a while, 606 00:30:33,090 --> 00:30:36,060 and you pick up an ultra-modern calculus book, 607 00:30:36,060 --> 00:30:40,440 you may be surprised to find that even elementary calculus 608 00:30:40,440 --> 00:30:44,940 begins with a preface to emphasizing 609 00:30:44,940 --> 00:30:47,970 what's called linear algebra or vector spaces. 610 00:30:47,970 --> 00:30:51,450 And the reason for this is the fact that all of calculus, 611 00:30:51,450 --> 00:30:53,400 going back to part 1 of our course, 612 00:30:53,400 --> 00:30:56,560 does deal with continuous and differentiable functions. 613 00:30:56,560 --> 00:30:59,490 And, in particular, continuous and differentiable functions, 614 00:30:59,490 --> 00:31:03,210 as we've just shown, obey the structure of a vector space. 615 00:31:03,210 --> 00:31:06,540 And, consequently, that's why many modern authors 616 00:31:06,540 --> 00:31:09,870 prefer to unify the approach to calculus 617 00:31:09,870 --> 00:31:12,810 and introduce vector spaces right at the very outset. 618 00:31:12,810 --> 00:31:15,750 But we'll talk about this more gradually as we go along. 619 00:31:15,750 --> 00:31:17,760 And until next time, then, goodbye. 620 00:31:21,690 --> 00:31:24,090 Funding for the publication of this video 621 00:31:24,090 --> 00:31:28,950 was provided by the Gabriella and Paul Rosenbaum Foundation. 622 00:31:28,950 --> 00:31:33,120 Help OCW continue to provide free and open access to MIT 623 00:31:33,120 --> 00:31:38,645 courses by making a donation at ocw.mit.edu/donate.