WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.560 continue to offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.210 --> 00:00:19.650 at ocw.mit.edu. 00:00:53.118 --> 00:00:54.620 HERBERT GROSS: Hi. 00:00:54.620 --> 00:00:58.200 We've sort of arrive at D-day in our course; 00:00:58.200 --> 00:01:01.530 that in a manner of speaking, everything that we've done up 00:01:01.530 --> 00:01:04.140 to now has been a rehearsal. 00:01:04.140 --> 00:01:07.050 That today we are going to come to grips 00:01:07.050 --> 00:01:09.870 with what the course is all about: namely, 00:01:09.870 --> 00:01:14.800 a real-valued function of several real variables. 00:01:14.800 --> 00:01:17.360 And before getting into that, what I'd like to do 00:01:17.360 --> 00:01:20.830 is to review very briefly what we've talked about 00:01:20.830 --> 00:01:23.500 so far in terms of functions, since we've 00:01:23.500 --> 00:01:25.860 introduced the vector notation. 00:01:25.860 --> 00:01:27.730 Namely, what we've mentioned is, is 00:01:27.730 --> 00:01:29.230 that our function machine could have 00:01:29.230 --> 00:01:32.640 either a vector or a scalar as its input, 00:01:32.640 --> 00:01:36.190 and either a vector or a scalar as its output. 00:01:36.190 --> 00:01:39.210 Part one of our course centered around the idea 00:01:39.210 --> 00:01:42.880 where both the input and the outputs were scalars. 00:01:42.880 --> 00:01:46.210 Whereas, the previous block of material 00:01:46.210 --> 00:01:49.670 concerned the case where our input was a scalar, 00:01:49.670 --> 00:01:52.630 and our output was a vector. 00:01:52.630 --> 00:01:55.070 Today what we're going to discuss 00:01:55.070 --> 00:02:00.910 is the situation that occurs when our input is a vector, 00:02:00.910 --> 00:02:04.240 and our output is a scalar. 00:02:04.240 --> 00:02:09.180 And I call today's lecture "n-dimensional Vector Spaces." 00:02:09.180 --> 00:02:13.250 Eventually, we'll discuss vector spaces in more detail. 00:02:13.250 --> 00:02:18.110 For the time being, I simply want to set the mood, 00:02:18.110 --> 00:02:20.460 and hopefully by the time I'm through, 00:02:20.460 --> 00:02:24.190 show you a rather peaceful coexistence between the worlds 00:02:24.190 --> 00:02:26.390 of the new mathematics, and the worlds 00:02:26.390 --> 00:02:28.090 of traditional mathematics. 00:02:28.090 --> 00:02:30.220 In fact, in many of our topics that we're 00:02:30.220 --> 00:02:32.690 going to tackle in this block of material, 00:02:32.690 --> 00:02:36.150 we will give both points of view. 00:02:36.150 --> 00:02:38.910 But to set the stage properly-- to get 00:02:38.910 --> 00:02:40.730 into the idea of what a vector space is all 00:02:40.730 --> 00:02:43.240 about-- and if that word frightens you, just 00:02:43.240 --> 00:02:45.460 don't worry about for a minute or two. 00:02:45.460 --> 00:02:48.450 Worry about it after that, but let's just get started. 00:02:48.450 --> 00:02:50.490 Let's consider the situation where 00:02:50.490 --> 00:02:54.660 I have a function machine where my input is a vector 00:02:54.660 --> 00:02:57.920 and my output is a scalar. 00:02:57.920 --> 00:03:01.840 As an example, let me just generically let v represent 00:03:01.840 --> 00:03:05.970 the vector whose form is x*i plus y*j, 00:03:05.970 --> 00:03:08.580 using i and j components. 00:03:08.580 --> 00:03:16.140 Let me define f of v to be pi times the square of the i 00:03:16.140 --> 00:03:18.200 component times the j component. 00:03:18.200 --> 00:03:20.210 And let's not worry right now about why I 00:03:20.210 --> 00:03:22.360 picked this particular recipe. 00:03:22.360 --> 00:03:26.640 Let's simply observe that once this recipe is chosen, 00:03:26.640 --> 00:03:33.040 f is a function which maps a two-dimensional vector here, 00:03:33.040 --> 00:03:37.880 in i and j components, into a number of pi x squared y. 00:03:37.880 --> 00:03:40.530 Just to illustrate this recipe, notice 00:03:40.530 --> 00:03:43.880 that if our vector were 3i plus 4j, 00:03:43.880 --> 00:03:47.240 the output of the f machine-- if this were the input-- 00:03:47.240 --> 00:03:48.150 would be what? 00:03:48.150 --> 00:03:53.010 Pi times the square of the first component-- the component of i, 00:03:53.010 --> 00:03:57.820 that's 3 squared-- times the component of j, which is 4. 00:03:57.820 --> 00:04:00.790 And that leads to 36*pi. 00:04:00.790 --> 00:04:03.650 By the way, observe that order does make a difference, 00:04:03.650 --> 00:04:07.040 namely if I reverse the roles of the coefficients, 00:04:07.040 --> 00:04:12.610 and feed the vector 4i plus 3j into my f machine, 00:04:12.610 --> 00:04:16.200 the output would be pi times 4 squared-- 00:04:16.200 --> 00:04:19.459 the square of the i component-- times 3, 00:04:19.459 --> 00:04:22.290 and that would be 48*pi. 00:04:22.290 --> 00:04:26.240 Notice also that we have allowed, already in our course, 00:04:26.240 --> 00:04:32.540 the abbreviation that a comma b would represent a*i plus b*j, 00:04:32.540 --> 00:04:36.860 or a comma b comma c would represent a*i plus b*j plus 00:04:36.860 --> 00:04:37.730 c*k. 00:04:37.730 --> 00:04:40.370 The idea is that if we now apply this shorthand 00:04:40.370 --> 00:04:45.100 notation to these two vectors, what we could say is what? 00:04:45.100 --> 00:04:50.680 f of this vector, in other words, f of 3 comma 4 is 36*pi, 00:04:50.680 --> 00:04:54.305 whereas f of 4 comma 3 is 48*pi. 00:04:54.305 --> 00:04:56.470 And that's exactly what we mean when 00:04:56.470 --> 00:04:59.900 we say that we may treat a two-dimensional vector 00:04:59.900 --> 00:05:01.850 as an ordered pair. 00:05:01.850 --> 00:05:05.150 You see, it's, not only a pair but the order 00:05:05.150 --> 00:05:06.720 does make a difference. 00:05:06.720 --> 00:05:09.310 Both in what the vector is and what 00:05:09.310 --> 00:05:12.280 the output of the f machine is. 00:05:12.280 --> 00:05:13.210 OK? 00:05:13.210 --> 00:05:16.090 Hopefully, let's say, so far so good, 00:05:16.090 --> 00:05:21.900 and let's tackle now a rather completely different problem. 00:05:21.900 --> 00:05:24.440 What I'd like to do now is the following. 00:05:24.440 --> 00:05:28.300 Let's consider the cylinder-- the right circular cylinder-- 00:05:28.300 --> 00:05:33.530 the radius of whose base is x, and whose height is y. 00:05:33.530 --> 00:05:40.910 Notice that the volume of this cylinder is pi x squared y. 00:05:40.910 --> 00:05:46.080 And if I use the same f that we used previously-- 00:05:46.080 --> 00:05:48.380 in other words, the same f that we 00:05:48.380 --> 00:05:52.540 were talking about over here-- notice that another way-- 00:05:52.540 --> 00:05:54.410 see how is f defined? 00:05:54.410 --> 00:05:59.220 Given that the input was x comma y the output was pi x 00:05:59.220 --> 00:06:04.380 squared y, notice that the volume is f of x, y. 00:06:04.380 --> 00:06:08.150 In particular, if I want the volume of the cylinder, 00:06:08.150 --> 00:06:13.700 the radius of whose base is 3, and whose height is 4-- 00:06:13.700 --> 00:06:15.320 see x is 3 and y is 4-- 00:06:15.320 --> 00:06:19.040 what I really want is f of 3 comma 4. 00:06:19.040 --> 00:06:23.190 That's pi times 3 squared times 4. 00:06:23.190 --> 00:06:27.010 f of 3 comma 4 is 36*pi. 00:06:27.010 --> 00:06:28.730 Now I'd like to pause for a second 00:06:28.730 --> 00:06:32.280 again, and return to our earlier remark. 00:06:32.280 --> 00:06:39.400 Namely, if I look at this, and if I look at this. 00:06:45.500 --> 00:06:49.060 Notice that these two expressions are identical. 00:06:49.060 --> 00:06:52.370 I cannot tell the difference between whether I'm looking 00:06:52.370 --> 00:06:56.370 at the vector 3i plus 4j, or whether I'm looking 00:06:56.370 --> 00:06:59.920 at the cylinder, the radius of whose base is 3, 00:06:59.920 --> 00:07:03.500 and whose height is 4; whether I look at this equation, 00:07:03.500 --> 00:07:06.040 or whether I look at this equation. 00:07:06.040 --> 00:07:09.830 The difference is that in the first case somehow or other, 00:07:09.830 --> 00:07:12.190 it was quite natural to think of 3 comma 00:07:12.190 --> 00:07:17.000 4 as being either an ordered pair, or an arrow. 00:07:17.000 --> 00:07:19.090 In this case, however, my contention 00:07:19.090 --> 00:07:22.300 is that when we think of the radius 00:07:22.300 --> 00:07:24.620 of the base of the cylinder, and the height, 00:07:24.620 --> 00:07:27.660 we do not tend to think in terms of arrows, 00:07:27.660 --> 00:07:30.530 but rather in terms of ordered pairs. 00:07:30.530 --> 00:07:36.410 In other words, the ordered pair x comma y, in the expression 00:07:36.410 --> 00:07:40.990 f of x, y, need not be viewed as an arrow, but as 00:07:40.990 --> 00:07:42.770 an ordered pair. 00:07:42.770 --> 00:07:46.870 And an ordered pair is called a 2-tuple. 00:07:46.870 --> 00:07:49.320 This leads to a generalization that I 00:07:49.320 --> 00:07:51.540 think is rather important, and I think 00:07:51.540 --> 00:07:57.310 you will see in a moment, where the idea of this approach 00:07:57.310 --> 00:08:00.410 comes into functions of several real variables. 00:08:00.410 --> 00:08:05.520 The topic I have in mind is something called an n-tuple. 00:08:05.520 --> 00:08:09.130 And let me read into that rather gradually as follows. 00:08:09.130 --> 00:08:15.220 Without giving you a specific physical example-- meaning 00:08:15.220 --> 00:08:18.290 I'll give you an illustration, but leave the numerical amounts 00:08:18.290 --> 00:08:19.390 out. 00:08:19.390 --> 00:08:23.370 Quite possibly if I'm studying temperature in a room, 00:08:23.370 --> 00:08:25.790 the temperature will in general what? 00:08:25.790 --> 00:08:28.960 It will depend on what position I'm at in the room, 00:08:28.960 --> 00:08:33.820 and also at what time I measure the temperature. 00:08:33.820 --> 00:08:36.710 It's fair to assume that in many applications 00:08:36.710 --> 00:08:39.809 the temperature is some function of the four 00:08:39.809 --> 00:08:43.750 independent variables x, y, z, and t, 00:08:43.750 --> 00:08:47.400 where x, y, and z are the Cartesian coordinates 00:08:47.400 --> 00:08:52.020 of three-dimensional space, and t represents time. 00:08:52.020 --> 00:08:56.740 What I'm driving at is I can now visualize this in terms 00:08:56.740 --> 00:08:58.810 of my function machine again. 00:08:58.810 --> 00:09:03.420 Namely to compute T, I think of feeding what? 00:09:03.420 --> 00:09:08.180 Specific values into the machine for x, y, z, and t. 00:09:08.180 --> 00:09:13.440 The f machine then performs on x, y, z, and t 00:09:13.440 --> 00:09:18.200 as indicated by f to compute t. 00:09:18.200 --> 00:09:21.050 The input of my f machine in this case 00:09:21.050 --> 00:09:25.490 is what I'm going to call a 4-tuple for the time being. 00:09:25.490 --> 00:09:29.700 I need four values-- x, y, z, and t. 00:09:29.700 --> 00:09:31.620 Order does make a difference. 00:09:31.620 --> 00:09:36.060 For example, if I interchange the x- and the y-coordinate, 00:09:36.060 --> 00:09:39.200 those x and y, what I'm doing is I'm interchanging the x- 00:09:39.200 --> 00:09:41.560 and the y-coordinate of the point in space, 00:09:41.560 --> 00:09:44.800 and that in general is going to change the point in space. 00:09:44.800 --> 00:09:48.010 The point, however, is that in this particular f machine, 00:09:48.010 --> 00:09:51.480 notice that my output is a scalar. 00:09:51.480 --> 00:09:54.020 Namely the temperature is a number, 00:09:54.020 --> 00:09:57.680 but the input is a 4-tuple. 00:09:57.680 --> 00:10:00.910 x, y, z, and t. 00:10:00.910 --> 00:10:03.100 Now the trouble with using symbolism 00:10:03.100 --> 00:10:07.280 like x, y, z, and t, I guess-- without going 00:10:07.280 --> 00:10:09.940 into a long philosophic discussion-- 00:10:09.940 --> 00:10:11.410 among other things, as soon as you 00:10:11.410 --> 00:10:14.800 have 27 or more independent variables, 00:10:14.800 --> 00:10:17.070 you run out of letters of the alphabet. 00:10:17.070 --> 00:10:20.730 As a result, it is quite common for one 00:10:20.730 --> 00:10:23.430 to adopt a new notation. 00:10:23.430 --> 00:10:26.040 instead of saying let (x, y, z, t) 00:10:26.040 --> 00:10:30.410 be a 4-tuple, what one usually does is chooses 00:10:30.410 --> 00:10:34.065 one symbol-- say x-- and then uses subscripts. 00:10:34.065 --> 00:10:38.570 Namely, a general 4-tuple would have the form what? 00:10:38.570 --> 00:10:43.560 x_1 comma x_2 comma x_3 comma x_4, where x_1, 00:10:43.560 --> 00:10:46.150 x_2, x_3 and x_4 are numbers. 00:10:46.150 --> 00:10:48.940 An expression like this is called the 4-tuple. 00:10:48.940 --> 00:10:51.280 What is this a generalization of? 00:10:51.280 --> 00:10:55.010 The 4-tuple is a generalization of the one-dimensional, 00:10:55.010 --> 00:10:57.910 two-dimensional, and three-dimensional arrow, 00:10:57.910 --> 00:11:00.450 so to speak, where we could think of what? 00:11:00.450 --> 00:11:06.200 The vector x_1*i as just needing one number to specify it. 00:11:06.200 --> 00:11:11.930 The vector x_1*i plus x_2*j could've been used to do what? 00:11:11.930 --> 00:11:15.580 It could've been abbreviated by the 2-tuple (x_1, x_2). 00:11:15.580 --> 00:11:20.150 And the vector x_1*i plus x_2*j plus x_3*k could've been 00:11:20.150 --> 00:11:24.470 abbreviated by the 3-tuple (x_1, x_2, x_3). 00:11:24.470 --> 00:11:26.610 It is conventional in one-, two-, 00:11:26.610 --> 00:11:30.240 or three-dimensional space to use x, y, and z, 00:11:30.240 --> 00:11:32.680 instead of x_1, x_2, and x_3. 00:11:32.680 --> 00:11:34.390 But that's just a convention. 00:11:34.390 --> 00:11:37.141 I think that it's because we learnt it that way that we do 00:11:37.141 --> 00:11:37.640 it. 00:11:37.640 --> 00:11:39.900 In general, I think the subscript 00:11:39.900 --> 00:11:44.440 notation is much nicer, but in general the idea is what? 00:11:44.440 --> 00:11:50.240 Given an ordered array of n numbers, x_1 up to x_n, 00:11:50.240 --> 00:11:52.590 we call that an n-tuple. 00:11:52.590 --> 00:11:55.120 And my friend and colleague John Fitch 00:11:55.120 --> 00:11:56.780 mentioned to me that if n is odd, 00:11:56.780 --> 00:11:58.770 like one, three, five, or seven, then 00:11:58.770 --> 00:12:01.120 it's known as an odd-tuple. 00:12:01.120 --> 00:12:03.290 Which isn't a very funny story, that's 00:12:03.290 --> 00:12:06.020 why I told you John told me that particular story. 00:12:06.020 --> 00:12:10.200 But the whole idea is this is an n-tuple. 00:12:10.200 --> 00:12:12.500 And the whole idea again is what? 00:12:12.500 --> 00:12:15.110 That an n-tuple makes sense, even when 00:12:15.110 --> 00:12:17.310 n is greater than three. 00:12:17.310 --> 00:12:20.210 The whole name of the game of functions 00:12:20.210 --> 00:12:26.200 of several real variables-- in terms of modern mathematics, 00:12:26.200 --> 00:12:28.800 in terms of the language of n-tuples-- 00:12:28.800 --> 00:12:32.360 is that a real-valued function of several-- 00:12:32.360 --> 00:12:35.950 where by several you mean more than one-- real variables is 00:12:35.950 --> 00:12:39.360 simply a function in which the input is 00:12:39.360 --> 00:12:43.730 an n-tuple and the output is a number. 00:12:43.730 --> 00:12:44.570 OK? 00:12:44.570 --> 00:12:46.530 That's what this whole thing is all about. 00:12:46.530 --> 00:12:50.530 And because of that, when we then abbreviate the n-tuple, 00:12:50.530 --> 00:12:53.150 we use x with a bar under it. 00:12:53.150 --> 00:12:54.570 Let's call it x-bar. 00:12:54.570 --> 00:12:57.360 Rather than x with the arrow over it, 00:12:57.360 --> 00:13:00.070 since arrows may be inappropriate. 00:13:00.070 --> 00:13:01.770 Now what do I mean by inappropriate? 00:13:01.770 --> 00:13:04.180 Well I mean that even in the case of one, two, 00:13:04.180 --> 00:13:06.530 or three dimensions, you might be thinking 00:13:06.530 --> 00:13:10.670 of, say, the radius and the height of a cylinder, 00:13:10.670 --> 00:13:12.500 rather than as an arrow. 00:13:12.500 --> 00:13:14.520 And in more than three dimensions-- 00:13:14.520 --> 00:13:17.670 for most of us at least-- it's difficult to visualize 00:13:17.670 --> 00:13:19.820 what we would mean by an arrow. 00:13:19.820 --> 00:13:22.460 So we just use the bar underneath. 00:13:22.460 --> 00:13:25.520 Now again, the major point is, notice this-- 00:13:25.520 --> 00:13:27.160 I keep saying the major point. 00:13:27.160 --> 00:13:29.720 I guess there's a lot of major points about this. 00:13:29.720 --> 00:13:34.120 Remember that we did not call arrows "vectors". 00:13:34.120 --> 00:13:38.630 We did not call arrows "vectors" until we defined a structure 00:13:38.630 --> 00:13:39.290 on the arrows. 00:13:39.290 --> 00:13:40.250 Remember what we did? 00:13:40.250 --> 00:13:42.790 We told what it meant for two arrows to be equal, 00:13:42.790 --> 00:13:45.350 we told how we added two arrows, and we 00:13:45.350 --> 00:13:48.230 told how we multiplied an arrow by a scalar. 00:13:48.230 --> 00:13:53.650 In a similar way, we will not call n-tuples a structure 00:13:53.650 --> 00:13:57.970 until we tell how to equate a pair of n-tuples, 00:13:57.970 --> 00:14:00.270 how to add a pair, and how to multiply 00:14:00.270 --> 00:14:01.960 an n-tuple by a number. 00:14:01.960 --> 00:14:04.670 By the way, the structure that we wind up with 00:14:04.670 --> 00:14:08.490 is then called an n-dimensional vector space, 00:14:08.490 --> 00:14:12.270 or more concisely, n-space. 00:14:12.270 --> 00:14:14.480 And the idea works like this-- let's 00:14:14.480 --> 00:14:16.690 pick a particular value of n. 00:14:16.690 --> 00:14:18.270 Lets just call it n. 00:14:18.270 --> 00:14:23.200 And let S sub n be the set of all n-tuples x_1 00:14:23.200 --> 00:14:25.410 up to x_n, in other words, the set of what? 00:14:25.410 --> 00:14:29.850 All n-tuples of numbers x_1 up to x_n. 00:14:29.850 --> 00:14:31.730 Let's pick two particular members 00:14:31.730 --> 00:14:34.830 of S sub n, which we'll call a-bar and b-bar. 00:14:34.830 --> 00:14:37.090 Where a-bar is simply an abbreviation 00:14:37.090 --> 00:14:42.860 for the n-tuple a_1 up to a_n, where the a_1, a_2 up to a_n, 00:14:42.860 --> 00:14:44.640 et cetera, are real numbers. 00:14:44.640 --> 00:14:48.100 And b-bar is an abbreviation for the n-tuple b_1, 00:14:48.100 --> 00:14:52.580 et cetera, b_n, where b_1 up through b_n 00:14:52.580 --> 00:14:54.870 are also real numbers. 00:14:54.870 --> 00:14:58.120 Now again, here's where structure comes into play. 00:14:58.120 --> 00:15:01.810 We have already defined an n-tuple arithmetic 00:15:01.810 --> 00:15:04.370 in terms of arrows for the case when 00:15:04.370 --> 00:15:07.640 n is either one, two, or three. 00:15:07.640 --> 00:15:10.910 Based on what happens when n is one, two, or three, 00:15:10.910 --> 00:15:13.480 we invent the following definitions. 00:15:13.480 --> 00:15:17.570 First of all, we invent the definition 00:15:17.570 --> 00:15:22.580 that a-bar equals b-bar means that the components-- meaning 00:15:22.580 --> 00:15:23.080 what? 00:15:23.080 --> 00:15:26.230 The individual members of the n-tuple-- the components 00:15:26.230 --> 00:15:30.680 of a-bar are equal to the components of b-bar, component 00:15:30.680 --> 00:15:31.570 by component. 00:15:31.570 --> 00:15:34.730 In other words, a_1 is equal to b_1, 00:15:34.730 --> 00:15:36.710 a_2 is equal to b_2, et cetera. 00:15:36.710 --> 00:15:37.980 All the way up to what? 00:15:37.980 --> 00:15:40.117 a_n is equal to b_n. 00:15:40.117 --> 00:15:41.950 Now, in other words again, what we're saying 00:15:41.950 --> 00:15:44.310 is that for two n-tuples to be equal, 00:15:44.310 --> 00:15:48.420 by definition, they should be equal component by component, 00:15:48.420 --> 00:15:50.550 and this is motivated by the fact 00:15:50.550 --> 00:15:52.300 that we already know that we've accepted 00:15:52.300 --> 00:15:55.580 this structural definition for the case of arrows. 00:15:55.580 --> 00:16:01.950 Similarly, given two n-tuples a-bar and b-bar, to add them, 00:16:01.950 --> 00:16:06.150 let me define that to be the n-tuple that I get by adding 00:16:06.150 --> 00:16:08.140 component by component. 00:16:08.140 --> 00:16:11.480 In other words, to find the first component of a-bar plus 00:16:11.480 --> 00:16:14.950 b-bar, I add the first component of a-bar 00:16:14.950 --> 00:16:17.080 to the first component of b-bar. 00:16:17.080 --> 00:16:20.970 Noticing of course, that this is what? 00:16:20.970 --> 00:16:21.690 Be careful here. 00:16:21.690 --> 00:16:23.530 This is one number. 00:16:23.530 --> 00:16:25.800 a_1 plus b_1 is one number. 00:16:25.800 --> 00:16:27.750 a_2 plus b_2 is a number. 00:16:27.750 --> 00:16:30.020 a_n plus b_n is a number. 00:16:30.020 --> 00:16:32.550 In other words, notice that by this definition, 00:16:32.550 --> 00:16:37.040 the sum of two n-tuples is again an n-tuple. 00:16:37.040 --> 00:16:41.780 And finally, to multiply a scalar by an n-tuple, 00:16:41.780 --> 00:16:44.330 I will agree to define that definition 00:16:44.330 --> 00:16:51.050 to mean that you multiply the n-tuple component by component 00:16:51.050 --> 00:16:54.500 by that particular scalar, or number. 00:16:54.500 --> 00:16:57.360 Notice again that all I have done here 00:16:57.360 --> 00:17:02.930 is I have obtained these three structural definitions 00:17:02.930 --> 00:17:07.020 from the equivalent situations of arrows. 00:17:07.020 --> 00:17:10.740 And since everything that was true about arrows followed 00:17:10.740 --> 00:17:13.310 from these three basic definitions, 00:17:13.310 --> 00:17:18.450 any set of n-tuples that obeys this particular structure will 00:17:18.450 --> 00:17:20.930 also behave like the arrows did. 00:17:20.930 --> 00:17:23.869 And that's why we call it a vector space. 00:17:23.869 --> 00:17:27.770 They behave like vectors even though they can no longer 00:17:27.770 --> 00:17:30.060 be viewed as arrows. 00:17:30.060 --> 00:17:32.010 And again there are creative people who 00:17:32.010 --> 00:17:33.780 view these things as arrows. 00:17:33.780 --> 00:17:35.750 I remember feeling very intimidated one day 00:17:35.750 --> 00:17:37.760 by my undergraduate professor the first time 00:17:37.760 --> 00:17:39.210 I learned vector spaces. 00:17:39.210 --> 00:17:42.084 I said, how do you visualize an n-dimensional vector space? 00:17:42.084 --> 00:17:44.250 And in full seriousness, without batting an eyelash, 00:17:44.250 --> 00:17:46.890 he says, "I visualize it like a porcupine with a bunch 00:17:46.890 --> 00:17:48.460 of quills coming out of it." 00:17:48.460 --> 00:17:51.020 And I knew that he knew what was visualizing it like, 00:17:51.020 --> 00:17:52.850 but didn't help me one bit. 00:17:52.850 --> 00:17:55.800 I'm saying, if you can visualize this things as arrows, 00:17:55.800 --> 00:17:56.650 be my guest. 00:17:56.650 --> 00:17:58.030 Feel free to do so. 00:17:58.030 --> 00:18:01.710 If you can't, notice that every one of these definitions 00:18:01.710 --> 00:18:05.130 stands on its own two feet. 00:18:05.130 --> 00:18:08.220 Subject to the condition that when n is one, two, or three, 00:18:08.220 --> 00:18:12.130 we happen to have a very nice geometric interpretation. 00:18:12.130 --> 00:18:15.010 By the way, I may have given you the impression 00:18:15.010 --> 00:18:18.320 that vector spaces were invented because of functions 00:18:18.320 --> 00:18:19.640 of several variables. 00:18:19.640 --> 00:18:22.280 Rather, the impression I would like to leave you with 00:18:22.280 --> 00:18:25.210 is, that in terms of motivating vector spaces, in terms 00:18:25.210 --> 00:18:27.610 of this course, that was the motivation 00:18:27.610 --> 00:18:29.050 that we elected to use. 00:18:29.050 --> 00:18:31.660 That the mathematician talked about vector spaces 00:18:31.660 --> 00:18:33.890 in many a different context from what 00:18:33.890 --> 00:18:35.994 we might even dream possible. 00:18:35.994 --> 00:18:38.160 In other words, I don't have to think of temperature 00:18:38.160 --> 00:18:41.460 being a function of the four variables x, y, z, and t. 00:18:41.460 --> 00:18:44.070 Let me give you a different kind of non-trivial example 00:18:44.070 --> 00:18:49.230 of a four-space that doesn't even bring functions into play. 00:18:49.230 --> 00:18:52.940 Let's suppose I invent the abbreviation, 00:18:52.940 --> 00:18:56.210 I write the 4-tuple (a_0, a_1, a_2, 00:18:56.210 --> 00:19:02.720 a_3) to denote the cubic polynomial a_0 plus a_1*x plus 00:19:02.720 --> 00:19:05.540 a_2 x squared plus a_3 x cubed. 00:19:05.540 --> 00:19:09.670 Notice that I can use these as a place value system. 00:19:09.670 --> 00:19:12.230 The first member tells me my constant term, 00:19:12.230 --> 00:19:14.260 the second member tells me the coefficient 00:19:14.260 --> 00:19:17.660 of x, the third member tells me the coefficient of x squared, 00:19:17.660 --> 00:19:20.890 and the fourth number gives me the coefficient of x cubed. 00:19:20.890 --> 00:19:24.790 Notice that for two polynomials to be identically equal, 00:19:24.790 --> 00:19:27.280 they must be equal, what? 00:19:27.280 --> 00:19:29.110 Coefficient by coefficient. 00:19:29.110 --> 00:19:30.870 That means what? 00:19:30.870 --> 00:19:32.700 Component by component. 00:19:32.700 --> 00:19:34.740 How do we add two polynomials? 00:19:34.740 --> 00:19:38.080 We add them coefficient by coefficient. 00:19:38.080 --> 00:19:39.200 We add like terms. 00:19:39.200 --> 00:19:42.830 In other words, given two polynomials, we add them what? 00:19:42.830 --> 00:19:44.230 Component by component. 00:19:44.230 --> 00:19:47.080 We add the two constant terms together, the two coefficients 00:19:47.080 --> 00:19:49.000 of x together, the two coefficients 00:19:49.000 --> 00:19:52.050 of x squared together, the two coefficients of x 00:19:52.050 --> 00:19:52.730 cubed together. 00:19:52.730 --> 00:19:53.630 You see? 00:19:53.630 --> 00:19:56.320 How do we multiply a polynomial by a scalar? 00:19:56.320 --> 00:19:59.420 We multiply each term by the scalar. 00:19:59.420 --> 00:20:02.340 That, in turn, is equivalent to multiplying each coefficient 00:20:02.340 --> 00:20:03.610 by that scalar. 00:20:03.610 --> 00:20:06.120 And that says, in terms of n-tuple notation, 00:20:06.120 --> 00:20:10.100 that we have multiplied each component by that scalar. 00:20:10.100 --> 00:20:12.450 The set of polynomials of degree n 00:20:12.450 --> 00:20:16.680 forms a very nice vector space in terms of our definition 00:20:16.680 --> 00:20:18.150 of a vector space. 00:20:18.150 --> 00:20:21.670 Now, of course, the danger is that one gets the idea 00:20:21.670 --> 00:20:25.250 that any set of n-tuples can be viewed as a vector space. 00:20:25.250 --> 00:20:27.080 An n-dimensional vector space. 00:20:27.080 --> 00:20:29.050 But this we have to be careful about. 00:20:29.050 --> 00:20:31.380 Remember, it is not the n-tuples, 00:20:31.380 --> 00:20:33.380 it is structure that they obey. 00:20:33.380 --> 00:20:36.600 Let me give you sort of a simple example over here. 00:20:36.600 --> 00:20:38.520 Let me consider the following situation. 00:20:38.520 --> 00:20:40.520 First of all, let me just emphasize a statement; 00:20:40.520 --> 00:20:42.450 I just made it, let me just read it with you. 00:20:42.450 --> 00:20:46.640 n-tuples are not automatically n-spaces. 00:20:46.640 --> 00:20:52.600 For example, let me invent the 2-tuple a comma b to represent 00:20:52.600 --> 00:20:54.940 the number a plus b. 00:20:54.940 --> 00:20:58.390 For example, if I define the 2-tuple a comma 00:20:58.390 --> 00:21:00.980 b to be an abbreviation for a plus b, 00:21:00.980 --> 00:21:05.127 What would 4 comma 5 denote? 00:21:05.127 --> 00:21:06.460 Remember the 2-tuple means what? 00:21:06.460 --> 00:21:08.910 To get the value of the 2-tuple is just 00:21:08.910 --> 00:21:10.760 the sum of the components. 00:21:10.760 --> 00:21:16.610 If I add a and b, in this case, 4 plus 5 happens to be 9. 00:21:16.610 --> 00:21:21.360 How about the 2-tuple 6 comma 3, what value would that have? 00:21:21.360 --> 00:21:24.210 That would also have the value 9. 00:21:24.210 --> 00:21:27.720 Therefore numerically, the 2-tuple 4 comma 5 00:21:27.720 --> 00:21:30.870 is equal to the 2-tuple 6 comma 3. 00:21:30.870 --> 00:21:33.740 Yet notice that the first component is not 00:21:33.740 --> 00:21:35.690 equal to the first component here. 00:21:35.690 --> 00:21:38.630 In other words 4 is not equal to 6. 00:21:38.630 --> 00:21:42.610 Nor is 5 is equal to 3, but if I were 00:21:42.610 --> 00:21:44.950 to choose this definition of equality, 00:21:44.950 --> 00:21:47.940 I could not say that these 2-tuples form 00:21:47.940 --> 00:21:51.770 a two-dimensional vector space, because it violates 00:21:51.770 --> 00:21:54.910 the first definition for a vector space, namely 00:21:54.910 --> 00:21:59.710 the definition of what it means for two vectors to be equal. 00:21:59.710 --> 00:22:01.790 Since we're going to let most of our material 00:22:01.790 --> 00:22:06.150 be covered by the exercises and the supplementary notes, 00:22:06.150 --> 00:22:09.850 and this is just to be an overview, let's move on now. 00:22:09.850 --> 00:22:13.020 Let's assume that we now know what n-dimensional vector 00:22:13.020 --> 00:22:14.200 spaces are like. 00:22:14.200 --> 00:22:17.850 We now know that we can view functions of several variables 00:22:17.850 --> 00:22:22.510 as functions that map n-tuples into numbers, and as a result, 00:22:22.510 --> 00:22:25.600 it now makes sense to talk about things like: 00:22:25.600 --> 00:22:28.620 suppose you were given the n-tuple (x_1, x_2, x_3, 00:22:28.620 --> 00:22:33.420 x_3) and suppose that under f, that n-tuple was mapped 00:22:33.420 --> 00:22:39.430 into x_1 cubed plus x_2 plus x_3 squared plus 2*x_4. 00:22:39.430 --> 00:22:46.230 For example, if I were to replace x_1 by 1, x_2 by 3, 00:22:46.230 --> 00:22:52.170 x_3 by 1, and x_4 by 2, I would arrive at the result what? 00:22:52.170 --> 00:22:57.130 1 cubed plus 3 plus 1 squared plus 2 times 2, 00:22:57.130 --> 00:22:59.130 and I can compute that output. 00:22:59.130 --> 00:23:01.370 Now the question that comes up in calculus is, 00:23:01.370 --> 00:23:03.250 can we talk about limits here? 00:23:03.250 --> 00:23:08.280 Instead of computing what f of 1 comma 3 comma 1 comma 2 is, 00:23:08.280 --> 00:23:13.000 can I compute the limit of this thing as (x_1, x_2, x_3, x_4) 00:23:13.000 --> 00:23:17.390 approaches 1 comma 3 comma 1 comma 2? 00:23:17.390 --> 00:23:21.320 I think intuitively it's clear that since equality means 00:23:21.320 --> 00:23:24.590 that you must have equality component by component, 00:23:24.590 --> 00:23:27.850 to say that this approaches this means 00:23:27.850 --> 00:23:30.280 that the first component here must approach 00:23:30.280 --> 00:23:31.810 the first component here. 00:23:31.810 --> 00:23:33.420 The second component here approaches 00:23:33.420 --> 00:23:35.540 is the second component here, et cetera. 00:23:35.540 --> 00:23:37.960 In other words, this could be replaced 00:23:37.960 --> 00:23:41.440 by the four separate linear one-dimensional limit 00:23:41.440 --> 00:23:47.400 problems: x_1 approaches 1, x_2 approaches 3, x_3 approaches 1, 00:23:47.400 --> 00:23:49.090 and x_4 approaches 2. 00:23:49.090 --> 00:23:51.360 And we would then be tempted to say what? 00:23:51.360 --> 00:23:59.290 We will replace x_1 by 1, x_2 by 3, x_3 by 1, x_4 by 2, 00:23:59.290 --> 00:24:01.610 see what happens to this expression. 00:24:01.610 --> 00:24:04.750 And we would then be tempted to say that this particular limit 00:24:04.750 --> 00:24:07.390 was equal to 9. 00:24:07.390 --> 00:24:11.020 Now the interesting point is this-- that traditionally, 00:24:11.020 --> 00:24:15.510 this particular problem was tackled long before anyone 00:24:15.510 --> 00:24:17.700 invented vector spaces. 00:24:17.700 --> 00:24:19.830 Or at least long before anybody was 00:24:19.830 --> 00:24:23.290 serious about vector spaces. 00:24:23.290 --> 00:24:25.490 People did say, why can't we reduce 00:24:25.490 --> 00:24:27.680 the study of four-dimensional space 00:24:27.680 --> 00:24:30.140 to four separate studies of one-dimensional space? 00:24:30.140 --> 00:24:33.880 In other words, let x_1 approach 1, x_2 approach 3, in that case 00:24:33.880 --> 00:24:35.030 you're allowing what? 00:24:35.030 --> 00:24:40.650 Four separate one-dimensional limits to be taking place here. 00:24:40.650 --> 00:24:43.540 But the insight that modern math gave 00:24:43.540 --> 00:24:46.720 us was that we can now go back to our traditional definition 00:24:46.720 --> 00:24:47.220 of limit. 00:24:47.220 --> 00:24:50.570 Remember what was our old structural definition of limit? 00:24:50.570 --> 00:24:53.140 Way back from the first time we had it. 00:24:53.140 --> 00:24:56.850 The limit of f of x as x approaches a equals L means, 00:24:56.850 --> 00:25:00.410 given epsilon greater than 0, we can find delta greater 00:25:00.410 --> 00:25:04.680 than zero, such that whenever the absolute value of x minus a 00:25:04.680 --> 00:25:06.960 is greater than 0 but less than delta, 00:25:06.960 --> 00:25:10.670 the absolute value of f of x minus L is less than epsilon 00:25:10.670 --> 00:25:12.740 Now here's what the new math said. 00:25:12.740 --> 00:25:15.130 The modern approach said look, let's just 00:25:15.130 --> 00:25:16.760 take our old structural definition-- 00:25:16.760 --> 00:25:19.580 the same as before-- and vectorize everything. 00:25:19.580 --> 00:25:22.010 Notice in this situation that we're dealing with, 00:25:22.010 --> 00:25:24.670 the input is where we have the several variables. 00:25:24.670 --> 00:25:28.230 The input is the n-tuple, the vector, OK? 00:25:28.230 --> 00:25:30.340 And the output is the scalar. 00:25:30.340 --> 00:25:33.100 So f is a scalar, L is a scalar. 00:25:33.100 --> 00:25:36.670 But x and a are vectors, so every place 00:25:36.670 --> 00:25:41.340 I see an x and an a, I have to put the bar underneath. 00:25:41.340 --> 00:25:44.380 And now I read this definition, and all of a sudden, 00:25:44.380 --> 00:25:46.940 as so often has happened in our course up to now, 00:25:46.940 --> 00:25:49.480 I come to something that I've never seen before. 00:25:49.480 --> 00:25:53.220 Namely, as soon as I look at this. 00:25:53.220 --> 00:25:57.200 This made very good sense when these were arrows. 00:25:57.200 --> 00:25:59.760 Namely, we talked about this earlier 00:25:59.760 --> 00:26:01.540 in our course and one of our lectures. 00:26:01.540 --> 00:26:04.410 That to say the two arrows were near each other 00:26:04.410 --> 00:26:06.790 was to say that their difference was small, 00:26:06.790 --> 00:26:09.290 and that in turn said if the arrows were placed tail 00:26:09.290 --> 00:26:12.480 to tail, we could make the distance between their heads 00:26:12.480 --> 00:26:14.130 as small as we wish. 00:26:14.130 --> 00:26:17.650 Now the price that we have to pay for higher dimensions 00:26:17.650 --> 00:26:20.010 is that if we have more dimensions than what 00:26:20.010 --> 00:26:23.240 we can draw arrows in, the problem that we're faced 00:26:23.240 --> 00:26:25.910 is that we have not defined what you 00:26:25.910 --> 00:26:31.380 mean by the magnitude of x minus a, where x and a happen 00:26:31.380 --> 00:26:33.710 to be n-tuples. 00:26:33.710 --> 00:26:36.480 And here again, we come back to our structure. 00:26:36.480 --> 00:26:40.120 But now for the first time the structure is not redundant. 00:26:40.120 --> 00:26:42.530 Let me tell you what I mean by that. 00:26:42.530 --> 00:26:44.920 In the one-dimensional case we define 00:26:44.920 --> 00:26:49.860 the magnitude of x minus a to be the square root of x_1 minus 00:26:49.860 --> 00:26:52.220 a_1 squared, where the vector x was 00:26:52.220 --> 00:26:57.640 the 1-tuple x_1, and the vector a was the n-tuple a_1. 00:26:57.640 --> 00:26:59.710 In the two-dimensional case, we said, 00:26:59.710 --> 00:27:03.910 OK, let's define the magnitude of the vector 00:27:03.910 --> 00:27:07.590 x minus a to be x_1 minus a_1 squared 00:27:07.590 --> 00:27:10.370 plus x_2 minus a_2 squared. 00:27:10.370 --> 00:27:11.820 And in the three-dimensional case, 00:27:11.820 --> 00:27:16.460 we said, let's define the magnitude of x-bar minus a-bar 00:27:16.460 --> 00:27:19.740 to be the square root of x_1 minus a_1 squared 00:27:19.740 --> 00:27:24.260 plus x_2 minus a_2 squared plus x_3 minus a_3 squared. 00:27:24.260 --> 00:27:28.690 At that time, I kept saying, notice that these recipes do 00:27:28.690 --> 00:27:32.620 not depend on a picture, that these are numerical results 00:27:32.620 --> 00:27:34.060 that we can compute without having 00:27:34.060 --> 00:27:35.550 to draw a picture at all. 00:27:35.550 --> 00:27:39.460 What happened of course was that in the one-dimensional case, 00:27:39.460 --> 00:27:42.890 in the two-dimensional case, in the three-dimensional case, 00:27:42.890 --> 00:27:46.360 it was easier to visualize the picture. 00:27:46.360 --> 00:27:49.240 Now here's where the real kicker comes in-- 00:27:49.240 --> 00:27:52.270 and this is the real crucial point-- structurally, 00:27:52.270 --> 00:27:54.780 can't you see what's happening over here? 00:27:54.780 --> 00:27:58.360 Can't you see how I can now define the absolute value 00:27:58.360 --> 00:28:04.850 of the vector x-bar minus a-bar, even if n is greater than 3, 00:28:04.850 --> 00:28:06.820 in such a way that the definition will 00:28:06.820 --> 00:28:11.620 make sense and still mimic everything that we're doing? 00:28:11.620 --> 00:28:14.100 I hope you are a step ahead of me 00:28:14.100 --> 00:28:17.730 on this except for some new notation I introduced here. 00:28:17.730 --> 00:28:20.290 It turns out that in the modern math book, 00:28:20.290 --> 00:28:24.960 one distinguishes between the absolute value 00:28:24.960 --> 00:28:28.290 of a number, and the magnitude of a vector, 00:28:28.290 --> 00:28:34.140 and it is frequently traditional to introduce a double bar 00:28:34.140 --> 00:28:39.010 on each side to represent the magnitude of the difference 00:28:39.010 --> 00:28:40.140 between two vectors, 00:28:40.140 --> 00:28:42.700 which I claim behaves like a distance. 00:28:42.700 --> 00:28:44.510 Let me show you what I mean by that. 00:28:44.510 --> 00:28:48.820 Let me define the magnitude of the n-tuple x-bar 00:28:48.820 --> 00:28:53.400 minus the n-tuple a-bar, written this way, to be 00:28:53.400 --> 00:28:57.110 the positive square root of x_1 minus a_1 squared, 00:28:57.110 --> 00:29:02.710 plus et cetera, plus x_n minus a_n squared. 00:29:02.710 --> 00:29:04.720 The thing that I would like you to notice here 00:29:04.720 --> 00:29:08.850 is that since each of these numbers are non-negative-- 00:29:08.850 --> 00:29:10.850 see they're squares of real numbers-- 00:29:10.850 --> 00:29:14.600 the only way this can be 0-- well the only way that the sum 00:29:14.600 --> 00:29:17.860 of squares of non-negative numbers can be 0 00:29:17.860 --> 00:29:20.080 is for each of the numbers to be 0. 00:29:20.080 --> 00:29:23.860 Consequently, the only way the magnitude of x-bar minus a-bar 00:29:23.860 --> 00:29:28.710 can equal 0 is if x_1 equals a_1, x_2 equals a_2, et cetera, 00:29:28.710 --> 00:29:31.060 and x_n equals a_n. 00:29:31.060 --> 00:29:35.900 In this vein, notice that the geometric phrase x-bar 00:29:35.900 --> 00:29:39.870 near a-bar still makes sense. 00:29:39.870 --> 00:29:41.760 It doesn't make sense pictorially, 00:29:41.760 --> 00:29:44.955 because we can't draw the arrows if n is greater than 3. 00:29:44.955 --> 00:29:46.330 But notice that what we're saying 00:29:46.330 --> 00:29:49.630 is that for x to be near a, all we're saying 00:29:49.630 --> 00:29:53.140 is that the magnitude-- defined this way-- 00:29:53.140 --> 00:29:57.420 the magnitude of x-bar minus a-bar is small. 00:29:57.420 --> 00:30:00.010 When you're adding up positive squares, 00:30:00.010 --> 00:30:03.700 the only way the sum can be small is if each of the factors 00:30:03.700 --> 00:30:04.640 are small. 00:30:04.640 --> 00:30:07.580 But notice what these factors are, except for the square, 00:30:07.580 --> 00:30:11.440 it's the difference between x_1 and a_1, x_2 and a_2, 00:30:11.440 --> 00:30:13.730 et cetera, x_n and a_n. 00:30:13.730 --> 00:30:17.320 In other words, to say that x-bar is near a-bar means 00:30:17.320 --> 00:30:21.400 that x_1 is near a_1, x_2 is near a_2, et cetera, 00:30:21.400 --> 00:30:24.170 and x_n is near a_n, which is exactly 00:30:24.170 --> 00:30:25.730 the traditional approach. 00:30:25.730 --> 00:30:28.960 And in fact, except for the fact that we can capitalize 00:30:28.960 --> 00:30:31.340 on structure, notice that once we 00:30:31.340 --> 00:30:34.840 define the magnitude of the difference between two 00:30:34.840 --> 00:30:37.170 n-tuples-- do you notice that by the way? 00:30:37.170 --> 00:30:41.490 The magnitude of the difference of two n-tuples is a number. 00:30:41.490 --> 00:30:45.240 Notice now if we replace this fancy phrase-- 00:30:45.240 --> 00:30:47.830 which we didn't know the meaning of before, 00:30:47.830 --> 00:30:52.340 but which we now know-- by its new definition, 00:30:52.340 --> 00:30:56.420 we obtain the traditional definition of limit. 00:30:56.420 --> 00:31:01.840 Namely the limit of f of x_1 up to x_n, as x_1 approaches a_1, 00:31:01.840 --> 00:31:03.630 et cetera, and x_n approaches a_n, 00:31:03.630 --> 00:31:07.830 equals L means that given epsilon greater than 0, 00:31:07.830 --> 00:31:10.930 we can find delta greater than 0 such 00:31:10.930 --> 00:31:14.040 that whenever the square root of x_1 minus a_1 00:31:14.040 --> 00:31:18.630 squared plus et cetera x_n minus a_n squared is less than delta 00:31:18.630 --> 00:31:21.460 but greater than 0, then the magnitude-- you see, 00:31:21.460 --> 00:31:23.330 these numbers here-- the magnitude 00:31:23.330 --> 00:31:28.010 of f of x_1 up to x_n minus L is less than epsilon. 00:31:28.010 --> 00:31:30.300 In, other words, this definition here 00:31:30.300 --> 00:31:33.940 happens to be the traditional definition. 00:31:33.940 --> 00:31:34.670 OK? 00:31:34.670 --> 00:31:38.340 But the point is that the traditional definition 00:31:38.340 --> 00:31:42.330 has exactly the same structure as the modern definition. 00:31:42.330 --> 00:31:44.820 And as a result, to make fun of the traditional math 00:31:44.820 --> 00:31:47.230 because it's not as pretty as the modern math 00:31:47.230 --> 00:31:49.070 is the wrong thing to say. 00:31:49.070 --> 00:31:51.340 It's like the fellow who once asked me at a PTA 00:31:51.340 --> 00:31:54.680 meeting, "how much is 8 plus 7 in the new mathematics?" 00:31:54.680 --> 00:31:56.550 That part hasn't changed. 00:31:56.550 --> 00:31:59.380 The beauty of using the n-tuple notation 00:31:59.380 --> 00:32:03.370 was that it allowed us to use the previous structure 00:32:03.370 --> 00:32:04.180 of limits. 00:32:04.180 --> 00:32:07.350 So that we can get all of our theorems, all of our formulas 00:32:07.350 --> 00:32:10.470 and what have you, to go through word for word, 00:32:10.470 --> 00:32:14.640 even though the higher the dimension, the more complex 00:32:14.640 --> 00:32:15.950 our computations are. 00:32:15.950 --> 00:32:19.600 But structurally, It essentially boiled down to, 00:32:19.600 --> 00:32:21.990 after you've seen one-dimensional space, 00:32:21.990 --> 00:32:23.160 you've seen them all. 00:32:23.160 --> 00:32:26.100 That was the big innovation with the modern approach 00:32:26.100 --> 00:32:28.100 to n-dimensional vector spaces. 00:32:28.100 --> 00:32:30.540 And to help put this in proper perspective, 00:32:30.540 --> 00:32:34.100 next time I shall introduce the calculus 00:32:34.100 --> 00:32:38.000 of several real variables in terms of the more 00:32:38.000 --> 00:32:39.430 traditional approach. 00:32:39.430 --> 00:32:44.220 But again, until next time, good bye. 00:32:44.220 --> 00:32:46.590 Funding for the publication of this video 00:32:46.590 --> 00:32:51.470 was provided by the Gabriella and Paul Rosenbaum foundation. 00:32:51.470 --> 00:32:55.640 Help OCW continue to provide free and open access to MIT 00:32:55.640 --> 00:33:03.350 courses by making a donation at MIT ocw.mit.edu/donate.