1 00:00:06,220 --> 00:00:09,770 GILBERT STRANG: Hi, I'm Gilbert Strang, and professor 2 00:00:09,770 --> 00:00:11,200 of mathematics at MIT. 3 00:00:11,200 --> 00:00:16,309 And I get a chance to say a few words 4 00:00:16,309 --> 00:00:18,970 about 18.06, Linear Algebra. 5 00:00:18,970 --> 00:00:21,100 It's one of the basic math courses. 6 00:00:24,590 --> 00:00:26,780 Can I say a little about linear algebra itself? 7 00:00:29,840 --> 00:00:34,650 Classes in linear algebra earlier years tended 8 00:00:34,650 --> 00:00:40,130 to be pretty much for pure math majors, and a lot of proofs, 9 00:00:40,130 --> 00:00:46,390 and usefulness of the subject kind of wasn't so clear. 10 00:00:46,390 --> 00:00:50,110 Whereas, it's an incredibly useful subject. 11 00:00:50,110 --> 00:00:52,010 Data is coming in all the time. 12 00:00:52,010 --> 00:00:55,360 We're in the century of data, and data 13 00:00:55,360 --> 00:01:01,930 tends to come in a matrix, in a rectangular array of numbers. 14 00:01:01,930 --> 00:01:07,370 And how to understand that data is a giant, giant problem. 15 00:01:07,370 --> 00:01:12,460 And people use matrices in solving differential equations 16 00:01:12,460 --> 00:01:14,760 in economics, everywhere. 17 00:01:14,760 --> 00:01:19,330 So the subject had to change to bring out 18 00:01:19,330 --> 00:01:24,910 this important aspect, that it's terrifically useful. 19 00:01:24,910 --> 00:01:27,560 Often networks are a great model, 20 00:01:27,560 --> 00:01:31,550 where you have like-- like the internet. 21 00:01:31,550 --> 00:01:36,600 Every website would be like a node in the network. 22 00:01:36,600 --> 00:01:41,570 And if one website is linked to another one, 23 00:01:41,570 --> 00:01:45,530 there would maybe be an edge in that network. 24 00:01:45,530 --> 00:01:48,190 So that's a network with a billion nodes. 25 00:01:48,190 --> 00:01:54,720 And a matrix describes all those links. 26 00:01:54,720 --> 00:02:00,710 Like when Google produces a PageRank, you enter-- well, 27 00:02:00,710 --> 00:02:04,070 you could enter linear algebra, and see what happens. 28 00:02:04,070 --> 00:02:05,150 I don't know. 29 00:02:05,150 --> 00:02:07,450 I hope something good. 30 00:02:07,450 --> 00:02:10,139 Well, anyway, thousands and millions of stuff 31 00:02:10,139 --> 00:02:14,230 would come up ranked in order, and that order 32 00:02:14,230 --> 00:02:18,050 comes from operating-- Google's very fast at it, 33 00:02:18,050 --> 00:02:23,350 very good at it-- operating on that giant matrix that 34 00:02:23,350 --> 00:02:25,520 describes the internet. 35 00:02:25,520 --> 00:02:29,820 OK, so a word about the course itself-- the MIT course. 36 00:02:29,820 --> 00:02:34,160 First of all, there will be students coming 37 00:02:34,160 --> 00:02:37,090 from all the departments. 38 00:02:37,090 --> 00:02:38,970 That includes management. 39 00:02:38,970 --> 00:02:42,400 Business data comes in matrix form 40 00:02:42,400 --> 00:02:44,390 just the way engineering data comes. 41 00:02:47,910 --> 00:02:52,930 So there is hardly a prerequisite for the course. 42 00:02:52,930 --> 00:02:56,552 There's no big reason why calculus has to come first. 43 00:03:00,540 --> 00:03:04,980 Probably most MIT students will know before the course starts-- 44 00:03:04,980 --> 00:03:07,760 they will have multiplied a matrix by a vector, 45 00:03:07,760 --> 00:03:09,820 or multiplied two matrices. 46 00:03:09,820 --> 00:03:13,020 So they've at least seen matrices before. 47 00:03:13,020 --> 00:03:17,310 But anybody could catch up on that quickly. 48 00:03:17,310 --> 00:03:20,080 And then, the course just takes off. 49 00:03:20,080 --> 00:03:24,960 Actually, we go back to ask, how do you understand multiplying 50 00:03:24,960 --> 00:03:26,540 a matrix by a vector? 51 00:03:26,540 --> 00:03:30,340 A key-- yeah, you guys will probably know how to do it, 52 00:03:30,340 --> 00:03:34,970 but let me say it another way-- A matrix times a vector 53 00:03:34,970 --> 00:03:39,930 produces a combination of the columns in that matrix, 54 00:03:39,930 --> 00:03:43,040 those column vectors in the matrix. 55 00:03:43,040 --> 00:03:46,970 So that's like the key step in linear algebra. 56 00:03:46,970 --> 00:03:51,460 What you can do with vectors is take linear combinations. 57 00:03:51,460 --> 00:03:56,040 Well, at MIT, the course is organized with three lectures 58 00:03:56,040 --> 00:03:58,670 a week. 59 00:03:58,670 --> 00:04:02,540 And I use the chalkboard. 60 00:04:02,540 --> 00:04:06,756 I hope you feel, in watching them, that that's OK. 61 00:04:06,756 --> 00:04:08,130 The nice thing about a chalkboard 62 00:04:08,130 --> 00:04:14,150 is you get to see-- what's written doesn't disappear. 63 00:04:14,150 --> 00:04:17,630 So your eye can continually check back 64 00:04:17,630 --> 00:04:24,970 and see how does it connect with what's happening at the moment. 65 00:04:24,970 --> 00:04:28,630 And then, there is one hour a week of recitation. 66 00:04:28,630 --> 00:04:31,430 Because that's a smaller class, it just 67 00:04:31,430 --> 00:04:34,050 means there's a teaching assistant 68 00:04:34,050 --> 00:04:39,710 there, who can help with problems, suggest new problems. 69 00:04:39,710 --> 00:04:45,020 It can be a problem-based hour, where my lectures are 70 00:04:45,020 --> 00:04:47,550 more explanation hours. 71 00:04:47,550 --> 00:04:49,770 So about the textbook. 72 00:04:49,770 --> 00:04:53,180 The homeworks come from the book mostly. 73 00:04:53,180 --> 00:04:56,590 Sometimes we add MATLAB problems, sort 74 00:04:56,590 --> 00:04:58,940 of specially constructed ones. 75 00:04:58,940 --> 00:05:02,720 But the central ideas of the subject 76 00:05:02,720 --> 00:05:05,870 are described in each section of the book, 77 00:05:05,870 --> 00:05:10,880 and then, naturally, exercises to practice with those ideas. 78 00:05:10,880 --> 00:05:13,720 And then, the neat thing about 18.06 Scholar 79 00:05:13,720 --> 00:05:20,220 is you get short lectures, short videos, from six different TAs, 80 00:05:20,220 --> 00:05:23,420 did about six problem-solving videos each. 81 00:05:23,420 --> 00:05:24,693 And they are neat. 82 00:05:24,693 --> 00:05:27,530 The TAs are good. 83 00:05:27,530 --> 00:05:31,780 And that's something that can happen in the recitation 84 00:05:31,780 --> 00:05:33,101 with a smaller group. 85 00:05:35,810 --> 00:05:39,370 There's chance for a discussion, whereas in the lecture-- well, 86 00:05:39,370 --> 00:05:41,610 I still ask questions in the lecture, 87 00:05:41,610 --> 00:05:43,970 as you'll probably see. 88 00:05:43,970 --> 00:05:48,110 But it's a little harder for students 89 00:05:48,110 --> 00:05:52,210 to shout out an answer, so they can shout all 90 00:05:52,210 --> 00:05:54,410 they want in their recitations. 91 00:05:54,410 --> 00:05:57,690 With each lecture, we produce a written summary 92 00:05:57,690 --> 00:05:59,780 of what it's about. 93 00:05:59,780 --> 00:06:02,950 So after you watch the lecture, you could look at that summary 94 00:06:02,950 --> 00:06:08,700 and it reinforces, remembering the key points of the lecture. 95 00:06:08,700 --> 00:06:13,170 And then we also added in some problems, four or five 96 00:06:13,170 --> 00:06:18,410 problems from the book that you can just look at and see, OK, 97 00:06:18,410 --> 00:06:20,570 do I know what the question is here? 98 00:06:20,570 --> 00:06:23,310 Do I know how to do it? 99 00:06:23,310 --> 00:06:27,740 I think, as a result, you're learning linear algebra. 100 00:06:27,740 --> 00:06:29,790 A thought or two about linear algebra 101 00:06:29,790 --> 00:06:33,950 worldwide, because it really is worldwide. 102 00:06:33,950 --> 00:06:36,930 The feedback comes from all over the world. 103 00:06:36,930 --> 00:06:40,380 It's really nice to get. 104 00:06:40,380 --> 00:06:42,550 Also, I enjoy going. 105 00:06:42,550 --> 00:06:48,170 So if somebody invites me to Egypt or Australia or China, 106 00:06:48,170 --> 00:06:51,850 I tend to go if I can. 107 00:06:51,850 --> 00:06:54,820 Because that's a lovely part about mathematics. 108 00:06:54,820 --> 00:06:57,580 It's really universal. 109 00:06:57,580 --> 00:07:01,110 It's a language almost of its own 110 00:07:01,110 --> 00:07:07,250 that everybody can learn to speak. 111 00:07:07,250 --> 00:07:11,160 And I hope these lectures help.