Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare

00:00:00.000 | The following is a conversation with Gilbert Strang.

00:00:03.320 | He's a professor of mathematics at MIT

00:00:05.760 | and perhaps one of the most famous

00:00:07.480 | and impactful teachers of math in the world.

00:00:10.600 | His MIT OpenCourseWare lectures on linear algebra

00:00:13.640 | have been viewed millions of times.

00:00:15.920 | As an undergraduate student,

00:00:17.480 | I was one of those millions of students.

00:00:19.960 | There's something inspiring about the way he teaches.

00:00:22.760 | There's at once calm, simple, and yet full of passion

00:00:26.120 | for the elegance inherent to mathematics.

00:00:29.080 | I remember doing the exercise in his book,

00:00:31.200 | Introduction to Linear Algebra,

00:00:33.000 | and slowly realizing that the world of matrices,

00:00:35.880 | of vector spaces, of determinants and eigenvalues,

00:00:39.360 | of geometric transformations and matrix decompositions

00:00:43.040 | reveal a set of powerful tools

00:00:45.120 | in the toolbox of artificial intelligence.

00:00:47.920 | From signals to images, from numerical optimization

00:00:51.080 | to robotics, computer vision, deep learning,

00:00:53.640 | computer graphics, and everywhere outside AI,

00:00:56.600 | including, of course, a quantum mechanical study

00:01:00.040 | of our universe.

00:01:01.520 | This is the Artificial Intelligence Podcast.

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00:01:12.360 | @LexFriedman, spelled F-R-I-D-M-A-N.

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00:03:40.960 | And now, here's my conversation with Gilbert Strang.

00:03:45.800 | How does it feel to be one of the modern day rock stars

00:03:50.440 | of mathematics?

00:03:51.720 | - I don't feel like a rock star.

00:03:53.840 | That's kind of crazy for an old math person,

00:03:57.280 | but it's true that the videos in linear algebra

00:04:02.280 | that I made way back in 2000, I think,

00:04:07.140 | have been watched a lot.

00:04:09.840 | And well, partly the importance of linear algebra,

00:04:14.400 | which I'm sure you'll ask me and give me a chance to say

00:04:17.720 | that linear algebra as a subject

00:04:20.080 | has just surged in importance.

00:04:22.600 | But also, it was a class that I taught a bunch of times,

00:04:26.480 | so I kind of got it organized and enjoyed doing it.

00:04:31.480 | The videos were just the class.

00:04:34.200 | So they're on OpenCourseWare and on YouTube and translated.

00:04:38.200 | That's fun.

00:04:39.040 | - But there's something about that chalkboard

00:04:41.120 | and the simplicity of the way you explain

00:04:44.480 | the basic concepts in the beginning.

00:04:46.800 | To be honest, when I went to undergrad--

00:04:50.440 | - You didn't do linear algebra, probably.

00:04:52.640 | - Of course I didn't do linear algebra.

00:04:54.400 | Yeah, yeah, yeah, of course.

00:04:55.440 | But before going through the course at my university,

00:05:00.440 | I was going through OpenCourseWare.

00:05:02.520 | You were my instructor for linear algebra.

00:05:04.640 | - Oh, I see, right, yeah.

00:05:05.480 | - And that, I mean, we were using your book.

00:05:07.560 | And I mean, the fact that there's thousands,

00:05:12.560 | you know, hundreds of thousands,

00:05:14.000 | millions of people that watch that video,

00:05:15.880 | I think that's really powerful.

00:05:18.240 | So how do you think the idea of putting lectures online,

00:05:23.240 | what really MIT OpenCourseWare has innovated?

00:05:27.280 | - That was a wonderful idea.

00:05:28.720 | You know, I think the story that I've heard

00:05:32.720 | is the committee was appointed by the president,

00:05:36.720 | President Vest at that time, a wonderful guy.

00:05:40.000 | And the idea of the committee was to figure out

00:05:43.440 | how MIT could be like other universities,

00:05:48.440 | market the work we were doing.

00:05:52.560 | And then they didn't see a way, and after a weekend,

00:05:55.680 | and they had an inspiration,

00:05:57.240 | came back to President Vest and said,

00:05:59.960 | what if we just gave it away?

00:06:02.240 | And he decided that was okay, good idea.

00:06:07.240 | So--

00:06:08.520 | - You know, that's a crazy idea.

00:06:09.640 | That's, if we think of a university

00:06:12.040 | as a thing that creates a product, isn't knowledge,

00:06:16.040 | - Right.

00:06:16.880 | - The, you know, the kind of educational knowledge

00:06:19.580 | isn't the product, and giving that away,

00:06:22.680 | are you surprised that it went through?

00:06:26.120 | - The result that he did it, well,

00:06:28.520 | knowing a little bit President Vest,

00:06:30.760 | it was like him, I think.

00:06:33.000 | And it was really the right idea, you know.

00:06:37.100 | MIT is a kind of, it's known for being high level,

00:06:43.200 | technical things, and this is the best way we can say,

00:06:48.200 | tell, we can show what MIT really is like.

00:06:52.600 | 'Cause in my case, those 1806 videos

00:06:57.380 | are just teaching the class.

00:06:59.960 | They were there in 26-100.

00:07:03.680 | They're kind of fun to look at.

00:07:04.880 | People write to me and say, oh, you've got a sense of humor,

00:07:08.300 | but I don't know where that comes through.

00:07:10.920 | Somehow I've been friendly with the class,

00:07:13.720 | I like students, and linear algebra,

00:07:18.720 | we gotta give the subject most of the credit.

00:07:22.840 | It really has come forward in importance in these years.

00:07:27.840 | - So let's talk about linear algebra a little bit,

00:07:32.740 | 'cause it is such a, it's both a powerful

00:07:34.620 | and a beautiful subfield of mathematics.

00:07:39.100 | So what's your favorite specific topic in linear algebra,

00:07:44.100 | or even math in general, to give a lecture on,

00:07:46.620 | to convey, to tell a story, to teach students?

00:07:50.220 | - Okay, well, on the teaching side,

00:07:53.480 | so it's not deep mathematics at all,

00:07:56.800 | but I'm kind of proud of the idea of the four subspaces,

00:08:01.800 | the four fundamental subspaces,

00:08:05.200 | which are, of course, known before,

00:08:09.640 | long before my name for them, but--

00:08:13.680 | - Can you go through them?

00:08:14.680 | Can you go through the four subspaces?

00:08:15.520 | - Sure I can, yeah.

00:08:17.040 | So the first one to understand is,

00:08:19.640 | so the matrix, maybe I should say the matrix--

00:08:22.920 | - What is a matrix?

00:08:23.760 | - What's a matrix?

00:08:24.880 | Well, so we have like a rectangle of numbers.

00:08:28.480 | So it's got N columns, got a bunch of columns,

00:08:32.240 | and also got an M rows, let's say,

00:08:36.500 | and the relation between, so of course,

00:08:38.560 | the columns and the rows, it's the same numbers,

00:08:41.600 | so there's gotta be connections there,

00:08:44.320 | but they're not simple.

00:08:46.600 | The columns might be longer than the rows,

00:08:50.080 | and they're all different, the numbers are mixed up.

00:08:53.720 | First space to think about is, take the columns,

00:08:57.760 | so those are vectors, those are points in N dimensions.

00:09:01.880 | - What's a vector?

00:09:02.720 | - So a physicist would imagine a vector,

00:09:05.680 | or might imagine a vector as a arrow in space,

00:09:10.680 | or the point it ends at in space.

00:09:14.840 | For me, it's a column of numbers.

00:09:17.280 | - You often think of, this is very interesting

00:09:20.960 | in terms of linear algebra, in terms of a vector,

00:09:23.960 | you think a little bit more abstract

00:09:26.680 | than how it's very commonly used, perhaps.

00:09:30.600 | You think this arbitrary space, multi-dimensional space--

00:09:34.720 | - Right away, I'm in high dimensions.

00:09:38.240 | - Dreamland.

00:09:39.080 | - Yeah, that's right, in the lecture, I try to,

00:09:42.440 | so if you think of two vectors in 10 dimensions,

00:09:46.760 | I'll do this in class, and I'll readily admit

00:09:50.440 | that I have no good image in my mind

00:09:54.360 | of a vector, of a arrow in 10-dimensional space,

00:09:58.260 | but whatever, you can add one bunch of 10 numbers

00:10:03.260 | to another bunch of 10 numbers,

00:10:05.320 | so you can add a vector to a vector,

00:10:08.080 | and you can multiply a vector by three,

00:10:10.320 | and that's, if you know how to do those,

00:10:12.160 | you've got linear algebra.

00:10:14.080 | - You know, 10 dimensions, there's this beautiful thing

00:10:18.120 | about math, if we look at string theory,

00:10:20.240 | and all these theories which are really fundamentally

00:10:22.840 | derived through math, but are very difficult to visualize,

00:10:25.960 | and how do you think about the things,

00:10:28.880 | like a 10-dimensional vector,

00:10:31.120 | that we can't really visualize?

00:10:33.020 | And yet, math reveals some beauty--

00:10:38.320 | - Oh, great beauty.

00:10:39.160 | - To define our world in that weird thing

00:10:43.040 | we can't visualize, how do you think about that difference?

00:10:46.000 | - Well, probably, I'm not a very geometric person,

00:10:48.880 | so I'm probably thinking in three dimensions,

00:10:51.680 | and the beauty of linear algebra is that

00:10:53.960 | it goes on to 10 dimensions with no problem.

00:10:58.160 | I mean, if you're just seeing what happens

00:11:01.200 | if you add two vectors in 3D,

00:11:04.600 | yeah, then you can add them in 10D,

00:11:06.440 | you're just adding the 10 components.

00:11:10.880 | So I can't say that I have a picture,

00:11:14.840 | but yet I try to push the class to think

00:11:17.320 | of a flat surface in 10 dimensions,

00:11:21.200 | so a plane in 10 dimensions,

00:11:23.440 | and so that's one of the spaces.

00:11:27.180 | Take all the columns of the matrix,

00:11:29.880 | take all their combinations,

00:11:31.800 | so much of this column, so much of this one,

00:11:35.180 | then if you put all those together,

00:11:36.800 | you get some kind of a flat surface

00:11:39.920 | that I call a vector space, space of vectors,

00:11:44.680 | and my imagination is just seeing

00:11:47.260 | like a piece of paper in 3D.

00:11:50.240 | But anyway, so that's one of the spaces,

00:11:54.600 | that's space number one, the column space of the matrix.

00:11:58.480 | And then there's the row space,

00:12:00.040 | which is, as I said, different,

00:12:02.160 | but came from the same numbers.

00:12:04.760 | So we got the column space,

00:12:07.080 | all combinations of the columns,

00:12:10.040 | and then we got the row space,

00:12:11.960 | all combinations of the rows.

00:12:14.560 | So those words are easy for me to say,

00:12:17.520 | and I can't really draw them on a blackboard,

00:12:20.240 | but I try with my thick chalk.

00:12:22.520 | Everybody likes that railroad chalk,

00:12:25.880 | and me too, I wouldn't use anything else now.

00:12:29.960 | - Yeah.

00:12:31.160 | - And then the other two spaces are perpendicular to those.

00:12:35.040 | So like if you have a plane in 3D,

00:12:39.080 | just a plane is just a flat surface in 3D,

00:12:43.520 | then perpendicular to that plane would be a line,

00:12:47.060 | so that would be the null space.

00:12:50.200 | So we've got two, we've got a column space, a row space,

00:12:54.000 | and there are two perpendicular spaces.

00:12:56.920 | So those four fit together

00:12:58.760 | in a beautiful picture of a matrix, yeah, yeah.

00:13:03.760 | It's sort of a fundamental, it's not a difficult idea.

00:13:06.720 | It comes pretty early in 1806, and it's basic.

00:13:11.720 | - So planes in these multidimensional spaces,

00:13:16.360 | how difficult of an idea is that to come to, do you think?

00:13:20.140 | If you look back in time,

00:13:23.640 | I think mathematically it makes sense,

00:13:26.800 | but I don't know if it's intuitive for us to imagine,

00:13:29.600 | just as we were talking about.

00:13:31.120 | It feels like calculus is easier to intuit.

00:13:34.800 | - Well, I have to admit, calculus came earlier,

00:13:38.400 | earlier than linear algebra.

00:13:39.860 | So Newton and Leibniz were the great men

00:13:42.200 | to understand the key ideas of calculus.

00:13:45.820 | But linear algebra, to me, is like,

00:13:49.520 | okay, it's the starting point,

00:13:51.640 | 'cause it's all about flat things.

00:13:54.160 | Calculus has got all the complications of calculus

00:13:57.160 | come from the curves, the bending, the curved surfaces.

00:14:02.160 | Linear algebra, the surfaces are all flat.

00:14:05.900 | Nothing bends in linear algebra.

00:14:08.040 | So it should have come first, but it didn't.

00:14:11.660 | And calculus also comes first in high school classes,

00:14:16.660 | in college class, it'll be freshman math,

00:14:20.880 | it'll be calculus, and then I say, enough of it.

00:14:24.360 | Like, okay, get to the good stuff.

00:14:27.520 | - Do you think linear algebra should come first?

00:14:30.920 | - Well, it really, I'm okay with it not coming first,

00:14:34.700 | but it should, yeah, it should.

00:14:37.120 | It's simpler.

00:14:38.080 | - 'Cause everything is flat.

00:14:40.160 | - Yeah, everything's flat.

00:14:41.640 | Well, of course, for that reason,

00:14:43.560 | calculus sort of sticks to one dimension,

00:14:46.600 | or eventually you do multivariate,

00:14:49.480 | but that basically means two dimensions.

00:14:52.240 | Linear algebra, you take off into 10 dimensions, no problem.

00:14:55.560 | - It just feels scary and dangerous

00:14:57.640 | to go beyond two dimensions, that's all.

00:14:59.800 | (both laughing)

00:15:01.120 | - If everything's flat, you can't go wrong.

00:15:03.700 | - So what concept or theorem in linear algebra,

00:15:08.280 | or in math, you find most beautiful,

00:15:12.700 | that gives you pause, that leaves you in awe?

00:15:15.420 | - Well, I'll stick with linear algebra here.

00:15:18.080 | I hope the viewer knows that really,

00:15:20.680 | mathematics is amazing, amazing subject,

00:15:23.400 | and deep, deep connections between ideas

00:15:28.400 | that didn't look connected, they turned out they were.

00:15:32.620 | But if we stick with linear algebra,

00:15:35.600 | so we have a matrix, that's like the basic thing,

00:15:38.800 | a rectangle of numbers, and it might be a rectangle of data.

00:15:42.680 | You're probably gonna ask me later about data science,

00:15:46.760 | where often data comes in a matrix.

00:15:50.400 | You have, maybe every column corresponds to a drug,

00:15:55.400 | and every row corresponds to a patient,

00:16:00.360 | and if the patient reacted favorably to the drug,

00:16:05.360 | then you put up some positive number in there.

00:16:09.180 | Anyway, rectangle of numbers, a matrix is basic.

00:16:15.160 | So the big problem is to understand all those numbers.

00:16:19.200 | You got a big, big set of numbers,

00:16:22.000 | and what are the patterns, what's going on?

00:16:24.640 | And so one of the ways to break down

00:16:29.640 | that matrix into simple pieces

00:16:32.680 | is uses something called singular values,

00:16:37.120 | and that's come on as fundamental in the last,

00:16:42.120 | and certainly in my lifetime.

00:16:45.520 | Eigenvalues, if you have viewers

00:16:48.560 | who've done engineering math or basic linear algebra,

00:16:53.560 | eigenvalues were in there,

00:16:56.120 | but those are restricted to square matrices,

00:17:00.160 | and data comes in rectangular matrices,

00:17:02.720 | so you gotta take that next step.

00:17:05.720 | I'm always pushing math faculty, get on, do it, do it,

00:17:10.720 | do it, singular values.

00:17:14.120 | So those are a way to break,

00:17:16.720 | to find the important pieces of the matrix,

00:17:22.720 | which add up to the whole matrix.

00:17:24.840 | So you're breaking a matrix into simple pieces,

00:17:29.040 | and the first piece is the most important part of the data,

00:17:33.240 | the second piece is the second most important part,

00:17:36.120 | and then often, so a data scientist will,

00:17:40.600 | like if a data scientist can find

00:17:44.800 | those first and second pieces, stop there.

00:17:47.840 | The rest of the data is probably round off,

00:17:52.840 | experimental error maybe,

00:17:57.640 | so you're looking for the important part.

00:18:00.160 | - So what do you find beautiful about singular values?

00:18:03.000 | - Well, yeah, I didn't give the theorem,

00:18:06.280 | so here's the idea of singular values.

00:18:09.400 | Every matrix, every matrix, rectangular, square, whatever,

00:18:14.400 | can be written as a product

00:18:17.000 | of three very simple special matrices,

00:18:20.040 | so that's the theorem.

00:18:21.320 | Every matrix can be written as a rotation times a stretch,

00:18:26.240 | which is just a matrix, a diagonal matrix,

00:18:30.320 | otherwise all zeros except on the one diagonal,

00:18:34.200 | and then the third factor is another rotation.

00:18:38.000 | So rotation, stretch, rotation is the breakup of any matrix.

00:18:43.000 | - The structure of that, the ability that you can do that,

00:18:48.440 | what do you find appealing?

00:18:49.840 | What do you find beautiful about it?

00:18:51.040 | - Well, geometrically, as I freely admit,

00:18:54.240 | the action of a matrix is not so easy to visualize,

00:18:59.240 | but everybody can visualize a rotation.

00:19:02.360 | Take two-dimensional space and just turn it

00:19:07.040 | around the center.

00:19:09.240 | Take three-dimensional space,

00:19:10.720 | so a pilot has to know about, well, what are the three,

00:19:15.200 | the yaw is one of them, I've forgotten all the three turns

00:19:19.400 | that a pilot makes.

00:19:20.740 | Up to 10 dimensions, you've got 10 ways to turn,

00:19:25.400 | but you can visualize a rotation.

00:19:28.500 | Take the space and turn it, and you can visualize a stretch.

00:19:32.080 | So to break a matrix with all those numbers in it

00:19:37.080 | into something you can visualize,

00:19:41.080 | rotate, stretch, rotate, is pretty neat.

00:19:44.840 | - Yeah. - Pretty neat.

00:19:45.760 | - That's pretty powerful.

00:19:47.660 | On YouTube, just consuming a bunch of videos

00:19:52.000 | and just watching what people connect with

00:19:54.000 | and what they really enjoy and are inspired by,

00:19:57.320 | math seems to come up again and again.

00:20:01.520 | I'm trying to understand why that is.

00:20:03.960 | Perhaps you can help give me clues.

00:20:06.480 | So it's not just the kinds of lectures that you give,

00:20:10.760 | but it's also just other folks, like with Numberphile,

00:20:14.160 | there's a channel where they just chat about things

00:20:16.940 | that are extremely complicated, actually.

00:20:19.880 | People nevertheless connect with them.

00:20:21.920 | What do you think that is?

00:20:24.560 | - It's wonderful, isn't it?

00:20:25.800 | I mean, I wasn't really aware of it.

00:20:28.480 | We're conditioned to think math is hard, math is abstract,

00:20:33.480 | math is just for a few people, but it isn't that way.

00:20:36.600 | A lot of people quite like math, and they like to,

00:20:41.400 | I get messages from people saying,

00:20:43.800 | you know, now I'm retired, I'm gonna learn some more math.

00:20:46.800 | I get a lot of those.

00:20:47.960 | It's really encouraging.

00:20:49.920 | And I think what people like is that there's some order,

00:20:53.080 | you know, a lot of order,

00:20:54.600 | or things are not obvious, but they're true.

00:20:59.600 | So it's really cheering to think that so many people

00:21:05.400 | really wanna learn more about math, yeah.

00:21:08.960 | - In terms of truth, again,

00:21:11.880 | sorry to slide into philosophy at times,

00:21:15.520 | but math does reveal pretty strongly what things are true.

00:21:20.520 | I mean, that's the whole point of proving things.

00:21:24.360 | And yet, sort of our real world is messy and complicated.

00:21:29.360 | What do you think about the nature of truth

00:21:33.240 | that math reveals?

00:21:34.680 | - Oh, wow.

00:21:35.520 | - Because it is a source of comfort, like you've mentioned.

00:21:38.000 | - Yeah, that's right.

00:21:39.560 | Well, I have to say, I'm not much of a philosopher.

00:21:43.040 | I just like numbers, you know, as a kid.

00:21:45.880 | This was before you had to go in,

00:21:52.160 | when you had a filly in your teeth,

00:21:54.040 | you had to kind of just take it.

00:21:56.080 | So what I did was think about math,

00:21:59.040 | you know, like take powers of two, two, four, eight, 16,

00:22:03.120 | up until the time the tooth stopped hurting

00:22:05.960 | and the dentist said you're through.

00:22:07.720 | Or counting, yeah.

00:22:10.800 | - So that was a source of peace, almost.

00:22:14.840 | What is it about math, do you think, that brings that?

00:22:19.760 | What is that?

00:22:20.960 | - Well, you know where you are, yeah.

00:22:22.920 | It's symmetry, it's certainty.

00:22:25.880 | The fact that, you know, if you multiply two by itself

00:22:29.880 | 10 times, you get 1,024, period.

00:22:33.280 | Everybody's gonna get that.

00:22:34.960 | - Do you see math as a powerful tool or as an art form?

00:22:39.120 | - So it's both.

00:22:40.800 | That's really one of the neat things.

00:22:42.480 | You can be an artist and like math.

00:22:46.440 | You can be an engineer and use math.

00:22:51.000 | - Which are you?

00:22:51.960 | Which-- - Which am I?

00:22:53.560 | - What did you connect with most?

00:22:54.880 | - Yeah, I'm somewhere between.

00:22:57.400 | I'm certainly not a artist-type, philosopher-type person.

00:23:01.600 | Might sound that way this morning, but I'm not.

00:23:04.120 | (both laughing)

00:23:06.480 | Yeah, I really enjoy teaching engineers

00:23:09.720 | because they go for an answer.

00:23:13.280 | And yeah, so probably within the MIT math department,

00:23:20.400 | most people enjoy teaching students

00:23:24.480 | who get the abstract idea.

00:23:26.920 | I'm okay with, I'm good with engineers

00:23:31.920 | who are looking for a way to find answers, yeah.

00:23:35.480 | - Actually, that's an interesting question.

00:23:37.720 | Do you think for teaching and in general,

00:23:41.360 | thinking about new concepts,

00:23:42.720 | do you think it's better to plug in the numbers

00:23:45.480 | or to think more abstractly?

00:23:49.040 | So looking at theorems and proving the theorems

00:23:53.640 | or actually building up a basic intuition of the theorem

00:23:58.240 | or the method, the approach,

00:23:59.920 | and then just plugging in numbers and seeing it work?

00:24:02.960 | - Yeah, well, certainly many of us like to see examples.

00:24:07.960 | First, we understand,

00:24:11.040 | it might be a pretty abstract-sounding example,

00:24:13.960 | like a three-dimensional rotation.

00:24:16.800 | How are you gonna understand a rotation in 3D or in 10D?

00:24:21.800 | And then some of us like to keep going with it

00:24:30.840 | to the point where you got numbers,

00:24:32.720 | where you got 10 angles, 10 axes, 10 angles.

00:24:37.120 | But the best, the great mathematicians probably,

00:24:43.120 | I don't know if they do that

00:24:44.760 | 'cause for them, an example

00:24:49.760 | would be a highly abstract thing to the rest of us.

00:24:54.680 | - Right, but nevertheless,

00:24:55.760 | working in the space of examples.

00:24:57.560 | - Yeah, examples. - It seems to--

00:24:59.200 | - Examples of structure.

00:25:01.840 | - Our brains seem to connect with that.

00:25:03.680 | - Yeah, yeah.

00:25:04.680 | - So I'm not sure if you're familiar with him,

00:25:07.240 | but Andrew Yang is a presidential candidate

00:25:11.840 | currently running with math in all capital letters

00:25:16.840 | and his hats as a slogan.

00:25:18.840 | - I see.

00:25:19.680 | - Stands for Make America Think Hard.

00:25:21.720 | - Okay, I'll vote for him.

00:25:23.880 | - And his name rhymes with yours, Yang, Strang.

00:25:28.680 | But he also loves math, and he comes from that world.

00:25:31.400 | But he also, looking at it,

00:25:35.520 | makes me realize that math, science, and engineering

00:25:38.600 | are not really part of our politics,

00:25:42.400 | political discourse about political,

00:25:44.600 | government in general.

00:25:46.160 | Why do you think that is?

00:25:49.480 | What are your thoughts on that in general?

00:25:51.200 | - Well, certainly somewhere in the system,

00:25:52.760 | we need people who are comfortable with numbers,

00:25:56.880 | comfortable with quantities.

00:25:59.480 | If you say this leads to that, they see it,

00:26:04.400 | it's undeniable.

00:26:06.000 | - But isn't it strange to you

00:26:07.400 | that we have almost no, I mean, I'm pretty sure

00:26:11.480 | we have no elected officials in Congress

00:26:15.920 | or obviously the president that either

00:26:20.720 | has an engineering degree or a math degree.

00:26:23.000 | - Yeah, well, that's too bad.

00:26:25.160 | A few who could make the connection,

00:26:30.720 | yeah, it would have to be people

00:26:32.560 | who understand engineering or science

00:26:36.840 | and at the same time can make speeches and lead, yeah.

00:26:41.840 | - And inspire people.

00:26:45.560 | - Inspire, yeah.

00:26:46.640 | - You were, speaking of inspiration,

00:26:49.320 | the president of the Society

00:26:50.600 | for Industrial and Applied Mathematics.

00:26:52.920 | - Oh, yes.

00:26:53.760 | - It's a major organization in math, in applied math.

00:26:57.960 | What do you see as a role of that society

00:27:00.560 | in our public discourse?

00:27:02.880 | - Right, yeah, so, well, it was fun to be president

00:27:06.440 | at the time.

00:27:07.280 | (laughing)

00:27:08.120 | - A couple years, a few years.

00:27:09.440 | - Two years, around 2000.

00:27:12.120 | So, that's the president of a pretty small society,

00:27:16.840 | but nevertheless, it was a time when math

00:27:19.640 | was getting some more attention in Washington.

00:27:24.400 | But yeah, I got to give a little 10 minutes

00:27:29.240 | to a committee of the House of Representatives

00:27:33.920 | talking about why math.

00:27:35.320 | And then, actually, it was fun

00:27:37.000 | because one of the members of the House

00:27:40.800 | had been a student, had been in my class.

00:27:44.880 | What do you think of that?

00:27:46.080 | Yeah, as you say, a pretty rare,

00:27:47.720 | most members of the House have had a different training,

00:27:51.880 | different background, but there was one from New Hampshire

00:27:56.360 | who was my friend, really, by being in the class.

00:28:02.480 | Yeah, so those years were good.

00:28:05.800 | Then, of course, other things take over

00:28:10.600 | in importance in Washington, and math,

00:28:13.600 | math just, at this point, is not so visible,

00:28:18.240 | but for a little moment, it was.

00:28:20.240 | - There's some excitement, some concern

00:28:23.780 | about artificial intelligence in Washington now.

00:28:26.320 | - Yes, sure. - About the future.

00:28:27.480 | - Yeah. - And I think at the core

00:28:28.820 | of that is math.

00:28:30.040 | - Well, it is, yeah.

00:28:32.040 | Maybe it's hidden, maybe it's wearing a different hat.

00:28:34.280 | - Well, artificial intelligence,

00:28:37.760 | and particularly, can I use the words deep learning?

00:28:41.640 | It's a deep learning, is a particular approach

00:28:44.280 | to understanding data.

00:28:47.620 | Again, you've got a big, whole lot of data

00:28:50.200 | where data is just swamping the computers of the world,

00:28:56.060 | and to understand it, out of all those numbers,

00:29:00.680 | to find what's important in climate, in everything.

00:29:05.200 | And artificial intelligence is two words

00:29:08.520 | for one approach to data.

00:29:11.700 | Deep learning is a specific approach there,

00:29:15.560 | which uses a lot of linear algebra.

00:29:17.440 | So I got into it.

00:29:19.280 | I thought, okay, I've gotta learn about this.

00:29:21.600 | - So maybe from your perspective,

00:29:24.120 | let me ask the most basic question.

00:29:27.520 | - Yeah. - How do you think

00:29:28.580 | of a neural network?

00:29:30.360 | What is a neural network?

00:29:31.720 | - Yeah, okay.

00:29:32.640 | So can I start with the idea about deep learning?

00:29:37.240 | What does that mean?

00:29:38.080 | - Sure.

00:29:38.900 | What is deep learning?

00:29:39.920 | - What is deep learning, yeah.

00:29:42.000 | So we're trying to learn, from all this data,

00:29:46.280 | we're trying to learn what's important.

00:29:47.880 | What's it telling us?

00:29:50.280 | So you've got data.

00:29:53.040 | You've got some inputs for which you know the right outputs.

00:29:57.620 | The question is, can you see the pattern there?

00:30:02.120 | Can you figure out a way for a new input,

00:30:04.640 | which we haven't seen,

00:30:06.200 | to understand what the output will be from that new input?

00:30:12.200 | So we've got a million inputs with their outputs.

00:30:15.960 | So we're trying to create some pattern,

00:30:19.240 | some rule that'll take those inputs,

00:30:22.200 | those million training inputs, which we know about,

00:30:25.560 | to the correct million outputs.

00:30:28.180 | And this idea of a neural net is part of the structure

00:30:33.180 | of our new way to create a rule.

00:30:39.820 | We're looking for a rule that will take these training inputs

00:30:45.700 | to the known outputs.

00:30:48.460 | And then we're gonna use that rule on new inputs

00:30:51.680 | that we don't know the output and see what comes.

00:30:56.080 | - Linear algebra is a big part of finding that rule.

00:30:59.120 | - That's right.

00:30:59.960 | Linear algebra is a big part.

00:31:01.860 | Not all the part.

00:31:03.520 | People were leaning on matrices.

00:31:06.200 | That's good, still do.

00:31:08.280 | Linear is something special.

00:31:10.320 | It's all about straight lines and flat planes.

00:31:13.560 | And data isn't quite like that.

00:31:18.880 | It's more complicated.

00:31:21.240 | So you gotta introduce some complication.

00:31:23.720 | So you have to have some function

00:31:25.480 | that's not a straight line.

00:31:27.480 | And it turned out-- - Non-linear.

00:31:28.960 | - Non-linear, non-linear, non-linear.

00:31:31.640 | And it turned out that it was enough to use the function

00:31:35.880 | that's one straight line and then a different one.

00:31:38.360 | Halfway, so piecewise linear.

00:31:40.920 | - Piecewise linear.

00:31:41.960 | - One piece has one slope, one piece,

00:31:45.000 | the other piece has the second slope.

00:31:47.360 | And so getting that non-linear,

00:31:52.360 | simple non-linearity in blew the problem open.

00:31:56.720 | - That little piece makes it sufficiently complicated

00:31:59.000 | to make things interesting.

00:32:00.480 | - 'Cause you're gonna use that piece

00:32:02.040 | over and over a million times.

00:32:03.840 | So it has a fold in the graph, the graph, two pieces.

00:32:08.840 | But when you fold something a million times,

00:32:13.720 | you've got a pretty complicated function

00:32:17.880 | that's pretty realistic.

00:32:19.280 | - So that's the thing about neural networks

00:32:21.120 | is they have a lot of these.

00:32:23.920 | - A lot of these, that's right.

00:32:25.200 | - So why do you think neural networks,

00:32:29.660 | by using a, sort of formulating an objective function,

00:32:34.660 | very not a plain function-- - Lots of folds.

00:32:39.440 | - Lots of folds of the inputs, the outputs.

00:32:42.380 | Why do you think they work to be able to find a rule

00:32:47.360 | that we don't know is optimal,

00:32:48.840 | but is just seems to be pretty good in a lot of cases?

00:32:53.360 | What's your intuition?

00:32:54.600 | Is it surprising to you as it is to many people?

00:32:58.200 | Do you have an intuition of why this works at all?

00:33:01.160 | - Well, I'm beginning to have a better intuition.

00:33:04.340 | This idea of things that are piecewise linear,

00:33:08.560 | flat pieces but with folds between them.

00:33:12.160 | Like think of a roof of a complicated,

00:33:15.000 | infinitely complicated house or something.

00:33:17.800 | That curved, it almost curved, but every piece is flat.

00:33:22.800 | That's been used by engineers.

00:33:26.840 | That idea's been used by engineers, is used by engineers,

00:33:31.120 | big time, something called the finite element method.

00:33:34.200 | If you wanna design a bridge, design a building,

00:33:38.160 | design an airplane, you're using this idea

00:33:43.160 | of piecewise flat as a good, simple,

00:33:48.880 | computable approximation.

00:33:52.400 | - But you have a sense that there's a lot

00:33:56.160 | of expressive power in this kind of piecewise linear--

00:33:58.560 | - Yeah, that's-- - Combined together.

00:34:00.000 | - You used the right word.

00:34:01.800 | If you measure the expressivity,

00:34:05.360 | how complicated a thing can this piecewise flat guys express,

00:34:10.360 | the answer is very complicated, yeah.

00:34:15.480 | - What do you think are the limits

00:34:17.480 | of such piecewise linear or just of neural networks,

00:34:22.720 | the expressivity of neural networks?

00:34:24.160 | - Well, you would have said a while ago

00:34:26.680 | that they're just computational limits.

00:34:31.440 | A problem beyond a certain size,

00:34:33.760 | a supercomputer isn't gonna do it.

00:34:36.120 | But those keep getting more powerful,

00:34:39.440 | so that limit has been moved

00:34:42.720 | to allow more and more complicated surfaces.

00:34:47.480 | - So in terms of just mapping from inputs to outputs,

00:34:52.480 | looking at data, what do you think of,

00:34:55.960 | in the context of neural networks in general,

00:35:00.560 | data is just tensor vectors, matrices, tensors.

00:35:04.240 | - Right.

00:35:05.080 | - How do you think about learning from data?

00:35:10.320 | How much of our world can be expressed in this way?

00:35:12.800 | How useful is this process?

00:35:16.560 | I guess that's another way to ask you,

00:35:18.000 | what are the limits of this approach?

00:35:19.400 | - Well, that's a good question, yeah.

00:35:21.440 | So I guess the whole idea of deep learning

00:35:24.240 | is that there's something there to learn.

00:35:26.260 | If the data is totally random,

00:35:28.520 | just produced by random number generators,

00:35:31.400 | then we're not gonna find a useful rule,

00:35:36.280 | 'cause there isn't one.

00:35:37.440 | So the extreme of having a rule

00:35:41.560 | is like knowing Newton's law, you know?

00:35:43.600 | If you hit a ball, it moves.

00:35:46.240 | So that's where you had laws of physics.

00:35:48.960 | Newton and Einstein and other great, great people

00:35:53.960 | have found those laws,

00:35:56.960 | and laws of the distribution of oil

00:36:01.960 | in an underground thing.

00:36:05.920 | I mean, so engineers, petroleum engineers,

00:36:10.920 | understand how oil will sit in an underground basin.

00:36:17.080 | So there were rules.

00:36:20.040 | Now, the new idea of artificial intelligence

00:36:24.720 | is learn the rules.

00:36:26.840 | Instead of figuring out the rules

00:36:29.920 | with help from Newton or Einstein,

00:36:32.720 | the computer is looking for the rules.

00:36:35.640 | So that's another step.

00:36:36.880 | But if there are no rules at all

00:36:39.880 | that the computer could find,

00:36:41.240 | if it's totally random data,

00:36:43.600 | well, you've got nothing.

00:36:45.320 | You've got no science to discover.

00:36:48.320 | - It's an automated search for the underlying rules.

00:36:51.400 | - Yeah, search for the rules, yeah, exactly.

00:36:54.960 | And there will be a lot of random parts,

00:36:57.800 | a lot, I'm not knocking random,

00:36:59.880 | 'cause that's there.

00:37:02.740 | There's a lot of randomness built in,

00:37:07.360 | but there's gotta be some basic--

00:37:09.400 | - It's almost always signal, right?

00:37:10.920 | In most-- - There's gotta be some signal,

00:37:12.600 | yeah, if it's all noise,

00:37:13.760 | then you're not gonna get anywhere.

00:37:17.440 | - Well, this world around us does seem to be,

00:37:19.920 | does seem to always have a signal of some kind

00:37:22.440 | to be discovered.

00:37:24.280 | - Right, that's it.

00:37:25.920 | - So what excites you more?

00:37:29.480 | We just talked about a little bit of application.

00:37:32.880 | What excites you more, theory

00:37:34.560 | or the application of mathematics?

00:37:38.400 | - Well, for myself, I'm probably a theory person.

00:37:43.400 | I'm speaking here pretty freely about applications,

00:37:49.720 | but I'm not the person who really,

00:37:53.240 | I'm not a physicist or a chemist or a neuroscientist.

00:37:57.040 | So for myself, I like the structure

00:38:03.080 | and the flat subspaces

00:38:06.480 | and the relation of matrices, columns to rows.

00:38:11.480 | That's my part in the spectrum.

00:38:17.280 | So really, science is a big spectrum

00:38:21.160 | of people from asking practical questions

00:38:25.760 | and answering them using some math,

00:38:28.760 | then some math guys like myself

00:38:31.880 | who are in the middle of it,

00:38:33.800 | and then the geniuses of math and physics and chemistry

00:38:38.800 | who are finding fundamental rules

00:38:43.360 | and then doing the really understanding nature.

00:38:50.120 | That's incredible.

00:38:51.880 | - At its lowest, simplest level.

00:38:55.000 | Maybe just a quick and broad strokes from your perspective.

00:38:58.640 | Where does linear algebra sit as a subfield of mathematics?

00:39:04.760 | What are the various subfields that you think about

00:39:09.760 | in relation to linear algebra?

00:39:12.200 | - So the big fields of math are algebra as a whole

00:39:18.040 | and problems like calculus and differential equations.

00:39:21.360 | So that's a second, quite different field.

00:39:24.360 | Then maybe geometry deserves to be thought of

00:39:28.800 | as a different field to understand the geometry

00:39:31.560 | of high dimensional surfaces.

00:39:34.120 | So I think, am I allowed to say this here?

00:39:39.720 | I think-- - Uh-oh, calculus.

00:39:41.640 | - This is where personal view comes in.

00:39:46.240 | I think math, thinking about undergraduate math,

00:39:51.240 | what millions of students study,

00:39:54.320 | I think we overdo the calculus at the cost of the algebra,

00:39:59.320 | at the cost of linear.

00:40:02.720 | - See, I have this talk titled Calculus Versus Linear Algebra.

00:40:05.360 | - That's right, that's right.

00:40:07.440 | - And you say that linear algebra wins.

00:40:09.480 | So can you dig into that a little bit?

00:40:13.840 | Why does linear algebra win?

00:40:17.080 | - Right, well, okay, the viewer is gonna think

00:40:21.200 | this guy is biased.

00:40:22.760 | Not true, I'm just telling the truth as it is.

00:40:27.080 | Yeah, so I feel linear algebra is just a nice part of math

00:40:31.920 | that people can get the idea of.

00:40:34.420 | They can understand something that's a little bit abstract

00:40:37.760 | 'cause once you get to 10 or 100 dimensions.

00:40:42.160 | And very, very, very useful.

00:40:44.400 | That's what's happened in my lifetime

00:40:47.840 | is the importance of data,

00:40:52.520 | which does come in matrix form,

00:40:54.680 | so it's really set up for algebra.

00:40:56.680 | It's not set up for differential equation.

00:40:59.280 | And let me fairly add probability,

00:41:03.320 | the ideas of probability and statistics

00:41:06.880 | have become very, very important, have also jumped forward.

00:41:11.240 | So, and that's different from linear algebra,

00:41:14.040 | quite different.

00:41:15.160 | So now we really have three major areas to me,

00:41:20.160 | calculus, linear algebra, matrices,

00:41:25.200 | and probability statistics.

00:41:29.000 | And they all deserve an important place.

00:41:34.000 | And calculus has traditionally had a lion's share

00:41:39.000 | of the time.

00:41:40.860 | - Disproportionate share.

00:41:42.280 | - Thank you, disproportionate, that's a good word.

00:41:44.760 | - Of the love and attention from the excited young minds.

00:41:50.120 | I know it's hard to pick favorites,

00:41:55.480 | but what is your favorite matrix?

00:41:57.720 | - What's my favorite matrix?

00:41:59.400 | Okay, so my favorite matrix is square, I admit it.

00:42:03.200 | It's a square bunch of numbers,

00:42:05.480 | and it has twos running down the main diagonal.

00:42:10.200 | And on the next diagonal,

00:42:13.060 | so think of top left to bottom right,

00:42:15.380 | twos down the middle of the matrix,

00:42:18.900 | and minus ones just above those twos,

00:42:22.140 | and minus ones just below those twos,

00:42:25.020 | and otherwise all zeros.

00:42:26.620 | So mostly zeros, just three non-zero diagonals coming down.

00:42:31.620 | - What is interesting about it?

00:42:34.380 | - Well, all the different ways it comes up.

00:42:37.200 | You see it in engineering,

00:42:39.220 | you see it as analogous in calculus to second derivative.

00:42:44.100 | So calculus learns about taking the derivative,

00:42:47.620 | figuring out how fast something's changing.

00:42:50.640 | But second derivative, now that's also important.

00:42:55.740 | That's how fast the change is changing,

00:42:58.760 | how fast the graph is bending,

00:43:02.060 | how fast it's curving.

00:43:06.460 | And Einstein showed that that's fundamental

00:43:10.140 | to understand space.

00:43:11.560 | So second derivatives should have a bigger place

00:43:15.980 | in calculus.

00:43:17.380 | Second, my matrices, which are like the linear algebra

00:43:22.380 | version of second derivatives, are neat in linear algebra.

00:43:29.380 | Yeah, just everything comes out right with those guys.

00:43:32.820 | - Beautiful.

00:43:35.220 | What did you learn about the process of learning

00:43:38.380 | by having taught so many students math over the years?

00:43:42.820 | - Ooh, that is hard.

00:43:44.820 | I'll have to admit here that I'm not really a good teacher

00:43:50.680 | because I don't get into the exam part.

00:43:55.540 | The exam's the part of my life that I don't like,

00:43:59.000 | and grading them, and giving the students A or B,

00:44:03.060 | or whatever.

00:44:04.360 | I do it because I'm supposed to do it,

00:44:08.260 | but I tell the class at the beginning,

00:44:11.900 | I don't know if they believe me, probably they don't,

00:44:14.580 | I tell the class, I'm here to teach you.

00:44:18.040 | I'm here to teach you math and not to grade you.

00:44:21.660 | But they're thinking, okay, this guy,

00:44:25.100 | when's he gonna, is he gonna give me an A minus,

00:44:28.820 | is he gonna give me a B plus, what?

00:44:31.420 | - What have you learned about the process of learning?

00:44:34.060 | - Of learning, yeah, well maybe,

00:44:36.860 | to give you a legitimate answer about learning,

00:44:41.400 | I should have paid more attention to the assessment,

00:44:45.620 | the evaluation part at the end.

00:44:47.720 | But I like the teaching part at the start,

00:44:50.020 | that's the sexy part, to tell somebody

00:44:53.120 | for the first time about a matrix, wow.

00:44:56.120 | - Is there, are there moments,

00:44:58.740 | so you are teaching a concept,

00:45:01.920 | are there moments of learning that you just see

00:45:05.460 | in the student's eyes, you don't need to look at the grades,

00:45:08.220 | but you see in their eyes that you hook them,

00:45:11.540 | that you connect with them in a way where,

00:45:16.260 | you know what, they fall in love with this beautiful world

00:45:20.980 | of math or this-- - They see that it's got

00:45:22.500 | some beauty there. - Yeah, yeah.

00:45:24.460 | - Or conversely, that they give up at that point,

00:45:28.060 | is the opposite, the dark is saying that math,

00:45:31.180 | I'm just not good at math, I don't wanna walk away.

00:45:33.340 | - Yeah, yeah, yeah.

00:45:34.300 | Maybe because of the approach in the past,

00:45:37.700 | they were discouraged, but don't be discouraged,

00:45:40.500 | it's too good to miss.

00:45:43.080 | Yeah, well if I'm teaching a big class,

00:45:48.420 | do I know when, I think maybe I do,

00:45:51.900 | sort of, I mentioned at the very start,

00:45:55.460 | the four fundamental subspaces and the structure

00:46:00.620 | of the fundamental theorem of linear algebra,

00:46:04.740 | the fundamental theorem of linear algebra,

00:46:06.780 | that is the relation of those four subspaces,

00:46:11.780 | those four spaces, yeah, so I think that,

00:46:15.580 | I feel that the class gets it.

00:46:17.740 | - When they see it. - Yeah.

00:46:19.940 | - What advice do you have to a student

00:46:22.420 | just starting their journey in mathematics today?

00:46:25.140 | How do they get started?

00:46:26.980 | (laughing)

00:46:27.820 | - Now-- - Yeah, that's hard.

00:46:30.100 | - Well, I hope you have a teacher,

00:46:33.980 | professor who is still enjoying what he's doing,

00:46:38.980 | what he's teaching, still looking for new ways

00:46:43.300 | to teach and to understand math,

00:46:46.680 | 'cause that's the pleasure,

00:46:49.620 | the moment when you see, oh yeah, that works.

00:46:54.980 | - So it's less about the material you--

00:46:57.180 | - Yeah. - You study,

00:46:58.500 | it's more about the source of the teacher

00:47:02.500 | being full of passion for--

00:47:03.940 | - Yeah, more about the fun, yeah.

00:47:06.260 | - The fun. - The moment of getting it.

00:47:10.540 | - But in terms of topics, linear algebra?

00:47:14.180 | - Well, that's my topic, but oh,

00:47:18.020 | there's beautiful things in geometry to understand.

00:47:21.260 | What's wonderful is that in the end,

00:47:25.700 | there's a pattern, there are rules

00:47:28.660 | that are followed in biology as there are in every field.

00:47:33.660 | - You describe the life of a mathematician

00:47:41.460 | as 100% wonderful.

00:47:44.300 | (laughing)

00:47:45.660 | Except for the grade stuff, having the good grades.

00:47:48.180 | - Except for grades.

00:47:49.020 | - Yeah, when you look back at your life,

00:47:52.180 | what memories bring you the most joy and pride?

00:47:55.980 | - Well, that's a good question.

00:47:58.260 | I certainly feel good when I,

00:48:01.580 | maybe I'm giving a class in 1806,

00:48:06.180 | that's MIT's linear algebra course that I started.

00:48:09.420 | So sort of, there's a good feeling that,

00:48:11.660 | okay, I started this course, a lot of students take it,

00:48:15.740 | quite a few like it, yeah.

00:48:17.780 | So I'm sort of happy when I feel I'm helping

00:48:24.100 | make a connection between ideas and students,

00:48:27.780 | between theory and the reader.

00:48:31.900 | Yeah, I get a lot of very nice messages

00:48:37.380 | from people who've watched the videos and it's inspiring.

00:48:43.580 | I just, I'll maybe take this chance to say thank you.

00:48:47.260 | - Well, there's millions of students who you've taught

00:48:51.540 | and I am grateful to be one of them.

00:48:54.300 | So Gilbert, thank you so much.

00:48:55.820 | It's been an honor.

00:48:56.660 | Thank you for talking today.

00:48:58.220 | - It was a pleasure, thanks.

00:48:59.820 | - Thank you for listening to this conversation

00:49:02.620 | with Gilbert Strang.

00:49:04.340 | And thank you to our presenting sponsor, Cash App.

00:49:07.460 | Download it, use code LexPodcast,

00:49:10.060 | you'll get $10 and $10 will go to FIRST,

00:49:12.860 | a STEM education nonprofit that inspires hundreds

00:49:15.820 | of thousands of young minds to learn

00:49:18.220 | and to dream of engineering our future.

00:49:20.780 | If you enjoy this podcast, subscribe on YouTube.

00:49:23.860 | We have five stars on Apple Podcast, support on Patreon

00:49:27.300 | or connect with me on Twitter.

00:49:29.380 | Finally, some closing words of advice

00:49:31.940 | from the great Richard Feynman.

00:49:34.020 | Study hard what interests you the most

00:49:36.380 | in the most undisciplined, irreverent

00:49:39.060 | and original manner possible.

00:49:41.300 | Thank you for listening and hope to see you next time.

00:49:44.340 | (upbeat music)

00:49:46.940 | (upbeat music)

00:49:49.540 | [BLANK_AUDIO]

Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare | Lex Fridman Podcast #52

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