The following is a conversation with Gilbert Strang.

He's a professor of mathematics at MIT and perhaps 1 of the most famous and impactful teachers of math in the world.

His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times.

As an undergraduate student, I was 1 of those millions of students.

There's something inspiring about the way he teaches.

There's at once calm, simple, and yet full of passion for the elegance inherent to mathematics.

I remember doing the exercises in his book, Introduction to Linear Algebra, and slowly realizing that the world of matrices, of vector spaces, of determinants and eigenvalues, of geometric transformations and matrix decompositions reveal a set of powerful tools in the toolbox of artificial intelligence.

From signals to images, from numerical optimization to robotics, computer vision, deep learning, computer graphics, and everywhere outside AI, including, of course, a quantum mechanical study of our universe.

This is the Artificial Intelligence Podcast.

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And now, here's my conversation with Gilbert Strang.

How does it feel to be 1 of the modern day rock stars of mathematics?

That's kind of crazy for an old math person.

But it's true that the videos in linear algebra that I made way back in 2000, I think.

And well, partly the importance of linear algebra, which I'm sure you'll ask me and give me a chance to say that linear algebra as a subject has just surged in importance.

But also, it was a class that I taught a bunch of times, so I kind of got it organized and enjoyed doing it.

It was just the videos were just the class.

So they're on OpenCourseWare and on YouTube and translated, that's fun.

But there's something about that chalkboard and the simplicity of the way you explain the basic concepts in the beginning.

But before going through the course at my university, I was going through open course where I was, you were my instructor

And that, I mean, we were using your book.

And I mean, the fact that there is thousands, you know, hundreds of thousands, millions of people that watch that video, I think that's really powerful.

So how do you think the idea of putting lectures online, what really MIT OpenCourseWare has innovated?

You know, I think The story that I've heard is the committee was appointed by the president, President Vest at that time, a wonderful guy.

And the idea of the committee was to figure out how MIT could be like other universities, market the work we were doing.

And then they didn't see a way, and after a weekend, and they had an inspiration, came back to the present vest and said, what if we just gave it away?

That's, if we think of a university as a thing that creates a product.

The kind of educational knowledge isn't the product and giving that away.

Well, knowing a little bit President vest, it was like him, I think.

MIT is a kind of, it's known for being high-level, technical things, And this is the best way we can say, tell, we can show what MIT really is like.

Because in my case, those 1806 videos are just teaching the class.

People write to me and say, oh, you've got a sense of humor, but I don't know where that comes through.

And linear algebra, the subject, we gotta give the subject most of the credit.

It really has come forward in importance in these years.

So let's talk about linear algebra a little bit, because it is such a, it's both a powerful and a beautiful subfield of mathematics.

So what's your favorite specific topic in linear algebra, or even math in general, to give a lecture on, to convey, to tell a story, to teach students.

Okay, well, on the teaching side, so it's not deep mathematics at all, but I'm kind of proud of the idea of the 4 subspaces, the 4 fundamental subspaces, which are of course known before, long before my name for them, but.

So the first 1 to understand is, so the matrix, maybe I should say the matrix.

Well, so we have like a rectangle of numbers.

So it's got N columns, got a bunch of columns, and also got an m rows, let's say, and the relation between, so of course the columns and the rows, it's the same numbers, so there's got to be connections there, but they're not simple.

The columns might be longer than the rows and they're all different, the numbers are mixed up.

First space to think about is, take the columns, so those are vectors, those are points in n dimensions.

So a physicist would imagine a vector or might imagine a vector as a arrow in space or the point it ends at in space.

You often think of, this is very interesting in terms of linear algebra, in terms of a vector.

You think a little bit more abstract than how it's very commonly used perhaps.

You think this arbitrary space, multidimensional space.

Yeah, that's right, in the lecture I try to, so if you think of 2 vectors in 10 dimensions, I'll do this in class, and I'll readily admit that I have no good image in my mind of a vector, of a arrow in 10 dimensional space, but whatever.

You can add 1 bunch of 10 numbers to another bunch of 10 numbers, so you can add a vector to a vector, and you can multiply a vector by 3, and that's, if you know how to do those, you've got linear algebra.

You know, There's this beautiful thing about math, if we look at string theory and all these theories which are really fundamentally derived through math, but are very difficult to visualize.

How do you think about the things, like a 10-dimensional vector, that we can't really visualize?

And yet, math reveals some beauty underlying our world in that weird thing we can't visualize.

Well, probably I'm not a very geometric person, so I'm probably thinking in 3 dimensions.

And the beauty of linear algebra is that it goes on to 10 dimensions with no problem.

I mean, if you're just seeing what happens if you add 2 vectors in 3D.

So I can't say that I have a picture, but yet I try to push the class to think of a flat surface in 10 dimensions.

Take all the columns of the matrix, take all their combinations, so much of this column, so much of this 1, then if you put all those together you get some kind of a flat surface that I call a vector space, space of vectors.

And my imagination is just seeing like a piece of paper in 3D.

But anyway, so that's 1 of the spaces, that's space number 1, the column space of the matrix.

And then there's the row space, which is, as I said, different, but came from the same numbers.

So we got the column space, all combinations of the columns, and then we got the row space, all combinations of the rows.

So Those words are easy for me to say and I can't really draw them on a blackboard, but I try with my thick chalk.

Everybody likes that railroad chalk and me too.

Then the other 2 spaces are perpendicular to those.

So like if you have a plane in 3D, just a plane is just a flat surface in 3D, then perpendicular to that plane would be a line.

So we've got 2, we've got a column space, a row space, and there are 2 perpendicular spaces.

So those 4 fit together in a Beautiful picture of a matrix, yeah, yeah.

It's sort of a fundamental, it's not a difficult idea.

It comes pretty early in 1806 and it's basic.

planes in these multidimensional spaces, how difficult of an idea is that to come to, do you think?

If you look back in time, I think mathematically it makes sense, but I don't know if it's intuitive for us to imagine, just as we were talking about.

It feels like calculus is easier to intuit.

Well, I have to admit calculus came earlier, earlier than linear algebra.

So Newton and Leibniz were the great men to understand the key ideas of calculus.

But linear algebra to me is like, okay, it's the starting point because it's all about flat things.

Calculus has got all the complications of calculus come from the curves, the bending, the curved surfaces.

Linear algebra, the surfaces are all flat.

So it should have come first, but it didn't.

And calculus also comes first in high school classes, in college class, it'll be freshman math, it'll be calculus, and then I say, enough of it, like okay, get to the good stuff.

Do you think linear algebra should come first?

Well it really, I'm okay with it not coming first, but it should, yeah, it should.

Well, of course, for that reason, calculus sort of sticks to 1 dimension, or eventually you do multivariate, but that basically means 2 dimensions.

Linear algebra, you take off into 10 dimensions, no problem.

It just feels scary and dangerous to go beyond 2 dimensions, that's all.

Well, If everything is flat, you can't go wrong.

So what concept or theorem in linear algebra or in math you find most beautiful that gives you pause that leaves you in awe?

Well, I'll stick with linear algebra here.

I hope the viewer knows that really, mathematics is amazing, amazing subject and deep, deep connections between ideas that didn't look connected, they turned out they were.

But if we stick with linear algebra, so we have a matrix, that's like the basic thing, a rectangle of numbers.

And it might be a rectangle of data, you're probably gonna ask me later about data science, where often data comes in a matrix, you have maybe every column corresponds to a drug and every row corresponds to a patient and if the patient reacted favorably to the drug, then you put up some positive number in there.

Anyway, rectangle of numbers, a matrix is basic.

So the big problem is to understand all those numbers.

And so 1 of the ways to break down that matrix into simple pieces is uses something called singular values.

And that's come on as fundamental in the last, certainly in my lifetime.

Eigenvalues, if you have viewers who've done engineering math or basic linear algebra, eigenvalues were in there.

But those are restricted to square matrices and data comes in rectangular matrices.

I'm always pushing math faculty, get on, do it, singular values.

So those are a way to find the important pieces of the matrix, which add up to the whole matrix.

You're breaking a matrix into simple pieces.

The first piece is the most important part of the data, the second piece is the second most important part.

Then often, so a data scientist will like, if a data scientist can find those first and second pieces, stop there, The rest of the data is probably round off, experimental error maybe.

So you're looking for the important part.

So what do you find beautiful about singular values?

Every matrix, every matrix, rectangular, square, whatever, can be written as a product of 3 very simple special matrices.

Every matrix can be written as a rotation times a stretch, which is just a matrix, a diagonal matrix, otherwise all zeros except on the 1 diagonal.

And then the third factor is another rotation.

So rotation, stretch, rotation is the breakup of any matrix.

The structure of that, the ability that you can do that, what do you find appealing?

Well, geometrically, as I freely admit, the action of a matrix is not so easy to visualize, but Everybody can visualize a rotation.

Take two-dimensional space and just turn it around the center.

So a pilot has to know about, well, what are the 3, The yaw is 1 of them.

I've forgotten all the 3 turns that a pilot makes.

Up to 10 dimensions, you've got 10 ways to turn.

So to break a matrix with all those numbers in it into something you can visualize, rotate, stretch, rotate, is pretty neat.

On YouTube, just consuming a bunch of videos and just watching what people connect with and what they really enjoy and are inspired by, math seems to come up again and again.

So it's not just the kinds of lectures that you give, but it's also just other folks, like with Numberphile, there's a channel, where they just chat about things that are extremely complicated, actually.

We're conditioned to think math is hard, math is abstract, math is just for a few people, but it isn't that way.

A lot of people quite like math, and they like to, I get messages from people saying, you know, now I'm retired, I'm gonna learn some more math.

I get a lot of those, it's really encouraging.

And I think what people like is that there's some order, a lot of order, and things are not obvious, but they're true.

So it's really cheering to think that so many people really want to learn more about math, yeah.

In terms of truth, again, sorry to slide into philosophy at times, but math does reveal pretty strongly what things are true.

I mean, that's the whole point of proving things.

And yet, sort of our real world is messy and complicated.

What do you think about the nature of truth that math reveals?

Because it is a source of comfort like you've mentioned.

Well, I have to say, I'm not much of a philosopher.

This was before you had to go in when you had a filly in your teeth, you had to kind of just take it.

You know, like take powers of 2248, 16, up until the time the tooth stopped hurting and the dentist said you're through.

So that was a source of just, source of peace almost.

The, What is it about math do you think that brings that?

The fact that if you multiply 2 by itself 10 times you get 1,024 period.

Do you see math as a powerful tool or as an art form?

So it's both, that's really 1 of the neat things.

You can be an artist and like math, you can be a engineer and use math.

I'm certainly not a artist-type, philosopher-type person.

Might sound that way this morning, but I'm not.

And Yeah, so probably within the MIT math department, most people enjoy teaching students who get the abstract idea.

I'm okay with, I'm good with engineers who are looking for a way to find answers.

So actually that's an interesting question.

Do you think for teaching and in general, thinking about new concepts, do you think it's better to plug in the numbers or to think more abstractly?

So looking at theorems and proving the theorems or actually building up a basic intuition of the theorem or the method, the approach, and then just plugging in numbers and seeing it work?

Yeah, well, certainly many of us like to see examples.

First, we understand it might be a pretty abstract sounding example, like a three-dimensional rotation.

How are you gonna understand a rotation in 3D?

But, and then some of us like to keep going with it to the point where you got numbers, where you got 10 angles, 10 axes, 10 angles.

But the best, the great mathematicians probably, I don't know if they do that, because For them, an example would be a highly abstract thing to the rest of it.

Right, but nevertheless, it's working within the space of examples.

So I'm not sure if you're familiar with him, but Andrew Yang is a presidential candidate, currently running, with math in all capital letters and his hats as a slogan.

So, and his name rhymes with yours, Yang Strang.

But he also loves math and he comes from that world.

But he also, looking at it, makes me realize that math, science, and engineering are not really part of our politics, political discourse about political, government in general.

What are your thoughts on that in general?

Well, certainly somewhere in the system, we need people who are comfortable with numbers, comfortable with quantities.

You know, if you say this leads to that, they see it, it's undeniable.

But isn't it strange to you that we have almost no, I mean I'm pretty sure we have no elected officials in Congress or obviously the president that either has an engineering degree or a math degree.

A few who could make the connection, yeah, it would have to be people who understand engineering or science and at the same time can make speeches and lead, yeah.

You were, speaking of inspiration, the president of the Society for Industrial and Applied Mathematics.

That's a major organization in math, in applied math.

What do you see as a role of that society in our public discourse?

so, well it was fun to be president at the time.

So that's president of a pretty small society, but nevertheless it was a time when math was getting some more attention in Washington, but yeah, I got to give a little 10 minutes to a committee of the House of Representatives talking about who I met.

And then, actually it was fun because 1 of the members of the House had been a student, had been in my class.

Yeah, as you say, a pretty rare, most members of the house have had a different training, different background, but there was 1 from New Hampshire who was my friend really by being in the class.

Then, of course, other things take over and importance in Washington, and math just at this point is not so visible, but for a little moment it was.

There's some excitement, some concern about artificial intelligence in Washington now.

Maybe it's hidden, maybe it's wearing a different hat.

Well, artificial intelligence, and particularly can I use the words deep learning?

It's a deep learning, it's a particular approach to understanding data.

Again, you've got a big whole lot of data, where data is just swamping the computers of the world.

To understand it out of all those numbers to find what's important, in climate, in everything.

Artificial intelligence is 2 words for 1 approach to data.

Deep learning is a specific approach there which uses a lot of linear algebra, so I got into it.

I thought, okay, I've gotta learn about this.

So maybe from your perspective, let me ask the most basic question.

Yeah, okay, so can I start with the idea about deep learning, what does that mean?

So we're trying to learn, From all this data, we're trying to learn what's important, what's it telling us.

So you've got data, you've got some inputs for which you know the right outputs.

The question is, Can you see the pattern there?

Can you figure out a way for a new input, which we haven't seen, to understand what the output will be from that new input?

So we've got a million inputs with their outputs.

So we're trying to create some pattern, some rule that'll take those inputs, those million training inputs which we know about, to the correct million outputs.

And this idea of a neural net is part of the structure of our new way to create a rule.

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Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare | Lex Fridman Podcast #52