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


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The following is a conversation with Gilbert Strang.
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He's a professor of mathematics at MIT
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and perhaps one of the most famous
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and impactful teachers of math in the world.
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His MIT OpenCourseWare lectures on linear algebra
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have been viewed millions of times.
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As an undergraduate student,
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I was one of those millions of students.
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There's something inspiring about the way he teaches.
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There's at once calm, simple, and yet full of passion
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for the elegance inherent to mathematics.
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I remember doing the exercise in his book,
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Introduction to Linear Algebra,
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and slowly realizing that the world of matrices,
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of vector spaces, of determinants and eigenvalues,
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of geometric transformations and matrix decompositions
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reveal a set of powerful tools
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in the toolbox of artificial intelligence.
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computer vision, deep learning, computer graphics,
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and everywhere outside AI,
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including, of course, a quantum mechanical study
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of our universe.
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And now, here's my conversation with Gilbert Strang.
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How does it feel to be one of the modern day rock stars
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of mathematics?
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I don't feel like a rock star.
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That's kind of crazy for an old math person.
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But it's true that the videos in linear algebra
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that I made way back in 2000, I think,
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have been watched a lot.
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And well, partly the importance of linear algebra,
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which I'm sure you'll ask me,
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and give me a chance to say that linear algebra
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as a subject has just surged in importance.
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But also, it was a class that I taught a bunch of times,
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so I kind of got it organized and enjoyed doing it.
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The videos were just the class.
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So they're on OpenCourseWare and on YouTube
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and translated, and it's fun.
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But there's something about that chalkboard
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and the simplicity of the way you explain
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the basic concepts in the beginning.
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To be honest, when I went to undergrad.
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You didn't do linear algebra, probably.
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Of course I didn't do linear algebra.
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You did.
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Yeah, yeah, yeah, of course.
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But before going through the course at my university,
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there was going through OpenCourseWare.
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You were my instructor for linear algebra.
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Right, yeah.
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And that, I mean, we're using your book.
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And I mean, the fact that there is thousands,
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hundreds of thousands, millions of people
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that watch that video, I think that's really powerful.
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So how do you think the idea of putting lectures online,
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what really MIT OpenCourseWare has innovated?
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That was a wonderful idea.
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I think the story that I've heard is the committee
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was appointed by the president, President Vest,
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at that time, a wonderful guy.
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And the idea of the committee was to figure out
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how MIT could be like other universities,
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market the work we were doing.
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And then they didn't see a way.
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And after a weekend, and they had an inspiration,
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came back to President Vest and said,
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what if we just gave it away?
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And he decided that was okay, good idea.
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So.
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You know, that's a crazy idea.
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If we think of a university as a thing
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that creates a product, isn't knowledge,
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the kind of educational knowledge,
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isn't the product and giving that away,
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are you surprised that it went through?
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The result that he did it,
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well, knowing a little bit President Vest, it was like him,
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I think, and it was really the right idea.
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MIT is a kind of, it's known for being high level,
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technical things, and this is the best way we can say,
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tell, we can show what MIT really is like,
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because in my case, those 1806 videos
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are just teaching the class.
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They were there in 26, 100.
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They're kind of fun to look at.
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People write to me and say, oh, you've got a sense of humor,
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but I don't know where that comes through.
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Somehow I'm friendly with the class, I like students.
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And then your algebra, the subject,
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we gotta give the subject most of the credit.
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It really has come forward in importance in these years.
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So let's talk about linear algebra a little bit,
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because it is such a, it's both a powerful
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and a beautiful subfield of mathematics.
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So what's your favorite specific topic in linear algebra,
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or even math in general to give a lecture on,
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to convey, to tell a story, to teach students?
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Okay, well, on the teaching side,
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so it's not deep mathematics at all,
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but I'm kind of proud of the idea of the four subspaces,
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the four fundamental subspaces,
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which are of course known before,
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long before my name for them, but.
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Can you go through them?
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Can you go through the four subspaces?
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Sure I can, yeah.
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So the first one to understand is,
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so the matrix is, maybe I should say the matrix is.
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What is a matrix?
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What's a matrix?
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Well, so we have like a rectangle of numbers.
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So it's got n columns, got a bunch of columns,
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and also got an m rows, let's say,
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and the relation between,
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so of course the columns and the rows,
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it's the same numbers.
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So there's gotta be connections there,
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but they're not simple.
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The columns might be longer than the rows,
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and they're all different, the numbers are mixed up.
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First space to think about is take the columns,
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so those are vectors, those are points in n dimensions.
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What's a vector?
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So a physicist would imagine a vector
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or might imagine a vector as a arrow in space
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or the point it ends at in space.
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For me, it's a column of numbers.
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You often think of, this is very interesting
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in terms of linear algebra, in terms of a vector,
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you think a little bit more abstract
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than how it's very commonly used, perhaps.
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You think this arbitrary multidimensional space.
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Right away, I'm in high dimensions.
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Dreamland.
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Yeah, that's right.
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In the lecture, I try to,
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so if you think of two vectors in 10 dimensions,
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I'll do this in class, and I'll readily admit
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that I have no good image in my mind
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of a vector of an arrow in 10 dimensional space,
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but whatever.
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You can add one bunch of 10 numbers
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to another bunch of 10 numbers,
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so you can add a vector to a vector,
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and you can multiply a vector by three,
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and that's, if you know how to do those,
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you've got linear algebra.
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10 dimensions, there's this beautiful thing about math,
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if we look at string theory and all these theories,
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which are really fundamentally derived through math,
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but are very difficult to visualize.
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How do you think about the things,
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like a 10 dimensional vector,
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that we can't really visualize?
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And yet, math reveals some beauty underlying our world
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in that weird thing we can't visualize.
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How do you think about that difference?
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Well, probably, I'm not a very geometric person,
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so I'm probably thinking in three dimensions,
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and the beauty of linear algebra is that
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it goes on to 10 dimensions with no problem.
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I mean, that if you're just seeing what happens
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if you add two vectors in 3D,
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yeah, then you can add them in 10D.
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You're just adding the 10 components.
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So, I can't say that I have a picture,
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but yet I try to push the class
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to think of a flat surface in 10 dimensions.
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So a plane in 10 dimensions,
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and so that's one of the spaces.
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Take all the columns of the matrix,
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take all their combinations,
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so much of this column, so much of this one,
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then if you put all those together,
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you get some kind of a flat surface
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that I call a vector space, space of vectors.
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And my imagination is just seeing
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like a piece of paper in 3D, but anyway,
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so that's one of the spaces, that's space number one,
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the column space of the matrix.
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And then there's the row space, which is, as I said,
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different, but came from the same numbers.
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So we got the column space,
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all combinations of the columns,
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and then we've got the row space,
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all combinations of the rows.
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So those words are easy for me to say,
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and I can't really draw them on a blackboard,
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but I try with my thick chalk.
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Everybody likes that railroad chalk, and me too.
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I wouldn't use anything else now.
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And then the other two spaces are perpendicular to those.
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So like if you have a plane in 3D,
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just a plane is just a flat surface in 3D,
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then perpendicular to that plane would be a line.
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So that would be the null space.
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So we've got two, we've got a column space, a row space,
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and there are two perpendicular spaces.
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So those four fit together in a beautiful picture
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of a matrix, yeah, yeah.
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It's sort of a fundamental, it's not a difficult idea.
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It comes pretty early in 1806, and it's basic.
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Planes in these multidimensional spaces,
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how difficult of an idea is that to come to, do you think?
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If you look back in time,
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I think mathematically it makes sense,
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but I don't know if it's intuitive for us to imagine,
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just as we were talking about.
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It feels like calculus is easier to intuit.
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Well, I have to admit, calculus came earlier,
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earlier than linear algebra.
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So Newton and Leibniz were the great men
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to understand the key ideas of calculus.
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But linear algebra to me is like, okay,
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it's the starting point,
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because it's all about flat things.
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Calculus has got, all the complications of calculus
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come from the curves, the bending, the curved surfaces.
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Linear algebra, the surfaces are all flat.
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Nothing bends in linear algebra.
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So it should have come first, but it didn't.
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And calculus also comes first in high school classes,
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in college class, it'll be freshman math,
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it'll be calculus, and then I say, enough of it.
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Like, okay, get to the good stuff.
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And that's...
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Do you think linear algebra should come first?
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Well, it really, I'm okay with it not coming first,
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but it should, yeah, it should.
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It's simpler.
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Because everything is flat.
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Yeah, everything's flat.
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Well, of course, for that reason,
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calculus sort of sticks to one dimension,
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or eventually you do multivariate,
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but that basically means two dimensions.
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Linear algebra, you take off into 10 dimensions, no problem.
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It just feels scary and dangerous
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to go beyond two dimensions, that's all.
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If everything's flat, you can't go wrong.
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So what concept or theorem in linear algebra or in math
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you find most beautiful,
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that gives you pause that leaves you in awe?
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Well, I'll stick with linear algebra here.
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I hope the viewer knows that really,
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mathematics is amazing, amazing subject
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and deep, deep connections between ideas
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that didn't look connected, they turned out they were.
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But if we stick with linear algebra...
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So we have a matrix.
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That's like the basic thing, a rectangle of numbers.
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And it might be a rectangle of data.
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You're probably gonna ask me later about data science,
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where often data comes in a matrix.
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You have maybe every column corresponds to a drug
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and every row corresponds to a patient.
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And if the patient reacted favorably to the drug,
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then you put up some positive number in there.
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Anyway, rectangle of numbers, a matrix is basic.
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So the big problem is to understand all those numbers.
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You got a big, big set of numbers.
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And what are the patterns, what's going on?
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And so one of the ways to break down that matrix
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into simple pieces is uses something called singular values.
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And that's come on as fundamental in the last,
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certainly in my lifetime.
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Eigenvalues, if you have viewers who've done engineering,
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math, or basic linear algebra, eigenvalues were in there.
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But those are restricted to square matrices.
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And data comes in rectangular matrices.
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So you gotta take that next step.
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I'm always pushing math faculty, get on, do it, do it.
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Singular values.
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So those are a way to break, to find the important pieces
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of the matrix, which add up to the whole matrix.
link |
00:17:22.140
So you're breaking a matrix into simple pieces.
link |
00:17:26.100
And the first piece is the most important part of the data.
link |
00:17:30.300
The second piece is the second most important part.
link |
00:17:33.100
And then often, so a data set is a matrix.
link |
00:17:38.100
And often, so a data scientist will like,
link |
00:17:41.540
if a data scientist can find those first and second pieces,
link |
00:17:46.820
stop there, the rest of the data is probably round off,
link |
00:17:55.660
experimental error maybe.
link |
00:17:57.660
So you're looking for the important part.
link |
00:18:00.180
So what do you find beautiful about singular values?
link |
00:18:03.020
Well, yeah, I didn't give the theorem.
link |
00:18:06.260
So here's the idea of singular values.
link |
00:18:09.420
Every matrix, every matrix, rectangular, square, whatever,
link |
00:18:15.180
can be written as a product
link |
00:18:16.980
of three very simple special matrices.
link |
00:18:20.020
So that's the theorem.
link |
00:18:21.340
Every matrix can be written as a rotation times a stretch,
link |
00:18:26.220
which is just a diagonal matrix,
link |
00:18:30.340
otherwise all zeros except on the one diagonal.
link |
00:18:34.220
And then the third factor is another rotation.
link |
00:18:37.980
So rotation, stretch, rotation
link |
00:18:41.420
is the breakup of any matrix.
link |
00:18:45.940
The structure of that, the ability that you can do that,
link |
00:18:48.420
what do you find appealing?
link |
00:18:49.860
What do you find beautiful about it?
link |
00:18:51.060
Well, geometrically, as I freely admit,
link |
00:18:54.220
the action of a matrix is not so easy to visualize,
link |
00:18:59.620
but everybody can visualize a rotation.
link |
00:19:02.380
Take two dimensional space and just turn it
link |
00:19:07.060
around the center.
link |
00:19:09.260
Take three dimensional space.
link |
00:19:10.740
So a pilot has to know about,
link |
00:19:13.220
well, what are the three, the yaw is one of them.
link |
00:19:16.860
I've forgotten all the three turns that a pilot makes.
link |
00:19:22.260
Up to 10 dimensions, you've got 10 ways to turn,
link |
00:19:25.460
but you can visualize a rotation.
link |
00:19:28.540
Take the space and turn it.
link |
00:19:30.300
And you can visualize a stretch.
link |
00:19:32.100
So to break a matrix with all those numbers in it
link |
00:19:38.660
into something you can visualize,
link |
00:19:41.100
rotate, stretch, rotate is pretty neat.
link |
00:19:44.860
It's pretty neat.
link |
00:19:45.740
That's pretty powerful.
link |
00:19:47.660
On YouTube, just consuming a bunch of videos
link |
00:19:51.980
and just watching what people connect with
link |
00:19:53.980
and what they really enjoy and are inspired by,
link |
00:19:57.300
math seems to come up again and again.
link |
00:19:59.580
I'm trying to understand why that is.
link |
00:20:03.940
Perhaps you can help give me clues.
link |
00:20:06.500
So it's not just the kinds of lectures that you give,
link |
00:20:10.740
but it's also just other folks like with Numberphile,
link |
00:20:14.180
there's a channel where they just chat about things
link |
00:20:16.940
that are extremely complicated, actually.
link |
00:20:19.860
People nevertheless connect with them.
link |
00:20:22.820
What do you think that is?
link |
00:20:24.580
It's wonderful, isn't it?
link |
00:20:25.820
I mean, I wasn't really aware of it.
link |
00:20:28.500
We're conditioned to think math is hard,
link |
00:20:32.020
math is abstract, math is just for a few people,
link |
00:20:35.300
but it isn't that way.
link |
00:20:36.620
A lot of people quite like math and they liked it.
link |
00:20:41.380
I get messages from people saying,
link |
00:20:44.100
now I'm retired, I'm gonna learn some more math.
link |
00:20:46.780
I get a lot of those.
link |
00:20:47.980
It's really encouraging.
link |
00:20:49.940
And I think what people like is that there's some order,
link |
00:20:53.460
a lot of order and things are not obvious, but they're true.
link |
00:21:00.380
So it's really cheering to think that so many people
link |
00:21:06.180
really wanna learn more about math.
link |
00:21:08.100
Yeah.
link |
00:21:08.940
And in terms of truth, again,
link |
00:21:11.820
sorry to slide into philosophy at times,
link |
00:21:15.500
but math does reveal pretty strongly what things are true.
link |
00:21:20.500
I mean, that's the whole point of proving things.
link |
00:21:23.500
It is, yeah.
link |
00:21:24.340
And yet, sort of our real world is messy and complicated.
link |
00:21:29.420
It is.
link |
00:21:30.260
What do you think about the nature of truth
link |
00:21:33.220
that math reveals?
link |
00:21:34.540
Oh, wow.
link |
00:21:35.500
Because it is a source of comfort like you've mentioned.
link |
00:21:37.980
Yeah, that's right.
link |
00:21:39.540
Well, I have to say, I'm not much of a philosopher.
link |
00:21:43.020
I just like numbers.
link |
00:21:44.740
As a kid, this was before you had to go in,
link |
00:21:52.100
when you had a filly in your teeth,
link |
00:21:54.020
you had to kind of just take it.
link |
00:21:56.060
So what I did was think about math,
link |
00:21:59.260
like take powers of two, two, four, eight, 16,
link |
00:22:03.100
up until the time the tooth stopped hurting
link |
00:22:05.900
and the dentist said you're through.
link |
00:22:08.700
Or counting.
link |
00:22:09.900
Yeah.
link |
00:22:10.740
So that was a source of just, source of peace almost.
link |
00:22:14.700
Yeah.
link |
00:22:15.540
What is it about math do you think that brings that?
link |
00:22:19.660
Yeah.
link |
00:22:20.500
What is that?
link |
00:22:21.340
Well, you know where you are.
link |
00:22:22.380
Yeah, it's symmetry, it's certainty.
link |
00:22:25.820
The fact that, you know, if you multiply two by itself
link |
00:22:29.820
10 times, you get 1,024 period.
link |
00:22:33.220
Everybody's gonna get that.
link |
00:22:34.900
Do you see math as a powerful tool or as an art form?
link |
00:22:39.020
So it's both.
link |
00:22:40.300
That's really one of the neat things.
link |
00:22:42.420
You can be an artist and like math,
link |
00:22:46.380
you can be an engineer and use math.
link |
00:22:50.940
Which are you?
link |
00:22:51.940
Which am I?
link |
00:22:53.500
What did you connect with most?
link |
00:22:54.820
Yeah, I'm somewhere between.
link |
00:22:57.300
I'm certainly not a artist type, philosopher type person.
link |
00:23:01.540
Might sound that way this morning, but I'm not.
link |
00:23:04.060
Yeah, I really enjoy teaching engineers
link |
00:23:09.060
because they go for an answer.
link |
00:23:13.260
And yeah, so probably within the MIT math department,
link |
00:23:20.380
most people enjoy teaching people,
link |
00:23:23.620
teaching students who get the abstract idea.
link |
00:23:26.940
I'm okay with, I'm good with engineers
link |
00:23:32.220
who are looking for a way to find answers.
link |
00:23:34.780
Yeah.
link |
00:23:35.620
Actually, that's an interesting question.
link |
00:23:37.700
Do you think for teaching and in general,
link |
00:23:41.340
thinking about new concepts,
link |
00:23:42.740
do you think it's better to plug in the numbers
link |
00:23:46.820
or to think more abstractly?
link |
00:23:49.060
So looking at theorems and proving the theorems
link |
00:23:53.620
or actually building up a basic intuition of the theorem
link |
00:23:58.220
or the method, the approach,
link |
00:23:59.900
and then just plugging in numbers and seeing it work.
link |
00:24:02.940
Yeah, well, certainly many of us like to see examples.
link |
00:24:09.220
First, we understand,
link |
00:24:11.060
it might be a pretty abstract sounding example,
link |
00:24:13.980
like a three dimensional rotation.
link |
00:24:16.820
How are you gonna understand a rotation in 3D?
link |
00:24:22.780
Or in 10D?
link |
00:24:28.100
And then some of us like to keep going with it
link |
00:24:30.860
to the point where you got numbers,
link |
00:24:32.740
where you got 10 angles, 10 axes, 10 angles.
link |
00:24:38.100
But the best, the great mathematicians probably,
link |
00:24:43.620
I don't know if they do that,
link |
00:24:44.740
because for them, an example would be a highly abstract thing
link |
00:24:53.980
to the rest of it.
link |
00:24:54.820
Right, but nevertheless, working in the space of examples.
link |
00:24:57.540
Yeah, examples.
link |
00:24:58.380
It seems to.
link |
00:24:59.220
Examples of structure.
link |
00:25:01.820
Our brains seem to connect with that.
link |
00:25:03.620
Yeah, yeah.
link |
00:25:04.620
So I'm not sure if you're familiar with him,
link |
00:25:07.180
but Andrew Yang is a presidential candidate
link |
00:25:11.820
currently running with math in all capital letters
link |
00:25:17.060
and his hats as a slogan.
link |
00:25:18.820
I see.
link |
00:25:19.660
Stands for Make America Think Hard.
link |
00:25:21.700
Okay, I'll vote for him.
link |
00:25:25.180
So, and his name rhymes with yours, Yang, Strang.
link |
00:25:28.660
But he also loves math and he comes from that world
link |
00:25:31.260
of, but he also, looking at it,
link |
00:25:35.500
makes me realize that math, science, and engineering
link |
00:25:38.580
are not really part of our politics, political discourse,
link |
00:25:43.300
about political government in general.
link |
00:25:46.140
Why do you think that is?
link |
00:25:48.620
Well.
link |
00:25:49.460
What are your thoughts on that in general?
link |
00:25:51.180
Well, certainly somewhere in the system,
link |
00:25:52.740
we need people who are comfortable with numbers,
link |
00:25:56.860
comfortable with quantities.
link |
00:25:58.340
You know, if you say this leads to that,
link |
00:26:02.460
they see it and it's undeniable.
link |
00:26:05.940
But isn't that strange to you that we have almost no,
link |
00:26:10.180
I mean, I'm pretty sure we have no elected officials
link |
00:26:14.420
in Congress or obviously the president
link |
00:26:18.380
that either has an engineering degree or a math degree.
link |
00:26:22.900
Yeah, well, that's too bad.
link |
00:26:25.820
A few could, a few who could make the connection.
link |
00:26:30.660
Yeah, it would have to be people who understand
link |
00:26:35.420
engineering or science and at the same time
link |
00:26:38.580
can make speeches and lead, yeah.
link |
00:26:44.420
Yeah, inspire people.
link |
00:26:45.540
Yeah, inspire, yeah.
link |
00:26:46.580
You were, speaking of inspiration,
link |
00:26:49.260
the president of the Society
link |
00:26:50.580
for Industrial and Applied Mathematics.
link |
00:26:52.860
Oh, yes.
link |
00:26:53.700
It's a major organization in math, applied math.
link |
00:26:57.940
What do you see as a role of that society,
link |
00:27:01.180
you know, in our public discourse?
link |
00:27:02.860
Right.
link |
00:27:03.700
In public.
link |
00:27:04.540
Yeah, so, well, it was fun to be president at the time.
link |
00:27:08.380
A couple years, a few years.
link |
00:27:09.420
Two years, around 2000.
link |
00:27:13.660
I just hope that's president of a pretty small society.
link |
00:27:16.820
But nevertheless, it was a time when math
link |
00:27:19.620
was getting some more attention in Washington.
link |
00:27:24.380
But yeah, I got to give a little 10 minutes
link |
00:27:29.220
to a committee of the House of Representatives
link |
00:27:33.900
talking about who I met.
link |
00:27:35.300
And then, actually, it was fun
link |
00:27:36.980
because one of the members of the House
link |
00:27:42.460
had been a student, had been in my class.
link |
00:27:44.860
What do you think of that?
link |
00:27:46.060
Yeah, as you say, pretty rare, most members of the House
link |
00:27:49.980
have had a different training, different background.
link |
00:27:52.860
But there was one from New Hampshire
link |
00:27:56.340
who was my friend, really, by being in the class.
link |
00:28:02.460
Yeah, so those years were good.
link |
00:28:05.780
Then, of course, other things take over in importance
link |
00:28:10.780
in Washington, and math just, at this point,
link |
00:28:16.980
is not so visible.
link |
00:28:18.260
But for a little moment, it was.
link |
00:28:20.220
There's some excitement, some concern
link |
00:28:23.780
about artificial intelligence in Washington now.
link |
00:28:26.300
Yes, sure. About the future.
link |
00:28:27.460
Yeah. And I think at the core
link |
00:28:28.820
of that is math.
link |
00:28:30.020
Well, it is, yeah.
link |
00:28:32.020
Maybe it's hidden.
link |
00:28:32.860
Maybe it's wearing a different hat.
link |
00:28:34.380
Well, artificial intelligence, and particularly,
link |
00:28:39.220
can I use the words deep learning?
link |
00:28:41.980
Deep learning is a particular approach
link |
00:28:44.260
to understanding data.
link |
00:28:47.580
Again, you've got a big, whole lot of data
link |
00:28:51.060
where data is just swamping the computers of the world.
link |
00:28:56.140
And to understand it, out of all those numbers,
link |
00:29:00.660
to find what's important in climate, in everything.
link |
00:29:05.180
And artificial intelligence is two words
link |
00:29:08.500
for one approach to data.
link |
00:29:11.700
Deep learning is a specific approach there,
link |
00:29:15.540
which uses a lot of linear algebra.
link |
00:29:17.420
So I got into it.
link |
00:29:19.300
I thought, okay, I've gotta learn about this.
link |
00:29:21.580
So maybe from your perspective,
link |
00:29:24.140
let me ask the most basic question.
link |
00:29:27.900
How do you think of a neural network?
link |
00:29:30.340
What is a neural network?
link |
00:29:31.700
Yeah, okay.
link |
00:29:32.660
So can I start with the idea about deep learning?
link |
00:29:37.220
What does that mean?
link |
00:29:38.860
What is deep learning?
link |
00:29:39.940
What is deep learning, yeah.
link |
00:29:41.980
So we're trying to learn, from all this data,
link |
00:29:46.300
we're trying to learn what's important.
link |
00:29:47.900
What's it telling us?
link |
00:29:50.260
So you've got data, you've got some inputs
link |
00:29:55.300
for which you know the right outputs.
link |
00:29:57.620
The question is, can you see the pattern there?
link |
00:30:02.100
Can you figure out a way for a new input,
link |
00:30:04.660
which we haven't seen, to understand
link |
00:30:09.740
what the output will be from that new input?
link |
00:30:12.220
So we've got a million inputs with their outputs.
link |
00:30:15.940
So we're trying to create some pattern,
link |
00:30:19.260
some rule that'll take those inputs,
link |
00:30:22.180
those million training inputs, which we know about,
link |
00:30:25.580
to the correct million outputs.
link |
00:30:28.180
And this idea of a neural net
link |
00:30:32.780
is part of the structure of our new way to create a rule.
link |
00:30:40.700
We're looking for a rule that will take
link |
00:30:43.900
these training inputs to the known outputs.
link |
00:30:48.460
And then we're gonna use that rule on new inputs
link |
00:30:51.660
that we don't know the output and see what comes.
link |
00:30:56.100
Linear algebra is a big part of finding that rule.
link |
00:30:59.140
That's right, linear algebra is a big part.
link |
00:31:01.860
Not all the part.
link |
00:31:03.500
People were leaning on matrices, that's good, still do.
link |
00:31:08.300
Linear is something special.
link |
00:31:10.300
It's all about straight lines and flat planes.
link |
00:31:13.980
And data isn't quite like that.
link |
00:31:18.860
It's more complicated.
link |
00:31:21.220
So you gotta introduce some complication.
link |
00:31:23.700
So you have to have some function
link |
00:31:25.460
that's not a straight line.
link |
00:31:27.460
And it turned out, nonlinear, nonlinear, not linear.
link |
00:31:31.620
And it turned out that it was enough to use the function
link |
00:31:35.860
that's one straight line and then a different one.
link |
00:31:38.340
Halfway, so piecewise linear.
link |
00:31:40.900
One piece has one slope,
link |
00:31:44.420
one piece, the other piece has the second slope.
link |
00:31:47.340
And so that, getting that nonlinear,
link |
00:31:52.380
simple nonlinearity in blew the problem open.
link |
00:31:56.700
That little piece makes it sufficiently complicated
link |
00:31:58.980
to make things interesting.
link |
00:32:00.460
Because you're gonna use that piece
link |
00:32:02.020
over and over a million times.
link |
00:32:03.820
So it has a fold in the graph, the graph, two pieces.
link |
00:32:10.740
But when you fold something a million times,
link |
00:32:13.700
you've got a pretty complicated function
link |
00:32:17.860
that's pretty realistic.
link |
00:32:19.260
So that's the thing about neural networks
link |
00:32:21.140
is they have a lot of these.
link |
00:32:23.900
A lot of these, that's right.
link |
00:32:25.220
So why do you think neural networks,
link |
00:32:29.660
by using sort of formulating an objective function,
link |
00:32:34.940
very not a plain function of the folds,
link |
00:32:39.380
lots of folds of the inputs, the outputs,
link |
00:32:42.340
why do you think they work to be able to find a rule
link |
00:32:47.300
that we don't know is optimal,
link |
00:32:48.780
but it just seems to be pretty good in a lot of cases?
link |
00:32:53.300
What's your intuition?
link |
00:32:54.580
Is it surprising to you as it is to many people?
link |
00:32:58.180
Do you have an intuition of why this works at all?
link |
00:33:01.140
Well, I'm beginning to have a better intuition.
link |
00:33:04.300
This idea of things that are piecewise linear,
link |
00:33:08.500
flat pieces but with folds between them.
link |
00:33:12.140
Like think of a roof of a complicated,
link |
00:33:14.980
infinitely complicated house or something.
link |
00:33:17.780
That curve, it almost curved, but every piece is flat.
link |
00:33:24.700
That's been used by engineers,
link |
00:33:26.820
that idea has been used by engineers,
link |
00:33:29.660
is used by engineers, big time.
link |
00:33:32.140
Something called the finite element method.
link |
00:33:34.220
If you want to design a bridge,
link |
00:33:36.980
design a building, design an airplane,
link |
00:33:40.860
you're using this idea of piecewise flat
link |
00:33:47.300
as a good, simple, computable approximation.
link |
00:33:52.260
But you have a sense that there's a lot of expressive power
link |
00:33:57.260
in this kind of piecewise linear.
link |
00:33:58.580
Yeah, you used the right word.
link |
00:34:01.820
If you measure the expressivity,
link |
00:34:04.460
how complicated a thing can this piecewise flat guys express?
link |
00:34:12.300
The answer is very complicated, yeah.
link |
00:34:15.500
What do you think are the limits of such piecewise linear
link |
00:34:20.380
or just of neural networks?
link |
00:34:22.660
The expressivity of neural networks.
link |
00:34:24.100
Well, you would have said a while ago
link |
00:34:26.660
that they're just computational limits.
link |
00:34:28.700
It's a problem beyond a certain size.
link |
00:34:33.740
A supercomputer isn't gonna do it.
link |
00:34:36.060
But those keep getting more powerful.
link |
00:34:39.420
So that limit has been moved
link |
00:34:44.260
to allow more and more complicated surfaces.
link |
00:34:47.420
So in terms of just mapping from inputs to outputs,
link |
00:34:52.940
looking at data, what do you think of,
link |
00:34:58.460
in the context of neural networks in general,
link |
00:35:00.500
data is just tensor, vectors, matrices, tensors.
link |
00:35:04.180
Right.
link |
00:35:05.820
How do you think about learning from data?
link |
00:35:09.380
How much of our world can be expressed in this way?
link |
00:35:12.780
How useful is this process?
link |
00:35:16.540
I guess that's another way to ask you,
link |
00:35:17.980
what are the limits of this approach?
link |
00:35:19.340
Well, that's a good question, yeah.
link |
00:35:21.380
So I guess the whole idea of deep learning
link |
00:35:24.220
is that there's something there to learn.
link |
00:35:26.220
If the data is totally random,
link |
00:35:28.500
just produced by random number generators,
link |
00:35:31.380
then we're not gonna find a useful rule
link |
00:35:36.220
because there isn't one.
link |
00:35:38.620
So the extreme of having a rule is like knowing Newton's law.
link |
00:35:43.620
If you hit a ball, it moves.
link |
00:35:46.220
So that's where you had laws of physics.
link |
00:35:48.940
Newton and Einstein and other great, great people
link |
00:35:54.140
have found those laws and laws of the distribution
link |
00:36:02.940
of oil in an underground thing.
link |
00:36:05.900
I mean, so engineers, petroleum engineers understand
link |
00:36:10.900
how oil will sit in an underground basin.
link |
00:36:18.180
So there were rules.
link |
00:36:20.060
Now, the new idea of artificial intelligence is
link |
00:36:25.620
learn the rules instead of figuring out the rules
link |
00:36:29.940
with help from Newton or Einstein.
link |
00:36:32.740
The computer is looking for the rules.
link |
00:36:35.660
So that's another step.
link |
00:36:36.900
But if there are no rules at all
link |
00:36:39.860
that the computer could find,
link |
00:36:41.220
if it's totally random data, well, you've got nothing.
link |
00:36:45.300
You've got no science to discover.
link |
00:36:48.300
It's an automated search for the underlying rules.
link |
00:36:51.380
Yeah, search for the rules.
link |
00:36:53.380
Yeah, exactly.
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00:36:54.780
And there will be a lot of random parts.
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00:36:57.820
A lot of, I mean, I'm not knocking random
link |
00:36:59.860
because that's there.
link |
00:37:05.580
There's a lot of randomness built in,
link |
00:37:07.340
but there's gotta be some basic.
link |
00:37:09.380
It's almost always signal, right?
link |
00:37:10.900
In most things.
link |
00:37:11.740
There's gotta be some signal, yeah.
link |
00:37:12.820
If it's all noise, then you're not gonna get anywhere.
link |
00:37:17.420
Well, this world around us does seem to be,
link |
00:37:19.900
does seem to always have a signal of some kind.
link |
00:37:22.420
Yeah, yeah, that's right.
link |
00:37:23.340
To be discovered.
link |
00:37:24.220
Right, that's it.
link |
00:37:25.900
So what excites you more?
link |
00:37:30.580
We just talked about a little bit of application.
link |
00:37:32.860
What excites you more, theory
link |
00:37:35.580
or the application of mathematics?
link |
00:37:38.380
Well, for myself, I'm probably a theory person.
link |
00:37:43.260
I'm not, I'm speaking here pretty freely about applications,
link |
00:37:49.700
but I'm not the person who really,
link |
00:37:53.220
I'm not a physicist or a chemist or a neuroscientist.
link |
00:37:58.100
So for myself, I like the structure
link |
00:38:03.220
and the flat subspaces
link |
00:38:06.460
and the relation of matrices, columns to rows.
link |
00:38:12.260
That's my part in the spectrum.
link |
00:38:17.860
So really, science is a big spectrum of people
link |
00:38:22.420
from asking practical questions
link |
00:38:25.740
and answering them using some math,
link |
00:38:28.740
then some math guys like myself who are in the middle of it
link |
00:38:33.740
and then the geniuses of math and physics and chemistry
link |
00:38:40.620
who are finding fundamental rules
link |
00:38:43.300
and then doing the really understanding nature.
link |
00:38:50.060
That's incredible.
link |
00:38:51.820
At its lowest, simplest level,
link |
00:38:54.980
maybe just a quick in broad strokes from your perspective,
link |
00:38:58.980
where does linear algebra sit as a subfield of mathematics?
link |
00:39:04.740
What are the various subfields that you think about
link |
00:39:10.300
in relation to linear algebra?
link |
00:39:12.180
So the big fields of math are algebra as a whole
link |
00:39:18.020
and problems like calculus and differential equations.
link |
00:39:21.340
So that's a second, quite different field.
link |
00:39:24.340
Then maybe geometry deserves to be thought of
link |
00:39:28.780
as a different field to understand the geometry
link |
00:39:31.540
of high dimensional surfaces.
link |
00:39:35.700
So I think, am I allowed to say this here?
link |
00:39:39.700
I think this is where personal view comes in.
link |
00:39:46.180
I think math, we're thinking about undergraduate math,
link |
00:39:51.980
what millions of students study.
link |
00:39:54.260
I think we overdo the calculus at the cost of the algebra,
link |
00:40:00.780
at the cost of linear.
link |
00:40:02.660
So you have this talk titled Calculus Versus Linear Algebra.
link |
00:40:05.300
That's right, that's right.
link |
00:40:07.380
And you say that linear algebra wins.
link |
00:40:09.420
So can you dig into that a little bit?
link |
00:40:13.780
Why does linear algebra win?
link |
00:40:17.020
Right, well, okay, the viewer is gonna think
link |
00:40:21.100
this guy is biased.
link |
00:40:22.700
Not true, I'm just telling the truth as it is.
link |
00:40:27.060
Yeah, so I feel linear algebra is just a nice part of math
link |
00:40:31.940
that people can get the idea of.
link |
00:40:34.420
They can understand something that's a little bit abstract
link |
00:40:37.780
because once you get to 10 or 100 dimensions
link |
00:40:42.140
and very, very, very useful,
link |
00:40:44.380
that's what's happened in my lifetime
link |
00:40:47.860
is the importance of data,
link |
00:40:52.540
which does come in matrix form.
link |
00:40:54.660
So it's really set up for algebra.
link |
00:40:56.660
It's not set up for differential equation.
link |
00:40:59.260
And let me fairly add probability,
link |
00:41:03.300
the ideas of probability and statistics
link |
00:41:06.860
have become very, very important, have also jumped forward.
link |
00:41:11.260
So, and that's different from linear algebra,
link |
00:41:14.060
quite different.
link |
00:41:15.220
So now we really have three major areas to me,
link |
00:41:20.220
calculus, linear algebra, matrices,
link |
00:41:26.180
and probability statistics.
link |
00:41:28.980
And they all deserve an important place.
link |
00:41:33.980
And calculus has traditionally had a lion's share
link |
00:41:40.020
of the time.
link |
00:41:40.860
A disproportionate share.
link |
00:41:41.900
It is, thank you, disproportionate, that's a good word.
link |
00:41:45.700
Of the love and attention from the excited young minds.
link |
00:41:50.020
Yeah.
link |
00:41:52.900
I know it's hard to pick favorites,
link |
00:41:55.500
but what is your favorite matrix?
link |
00:41:57.700
What's my favorite matrix?
link |
00:41:59.380
Okay, so my favorite matrix is square, I admit it.
link |
00:42:03.220
It's a square bunch of numbers
link |
00:42:05.460
and it has twos running down the main diagonal.
link |
00:42:10.180
And on the next diagonal,
link |
00:42:13.020
so think of top left to bottom right,
link |
00:42:15.380
twos down the middle of the matrix
link |
00:42:18.900
and minus ones just above those twos
link |
00:42:22.140
and minus ones just below those twos
link |
00:42:25.020
and otherwise all zeros.
link |
00:42:26.620
So mostly zeros, just three nonzero diagonals coming down.
link |
00:42:32.900
What is interesting about it?
link |
00:42:34.380
Well, all the different ways it comes up.
link |
00:42:37.180
You see it in engineering,
link |
00:42:39.260
you see it as analogous in calculus to second derivative.
link |
00:42:44.100
So calculus learns about taking the derivative,
link |
00:42:47.180
the figuring out how much, how fast something's changing.
link |
00:42:51.500
But second derivative, now that's also important.
link |
00:42:55.740
That's how fast the change is changing,
link |
00:42:58.740
how fast the graph is bending, how fast it's curving.
link |
00:43:06.460
And Einstein showed that that's fundamental
link |
00:43:10.140
to understand space.
link |
00:43:11.540
So second derivatives should have a bigger place in calculus.
link |
00:43:17.380
Second, my matrices,
link |
00:43:21.020
which are like the linear algebra version
link |
00:43:24.980
of second derivatives are neat in linear algebra.
link |
00:43:30.020
Yeah, just everything comes out right with those guys.
link |
00:43:34.020
Beautiful.
link |
00:43:35.220
What did you learn about the process of learning
link |
00:43:38.380
by having taught so many students math over the years?
link |
00:43:42.820
Ooh, that is hard.
link |
00:43:45.700
I'll have to admit here that I'm not really a good teacher
link |
00:43:51.260
because I don't get into the exam part.
link |
00:43:55.540
The exam is the part of my life that I don't like
link |
00:43:59.020
and grading them and giving the students A or B or whatever.
link |
00:44:04.380
I do it because I'm supposed to do it,
link |
00:44:08.260
but I tell the class at the beginning,
link |
00:44:11.900
I don't know if they believe me.
link |
00:44:13.180
Probably they don't.
link |
00:44:14.580
I tell the class, I'm here to teach you.
link |
00:44:18.020
I'm here to teach you math and not to grade you.
link |
00:44:22.700
But they're thinking, okay, this guy is gonna,
link |
00:44:26.300
when is he gonna give me an A minus?
link |
00:44:28.820
Is he gonna give me a B plus?
link |
00:44:30.580
What?
link |
00:44:31.420
What have you learned about the process of learning?
link |
00:44:34.060
Of learning.
link |
00:44:34.940
Yeah, well, maybe to give you a legitimate answer
link |
00:44:40.220
about learning, I should have paid more attention
link |
00:44:43.900
to the assessment, the evaluation part at the end.
link |
00:44:47.660
But I like the teaching part at the start.
link |
00:44:49.980
That's the sexy part.
link |
00:44:52.060
To tell somebody for the first time about a matrix, wow.
link |
00:44:56.060
Is there, are there moments,
link |
00:44:58.700
so you are teaching a concept,
link |
00:45:01.900
are there moments of learning that you just see
link |
00:45:05.500
in the student's eyes?
link |
00:45:06.460
You don't need to look at the grades.
link |
00:45:08.220
But you see in their eyes that you hook them,
link |
00:45:11.540
that you connect with them in a way where,
link |
00:45:16.260
you know what, they fall in love
link |
00:45:18.620
with this beautiful world of math.
link |
00:45:21.180
They see that it's got some beauty there.
link |
00:45:24.460
Or conversely, that they give up at that point
link |
00:45:28.060
is the opposite.
link |
00:45:29.140
The dark could say that math, I'm just not good at math.
link |
00:45:32.420
I don't wanna walk away.
link |
00:45:33.260
Yeah, yeah, yeah.
link |
00:45:34.300
Maybe because of the approach in the past,
link |
00:45:37.700
they were discouraged, but don't be discouraged.
link |
00:45:40.500
It's too good to miss.
link |
00:45:44.420
Yeah, well, if I'm teaching a big class,
link |
00:45:48.420
do I know when, I think maybe I do.
link |
00:45:51.900
Sort of, I mentioned at the very start,
link |
00:45:55.460
the four fundamental subspaces
link |
00:45:59.340
and the structure of the fundamental theorem
link |
00:46:03.100
of linear algebra.
link |
00:46:04.740
The fundamental theorem of linear algebra.
link |
00:46:06.780
That is the relation of those four subspaces,
link |
00:46:11.740
those four spaces.
link |
00:46:13.420
Yeah, so I think that, I feel that the class gets it.
link |
00:46:17.740
At length.
link |
00:46:18.580
Yeah.
link |
00:46:19.940
What advice do you have to a student
link |
00:46:22.420
just starting their journey in mathematics today?
link |
00:46:25.140
How do they get started?
link |
00:46:27.060
Oh, yeah, that's hard.
link |
00:46:30.100
Well, I hope you have a teacher, professor,
link |
00:46:34.780
who is still enjoying what he's doing,
link |
00:46:39.860
what he's teaching.
link |
00:46:41.380
They're still looking for new ways to teach
link |
00:46:44.020
and to understand math.
link |
00:46:47.940
Cause that's the pleasure,
link |
00:46:51.140
the moment when you see, oh yeah, that works.
link |
00:46:54.980
So it's less about the material you study,
link |
00:46:58.500
it's more about the source of the teacher
link |
00:47:02.460
being full of passion.
link |
00:47:03.900
Yeah, more about the fun.
link |
00:47:05.740
Yeah, the moment of getting it.
link |
00:47:10.500
But in terms of topics, linear algebra?
link |
00:47:14.140
Well, that's my topic,
link |
00:47:16.940
but oh, there's beautiful things in geometry to understand.
link |
00:47:21.220
What's wonderful is that in the end,
link |
00:47:24.140
there's a pattern, there are rules
link |
00:47:28.620
that are followed in biology as there are in every field.
link |
00:47:37.260
You describe the life of a mathematician
link |
00:47:41.420
as 100% wonderful.
link |
00:47:44.260
No.
link |
00:47:45.620
Except for the grade stuff.
link |
00:47:47.140
Yeah.
link |
00:47:47.980
And the grades.
link |
00:47:48.820
Except for grades.
link |
00:47:49.660
Yeah, when you look back at your life,
link |
00:47:52.140
what memories bring you the most joy and pride?
link |
00:47:55.980
Well, that's a good question.
link |
00:47:59.500
I certainly feel good when I,
link |
00:48:01.620
maybe I'm giving a class in 1806,
link |
00:48:06.140
that's MIT's linear algebra course that I started.
link |
00:48:09.380
So sort of, there's a good feeling that,
link |
00:48:11.620
okay, I started this course,
link |
00:48:13.740
a lot of students take it, quite a few like it.
link |
00:48:17.340
Yeah, so I'm sort of happy
link |
00:48:21.380
when I feel I'm helping make a connection
link |
00:48:25.060
between ideas and students,
link |
00:48:27.740
between theory and the reader.
link |
00:48:32.980
Yeah, it's, I get a lot of very nice messages
link |
00:48:38.540
from people who've watched the videos and it's inspiring.
link |
00:48:43.460
I just, I'll maybe take this chance to say thank you.
link |
00:48:48.060
Well, there's millions of students
link |
00:48:50.380
who you've taught and I am grateful to be one of them.
link |
00:48:54.220
So Gilbert, thank you so much, it's been an honor.
link |
00:48:56.540
Thank you for talking today.
link |
00:48:58.140
It was a pleasure, thanks.
link |
00:49:00.700
Thank you for listening to this conversation
link |
00:49:02.500
with Gilbert Strang.
link |
00:49:04.220
And thank you to our presenting sponsor, Cash App.
link |
00:49:07.380
Download it, use code LexPodcast,
link |
00:49:09.940
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link |
00:49:12.780
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link |
00:49:14.500
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link |
00:49:17.500
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link |
00:49:20.660
If you enjoy this podcast, subscribe on YouTube.
link |
00:49:23.780
We have five stars in Apple Podcast,
link |
00:49:25.900
support on Patreon or connect with me on Twitter.
link |
00:49:29.300
Finally, some closing words of advice
link |
00:49:31.860
from the great Richard Feynman.
link |
00:49:33.940
Study hard what interests you the most
link |
00:49:36.300
in the most undisciplined, irreverent
link |
00:49:38.980
and original manner possible.
link |
00:49:41.220
Thank you for listening and hope to see you next time.