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

Lex Fridman and Gilbert Strang on gilbert Strang on Linear Algebra’s Power, Beauty, and Global Classroom.

Lex FridmanhostGilbert Strangguest
Nov 25, 201949mWatch on YouTube ↗

EVERY SPOKEN WORD

  1. 0:00 – 5:17

    Gilbert Strang’s OpenCourseWare impact and the rise of linear algebra

    1. LF

      The following is a conversation with Gilbert Strang. He's a professor of mathematics at MIT, and perhaps one of the most famous and impactful teachers of math in the world. His MIT Open Courseware lectures on linear algebra have been viewed millions of times. As an undergraduate student, I was one of those millions of students. There's something inspiring about the way he teaches, that is 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. If you enjoy it, subscribe on YouTube, give it five stars on Apple Podcasts, support it on Patreon, or simply connect with me on Twitter @lexfridman, spelled F-R-I-D-M-A-N. This podcast is supported by ZipRecruiter. Hiring great people is hard, and to me, is the most important element of a successful mission-driven team. I've been fortunate to be a part of and to lead several great engineering teams. The hiring I've done in the past was mostly through tools that we built ourselves, but reinventing the wheel was painful. ZipRecruiter's a tool that's already available for you. It seeks to make hiring simple, fast, and smart. For example, Codable co-founder, Gretchen Huebner, used ZipRecruiter to find a new game artist to join her education tech company. By using ZipRecruiter's screening questions to filter candidates, Gretchen found it easier to focus on the best candidates and finally hiring the perfect person for the role in less than two weeks from start to finish. ZipRecruiter, the smartest way to hire. See why ZipRecruiter's effective for businesses of all sizes by signing up, as I did, for free at ziprecruiter.com/lexpod. That's ziprecruiter.com/lexpod. This show is presented by Cash App, the number one finance app in the App Store. I personally use Cash App to send money to friends, but you can also use it to buy, sell, and deposit Bitcoin. Most Bitcoin exchanges take days for a bank transfer to become investable. Through Cash App, it takes seconds. Cash App also has a new investing feature. You can buy fractions of a stock, which to me is a really interesting concept. So you can buy of $1 worth, no matter what the stock price is. Brokerage services are provided by Cash App Investing, a subsidiary of Square and member SIPC. I'm excited to be working with Cash App to support one of my favorite organizations that many of you may know and have benefited from called FIRST, best known for their FIRST Robotics and Lego competitions. They educate and inspire hundreds of thousands of students in over 110 countries and have a perfect rating on Charity Navigator, which means the donated money is used to maximum effectiveness. When you get Cash App from the App Store or Google Play and use code LEXPODCAST, you'll get $10 and Cash App will also donate $10 to FIRST, which again is an organization that I've personally seen inspire girls and boys to dream of engineering a better world. And now here's my conversation with Gilbert Strang. How does it feel to be one of the, uh, modern day rockstars of mathematics?

    2. GS

      (laughs) I don't feel like a rockstar. That's kind of crazy for old math person. But, uh, it's true that, um, the videos in linear algebra that I made way back in 2000, I think, uh, have been watched a lot. And, uh, well, partly the importance of linear algebra, uh, which we- 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, I, it was a class that I taught a bunch of times, so I kind of got it organized and, uh, an- and enjoyed doing it. It was just the videos were just the class, so they're on Open Courseware and on YouTube and translated-

    3. LF

      But th-

    4. GS

      ... as one.

    5. LF

      But there's something about that chalkboard and the, and the simplicity of the way you explain the basic concepts in the beginning. I, you know, to be honest, when I went to undergrad, you know...

    6. GS

      You didn't do linear algebra probably.

    7. LF

      Of course, I did linear algebra.

    8. GS

      You did? Okay, yeah.

    9. LF

      Yeah, yeah, yeah, of course.

    10. GS

      Right.

    11. LF

      But I, before going through the course at my university, I li- there was going through Open Course where I was, you were my instructor for linear algebra.

    12. GS

      Oh, I see. Right, yeah.

    13. LF

      (laughs) And that, uh, I mean, we were using your book, and, I mean, that, that, the fact that there is thousands, you know, hundreds of thousands, millions of people that watch that video, I think that's-

    14. GS

      Yeah.

  2. 5:17 – 7:30

    MIT OpenCourseWare’s ‘give it away’ philosophy

    1. LF

      ... that's really powerful. So, uh, how do you think the idea of putting lectures online wo- would really... MIT Open Courseware has innovated?

    2. GS

      That was a wonderful idea. You know, I think, uh, uh, the story that I've heard is the committee, uh, committee was appointed by the president, President Vest at that time, a wonderful guy. And, uh, the idea of the committee was to figure out how MIT could make, uh, be like other universities market, uh, market the work we were doing. And then they didn't see a way and after a weekend and they had an inspiration and came back to the President Vest and said, "What if we just gave it away?" And, uh, he decided that was g- okay, good idea.... so...

    3. LF

      You know, that's a crazy idea, that's, uh-

    4. GS

      Yeah.

    5. LF

      ... if we think of a university as a thing that creates a product-

    6. GS

      Yes.

    7. LF

      ... isn't knowledge-

    8. GS

      Right.

    9. LF

      ... the, uh, you know, the kind of educational knowledge, isn't the product, and giving that away?

    10. GS

      Yeah.

    11. LF

      Are you surprised that (laughs) -

    12. GS

      Th- the-

    13. LF

      ... that it went through?

    14. GS

      ... uh, th- the result that it w- that he did it? Well, knowing a little bit, President Vest, it was like him, I think.

    15. LF

      (laughs)

    16. GS

      And, uh, and it was really the right idea. You know, uh, um, MIT is a kind of... It's known for being high-level technical things. And, and this is the best way we can, say, w- tell, we can show what MIT really is like, uh, 'cause th- the, the, v- in my case, those 18.06 videos are just teaching the class. They were there in 26.100. They're kind of fun to look at. People write to me and say, "Oh, you've got a sense of humor," but I, I don't know where (laughs) that comes through. Somehow, I've been friendly with the class. I like students-

    17. LF

      Yeah.

    18. GS

      ... and, uh... And linear algebra, uh, uh, the subje- we gotta give the subject most of the credit. It, it, it really has come forward in importance, uh, in, in these years.

  3. 7:30 – 10:14

    The ‘four fundamental subspaces’: a mental model for matrices

    1. LF

      So let's talk about linear algebra a little bit 'cause it is such a... It's both a powerful and a beautiful, uh, s- uh, subfield of mathematics. So what's your s- favorite specific topic in linear algebra, or even math in general, to give a lecture on, to, to-

    2. GS

      I see.

    3. LF

      ... convey, to tell a story-

    4. GS

      I see.

    5. LF

      ... to teach students?

    6. GS

      Okay. Well, on the teaching side, so it's not deep mathematics at all, but I, I'm kind of proud of the idea of the four subspaces, the, the four fundamental subspaces, w- which were, of course, known before, long before m- my name for them. But, uh, th-

    7. LF

      Can you go through them? Can you go through the four subspaces?

    8. GS

      Sure I can, yeah. So the first one to understand is, so the matrix is... Maybe I should say the matrix is-

    9. LF

      What is a matrix?

    10. GS

      What's a matrix? Well, so we have a, like, a rectangle of numbers. So it's got N columns, it's got a bunch of columns, and also got, um, 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 gotta be connections there. But they're not simple. The, uh, they're, they're mi- 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.

    11. LF

      What's a vector?

    12. GS

      So a physicist would imagine a vector, or might imagine a vector as a arrow, you know, in space, or the point it ends at i- in space. For me, it's a column of numbers that-

    13. LF

      This is- i- it... You often think of... This is very interesting in terms of linear algebra-

    14. GS

      Yeah.

    15. LF

      ... in terms of a vector. You think a little bit more abstract than the... how it's very commonly used perhaps.

    16. GS

      Yeah.

    17. LF

      You, you think this arbitrary s- spa- multidimensional space, that there's a-

    18. GS

      Yeah. I'm, I'm... Right away, I'm in high dimensions and in the-

    19. LF

      (laughs) Dreamland (laughs) .

    20. GS

      Yeah, that's right. In a lecture, I try to... So if you think of two 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. Y- you can, uh, you can add one bunch of ten numbers to another bunch of ten numbers. So you can add a vector to a vector, and you can multiply a vector by three. And that's... If you know how to do those, you've got linear algebra.

  4. 10:14 – 13:11

    Thinking beyond 3D: intuition for high-dimensional ‘flat’ spaces

    1. LF

      You know, 10 dimensions-

    2. GS

      Yeah.

    3. LF

      ... you know, there's this beautiful thing about math if we look at string theory and-

    4. GS

      Yeah.

    5. LF

      ... all these theories which are really fundamentally derived through math-

    6. GS

      Yes.

    7. LF

      ... but are very difficult to visualize. And-

    8. GS

      Yeah.

    9. LF

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

    10. GS

      Yeah.

    11. LF

      Do you, do you... And, and yet math reveals some beauty-

    12. GS

      Oh, great beauty, yeah.

    13. LF

      ... underlying our world-

    14. GS

      Yeah.

    15. LF

      ... in that weird thing we can't visualize.

    16. GS

      Yeah.

    17. LF

      How do you think about that difference?

    18. GS

      Well, probably, I'm not a very geometric person, so I'm probably thinking in three dimensions. And the beauty of linear algebra is that, m- is that it goes on to 10 dimensions with no problem. I mean, that, uh, that if you're just seeing what happens if you add two vectors in 3D, yeah, then you can add them in 10D. You're, you're just adding the 10 components. So, uh, so I, uh, uh, 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, so a plane in 10 dimensions. And, and, uh... So that's one of the spaces. Take all the columns of the matrix, take all their combinations, so, um, so much of this column, so much of this one. 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, and my imagination is just seeing, like, a piece of paper (laughs) in 3D. Uh, but, uh, anyway, so that's one of the spaces. Uh, that's space number one, the column space of the matrix. And then there's the row space, which is, as I said, different, but came, 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, uh, th- those words are easy for me to say and, uh, I can't really draw them on a blackboard but I try with my thick chalk.

    19. LF

      (laughs)

    20. GS

      Everybody, everybody likes that railroad chalk.

    21. LF

      (laughs)

    22. GS

      And me too. I, I wouldn't use anything else now.

    23. LF

      Yeah.

    24. GS

      And, uh, and then the other two spaces are perpendicular to those. So like if you have a plane in 3D, just a... A plane is just a flat surface, uh, in 3D. Then perpendicular to that plane would be a line, so that would be the null space. So we've got two... We've got a column space, a row space, and their two perpendicular spaces. So those four fit together in the, in the b- beautiful picture of a matrix. Yeah, yeah. It- it's sort of a fundamental... It's not a difficult idea. It comes, comes pretty early in 1806 and it's basic.

  5. 13:11 – 15:03

    Why linear algebra should come earlier than calculus

    1. LF

      So planes in these multidimensional spaces, how, how difficult of an idea is that to, to come to? Do you think if you-

    2. GS

      Yeah.

    3. LF

      ... if you look back in time-

    4. GS

      Yeah.

    5. LF

      ... I, I think, uh, mathematically, it makes sense. But I don't know if it's intuitive for our... uh, us to imagine-

    6. GS

      Right.

    7. LF

      ... just as we were talking about. It feels like calculus is easier to-

    8. GS

      I see.

    9. LF

      ... intuit.

    10. GS

      Well, calcul- I have to admit, calculus came earlier-

    11. LF

      (laughs)

    12. GS

      ... uh, 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 'cause it's all about flat things.

    13. LF

      (laughs)

    14. GS

      Calculus has got... All the complications of calculus come from the curves-

    15. LF

      Yeah.

    16. GS

      ... the bending, the s- the curved surfaces. Linear algebra, the surfaces are all flat.

    17. LF

      (laughs)

    18. GS

      Nothing bends in linear algebra.

    19. LF

      (laughs)

    20. GS

      So, uh, it should've come first but it didn't. And calculus also comes first in, in high school classes, in, in college class. It'll be freshman math, it'll be calculus. And then I say, "Enough of it." Like, "Okay, get to, get to the good stuff." And, uh, that-

    21. LF

      Do you think linear algebra should come first?

    22. GS

      Well, it really... Uh, I- I'm okay with it not coming first but it should. Yeah, it should. It's simpler. It- i- i-

    23. LF

      'Cause everything is flat (laughs) .

    24. GS

      Yeah, everything is flat.

    25. LF

      (laughs)

    26. GS

      Well, of course, for that reason, your calculus sort of sticks to one dimension or s- or eventually you do multivariate, but that basically means two dimensions. Linear algebra, you take off into 10 dimensions, no problem.

    27. LF

      It just feels scary and dangerous to go beyond two dimensions.

    28. GS

      Well-

    29. LF

      That's all. (laughs)

    30. GS

      ... I mean if everything is flat, you can't go wrong.

  6. 15:03 – 19:45

    Singular Value Decomposition: ‘rotate–stretch–rotate’ as a universal lens

    1. LF

      So w- what concept or theorem in linear algebra or in math you find most beautiful, that gives you pause-

    2. GS

      Most beautiful.

    3. LF

      ... that leaves you in awe?

    4. GS

      Well, I'll stick with linear algebra here. Uh, I hope the viewer knows that really mathematics is amazing, amazing subject and deep, deep, uh, connections between ideas that didn't look connected. Some, they turned out they were. But if we stick with linear algebra, so we have a matrix. That, that's like the basic thing, a rectangle of numbers, and might be a rectangle of data. You're probably gonna ask me later about data science-

    5. LF

      Yeah.

    6. GS

      ... where an often data comes in a matrix. You have, you know, the... uh, maybe every column corresponds to a, to a drug, and every row corresponds to a patient. And, and, uh, if the patient, uh, uh, reacted favorably to the drug then you put up some positive number in there. Anyway, m- m- rectangle of n- of numbers, a matrix is basic. So, uh, the big problem is to understand all those numbers. You got a big, big set of numbers and what are the patterns, what's going on? And, uh, so one of the ways to break down that matrix into simple pieces is uses something called singular values.

    7. LF

      Mm-hmm.

    8. GS

      And that's come on as fundamental in the last... in, certainly in my lifetime. Uh, eigenvalues pro- if you have viewers who've done engineering math or, or, uh, or basic linear algebra, eigenvalues were in there. Uh, but those are restricted to square matrices. And data comes in rectangular matrices, so you gotta take that... you gotta take that next step.

    9. LF

      (laughs)

    10. GS

      I'm, I'm always pushing-

    11. LF

      (laughs)

    12. GS

      ... math faculty, "Get on, do, do, do it. Do it," uh, singular values. So those are a way to break, to, to make, to find the es- the important pieces of the matrix w- which add up to the whole matrix. So, so you're breaking a matrix into simple pieces and, uh, the first piece is the most important part of the data, the second piece is the second most important part. And, uh, then often... So a data scientist will have to like... if you, if a data scientist can find those first and second pieces, stop there, the rest of, of the data is probably round off, you know, er- um, experimental error maybe. So you're looking for the important part.

    13. LF

      Yeah. So what do you find beautiful about singular values? What, what is the problem-

    14. GS

      Well, yeah, I didn't give the theorem. Yeah, so here's the, here's the idea of singular values. Every matrix, every matrix, uh, rectangular, square, whatever-... can be written as a product of three very simple special matrices. So that's the theorem. Every matrix can be written as a rotation, times a stretch, which is a s- just a matrix, a diagonal matrix, otherwise all zeros except on the one diagonal, and then a thir- and the third factor is another rotation. So rotation, stretch, rotation is the breakup of a, of a, of any matrix.

    15. LF

      The structure that, uh, the ability that you can do that, what- what- what do you find appealing? What do you find beautiful about it?

    16. GS

      Well, geometrically, as I freely admit, the- the ma- action of a matrix, this is not so easy to visualize. But everybody can visualize a rotation. Take- take- take two-dimensional space and just turn it around the, around the center. Take three-dimensional space. So a pilot has to know about, well, what are the three, the yaw is one of them. I've forgotten all of the three turns that a pilot makes. Uh, up to ten dimensions, you got ten ways to turn. But, uh, you can visualize a rotation. Take the space and turn it. And you can visualize a stretch. So to break a- a- a- a matrix with all those numbers in it into something you can visualize, rotate, stretch, rotate, is pretty neat.

    17. LF

      Yeah.

    18. GS

      Pretty neat.

  7. 19:45 – 21:08

    Why people love math online: order, certainty, and lifelong curiosity

    1. LF

      That's pretty powerful. 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. I- I'm trying to understand why that is. Perhaps you can help-

    2. GS

      Yeah.

    3. LF

      ... me, give me clues. So it's not just the lec- the kinds of lectures that y- you give, but it's also just the other folks, like with Numberphile, there's a channel-

    4. GS

      Yeah.

    5. LF

      ... where they just chat about things that are extremely complicated, actually.

    6. GS

      Yeah.

    7. LF

      People, nevertheless, connect with them.

    8. GS

      Yeah.

    9. LF

      What do you think that is? What-

    10. GS

      It's wonderful, isn't it?

    11. LF

      Yeah.

    12. GS

      I mean, I wasn't really aware of it. Do s- we're- 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, you know, a lot of order and org- you know, things are, uh, not obvious, but they're true. So it's really cheering to think that, that, uh, so many people really want to learn more about math. Yeah.

  8. 21:08 – 22:35

    Math as comfort and certainty: symmetry, truth, and the ‘powers of two’ story

    1. LF

      Well, in terms of truth-

    2. GS

      Yeah.

    3. LF

      ... again, sorry to, uh, slide into philosophy at times-

    4. GS

      Yeah.

    5. LF

      ... but, uh, math does reveal pretty strongly what things are true.

    6. GS

      Yeah.

    7. LF

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

    8. GS

      It is, yeah.

    9. LF

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

    10. GS

      It is.

    11. LF

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

    12. GS

      Oh, wow. (laughs)

    13. LF

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

    14. GS

      Yeah.

    15. LF

      ... in our lives.

    16. GS

      That's right. Well, I have to say, I'm- I'm not much of a philosopher. I just like numbers, you know?

    17. LF

      (laughs) Yeah.

    18. GS

      As a kid, I would, you'd... This was before you had, uh, uh, you had to go in when you're, when you had a filling in your teeth. You had to kind of just take it. Yeah. So I, what I did was think about math, you know, like take powers of two, two, four, eight, 16, up until the time the tooth stopped hurting-

    19. LF

      (laughs)

    20. GS

      ... and the dentist said you're through. Uh, or counting. Yeah. So it...

    21. LF

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

    22. GS

      Yeah.

    23. LF

      The... What- what- what- what is it-

    24. GS

      Yeah.

    25. LF

      ... about math do you think that brings that?

    26. GS

      Yeah.

    27. LF

      What- what is that?

    28. GS

      Just, well, you know-

    29. LF

      The symmetry, the-

    30. GS

      ... where you are. Yeah.

  9. 22:35 – 25:04

    Tool vs art—and how engineers learn: examples, answers, and intuition

    1. LF

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

    2. GS

      So it's both. Uh, that's- that's really one of the neat things. You can, you can be an artist and- and like math. You can be a, uh, engineer and- and use math.

    3. LF

      Which are you? Which-

    4. GS

      Which am I? Yeah.

    5. LF

      What- what- what did you connect with most-

    6. GS

      Yeah. I'm somewhere-

    7. LF

      ... in- in your-

    8. GS

      ... between. I'm certainly not a artist type, philosopher type person. Might sound that way this morning, but I'm not.

    9. LF

      (laughs)

    10. GS

      (laughs) Uh, uh, yeah. I- I really enjoy teaching engineers-

    11. LF

      Yeah.

    12. GS

      ... because they- they- they go for an answer. And, uh, um, yeah. So probably within the ma- you know, MIT math department, uh, most people enjoy teaching people, teaching students who get the abstract idea. I'm okay with- with, uh, I'm good with engineers who are looking for a way to find answers. Yeah.

    13. LF

      Well, actually that's a interesting question. Do you think, do you think for teaching and in general, thinking about new concepts, do you think it's better to plug in the numbers-

    14. GS

      (laughs)

    15. LF

      ... or to think more abstractly? So, uh, l- looking at theorems and proving the theorems, or actually, you know, building up a basic intuition of the theorem or the method, the approach, and then just plugging in numbers and seeing it work?

    16. GS

      Yeah. Well, certainly, uh, uh, many of us like to see examples first. We- we understand it might be a pretty abstract sounding example, like a three-dimensional rotation. How- how are you gonna, how are you gonna understand a rotation in 3D, uh, or in 10D?... or, uh, uh, but, uh... The, and then some of us like to keep going with it to the point where you got numbers, where you got-

    17. LF

      Mm-hmm.

    18. GS

      ... 10 angles, 10 axes, 10 angles, uh, but, uh, the best, the great mathematicians probably... I don't know if they do that 'cause they, they... for them, uh, uh, uh, uh, uh, for them, uh, an example would be a highly abstract thing to the rest of us.

    19. LF

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

    20. GS

      Yeah, examples.

    21. LF

      It seems to-

    22. GS

      Examples of, of structure.

    23. LF

      Our brains seem to connect with that.

    24. GS

      Yeah, yeah.

  10. 25:04 – 28:20

    Math, politics, and SIAM: why quantitative thinking is underrepresented

    1. LF

      So, uh, not sure if you're familiar with him, but A- Andrew Yang is a presidential candidate currently running-

    2. GS

      Yeah.

    3. LF

      ... with, uh, "MATH" in all capital letters in his hats as a slogan.

    4. GS

      I see.

    5. LF

      Stands for "Make America Think Hard."

    6. GS

      Okay.

    7. LF

      (laughs)

    8. GS

      I'll vote for him. (laughs) .

    9. LF

      (laughs) So, uh... And his name rhymes with yours, Yang, Strang, so. But he also loves math-

    10. GS

      Yeah.

    11. LF

      And he comes from that world of-

    12. GS

      Right.

    13. LF

      But he also... Looking at it makes me realize that math, science, and engineering are not really part of our politics, uh-

    14. GS

      Right.

    15. LF

      ... political discourse, about political life-

    16. GS

      Yeah.

    17. LF

      ... government in general.

    18. GS

      Yeah.

    19. LF

      Why do you think that is?

    20. GS

      Well-

    21. LF

      What are your thoughts on that in general?

    22. GS

      Well, certainly somewhere in the system we need people who are comfortable with numbers, comfortable with quantities. You know, if you, if you say this leads to that, they see it and it's, it's undeniable.

    23. LF

      But isn't that strange to you that we have almost no... I mean, I'm pretty sure we have-

    24. GS

      Yeah.

    25. LF

      ... no elected officials in Congress or obviously the pr- president-

    26. GS

      Yeah.

    27. LF

      ... that is a... either has an engineering degree or a math degree (laughs) .

    28. GS

      Yeah, well, uh, that's too bad, in... No, a few could, uh... a few who could make the connection. Yeah, it would have to be people who are at the... who are, uh, who understand engineering or science and at the same time can make speeches and, uh, and, uh, lead, yeah.

    29. LF

      And inspire people.

    30. GS

      Yeah, inspire, yeah.

  11. 28:20 – 33:01

    Deep learning in plain terms: learning rules from data with linear algebra + nonlinearity

    1. LF

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

    2. GS

      Yeah, sure.

    3. LF

      ... about the future.

    4. GS

      Yeah, uh-

    5. LF

      And I think at the core of that is math (laughs) .

    6. GS

      Well, it is, yeah.

    7. LF

      Yeah.

    8. GS

      But-

    9. LF

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

    10. GS

      Well, that's right.

    11. LF

      But, uh-

    12. GS

      Well, uh, artificial intelligence and, and particularly, kinda, I use the words "deep learning."

    13. LF

      Yeah.

    14. GS

      It's a... Deep learning is a particular approach to understanding data. Again, you've got a big, whole lot of data where data is just swamping the, the computers of the world and, uh, and to un- understand it, to out of all those numbers, to find what's important, you know, in, in climate, in everything. And artificial intelligence is too words for, uh, for one approach to data. Deep learning is a specific, uh, approach there which uses a lot of linear algebra, so I got into it. I thought, "Okay, I've gotta learn about this."

    15. LF

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

    16. GS

      Yeah.

    17. LF

      H- how do you think of a neural network? What is a neural network?

    18. GS

      What... Yeah, okay. So can I s- start with the, uh, i- idea about deep learning? What does that mean?

    19. LF

      Sure.

    20. GS

      Uh...

    21. LF

      What is deep learning? (laughs)

    22. GS

      What is deep learning? Yeah. So, so w- we're trying to learn... From all this data, we're trying to learn what's important, what, what's, um, what's it telling us. So you've, 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, to get the, to, 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 will take those inputs, those million training inputs which we know about, to the correct million outputs.... and, uh, this idea of a neural net is part of the structure of the, of our new way to create a, create a rule. We're looking for a rule that will take these training inputs to the known outputs, and then we're gonna use that rule on new inputs that we don't know the output and, and see what comes.

    23. LF

      Linear algebra is a big part of defining- of finding that rule.

    24. GS

      That's right. Linear algebra is a big part, not all the part. People were leaning on matrices, that's good, still do. Linear is something special. It's, it's all about straight lines and flat planes, and, uh, and, and data isn't quite like that, you know? It's, uh, it's, it's more complicated. So you gotta introduce some complication, so you have to have some function that's not a straight line-

    25. LF

      Nonlinear.

    26. GS

      ... and it turned out that... Nonlinear, nonlinear-

    27. LF

      Scary.

    28. GS

      ... not linear. And it turned out that, uh, it was enough to use the function that's one straight line and then a different one halfway-

    29. LF

      That's- (laughs)

    30. GS

      ... so piecewise linear.

  12. 33:01 – 38:51

    Expressivity, finite elements, and the limits of neural networks

    1. GS

      Well, I'm beginning to have a better intuition. This idea of things that are piecewise linear, flat pieces but, but with folds between them, like think of a roof of a complicated, i- infinitely complicated house or something, that, that, that curve, it almost curve, but it- but every piece is flat. Uh, that, that's been used by engineers, that idea has been used by engineers, uh, is used by engineer- big time, something called the finite-element method. If you wanna, if you wanna design a bridge, design a building, d- design a pl- airplane, you're, you're using this idea of piecewise flat as, as, as a good, a simple computable approximation.

    2. LF

      So, but you're- you have a sense that, um, that there's a lot of expressive power in this kind of piecewise linear-

    3. GS

      Yeah. That's-

    4. LF

      ... functions combined together?

    5. GS

      That- you used the right word. Ex- if you measure the expressivity-

    6. LF

      Yeah.

    7. GS

      ... how, how ma- how complicated a thing can, can this piecewise flat guys express? The answer is very complicated. Yeah, so...

    8. LF

      What do you think are the limits of such piecewise linear or just-

    9. GS

      Yeah.

    10. LF

      ... of neural networks, the expressivity of neural networks?

    11. GS

      Well, you would have said a while ago that they're just computational limits. It- y- you know, you, y- uh, it's a problem beyond a certain size, a supercomputer isn't gonna do it. But that- those keep getting more powerful, so that's, uh, that limit has been moved, uh, y- to allow more and more complicated surfaces.

    12. LF

      So in terms of just mapping, uh, from inputs to outputs-

    13. GS

      Yeah.

    14. LF

      ... looking at data-

    15. GS

      Yeah.

    16. LF

      ... what do you think of, uh, you know, in, in the context of neural networks in general, data is just tensor vectors, matrices-

    17. GS

      Right, yeah.

    18. LF

      ... tensors.

    19. GS

      Right.

    20. LF

      W- how do you think about learning from data? What- how much of our world can be expressed in this way? How useful is this process? Is- I guess that's another way of asking what are the limits of this approach?

    21. GS

      Well, that's a good question, yeah. So I guess the whole idea of deep learning is that there's something there to learn. If the data is totally random, just produced by random number generators, then th- we're not gonna find a useful rule 'cause there isn't one. So, uh, uh, the extreme of having a rule is like knowing Newton's Law, you know? If you hit a, hit a ball, it moves. So that's where you had laws of physics, Newton and, and Einstein and other great, great people have, have found those laws and laws of, uh, the, the, the, the distribution of oil in a, in a underground thing. I mean, that- so, so, uh, engineers, petroleum engineers, uh, uh, understand how, how oil will sit in a, in a underground basin. Uh, so there were rules. Now, now the, the new idea of artificial intelligence is learn the rules instead of s- instead of figuring out the rules by- with help from Newton or, or Einstein-... the computer is looking for the rules. So that's another step. But if there are no rules at all for, uh, that the computer could find, if it's totally random data, well, y- you got nothing, you got no science to, to discover.

    22. LF

      It's a automated search for the underlying rules.

    23. GS

      Yeah. Search-

    24. LF

      Um...

    25. GS

      ... search for the rules, yeah, exactly.

    26. LF

      Do you think-

    27. GS

      And there will be a lot of random parts, a lot of... I mean, I'm not knocking random 'cause-

    28. LF

      (laughs)

    29. GS

      ... you know, the, that's there. The, the, uh, the, uh, there's a lot of randomness built in, but there's gotta be some basic, uh-

    30. LF

      It's almost always signal, right?

  13. 38:51 – 41:53

    Calculus vs linear algebra (again): rebalancing the undergraduate math triad

    1. LF

      At its lowest, uh, simplest level. Maybe just a quick, in broad strokes, from your perspective, what is, uh, where does l- linear algebra sit as a sub-field of mathematics? What, what are the various sub-fields that you're-

    2. GS

      Okay.

    3. LF

      ... that you think about in relation to linear algebra?

    4. GS

      So the big fields of math are algebra as a whole, and problems like c- calculus and differential equations, so that's a second, quite different field. Then maybe geometry, uh, deserves to be under- thought of as a different field under "Stand the geometry of high-dimensional surfaces." So I think... Am I allowed to say this here? I- I think-

    5. LF

      Uh-oh.

    6. GS

      ... m-

    7. LF

      Calculus?

    8. GS

      ... this, this is, this is-

    9. LF

      Discussion? (laughs)

    10. GS

      ... this is where, uh, personal view comes in. I think, uh, m- math, if we're thinking about undergraduate math-

    11. LF

      Mm-hmm.

    12. GS

      ... what, what millions of students study, I think we o- uh, overdo the calculus at, at the cost of the algebra, at the cost of linear.

    13. LF

      So you have this talk-

    14. GS

      Linear.

    15. LF

      ... titled Calculus Versus Linear Algebra.

    16. GS

      That's right. That's right.

    17. LF

      No, and s- and, and you say that linear algebra wins, so-

    18. GS

      Uh, well...

    19. LF

      ... can you, can you, uh, can you dig into that a little bit?

    20. GS

      Yeah.

    21. LF

      Why does linear algebra win? (laughs)

    22. GS

      (laughs) Right. Well, okay, I'm, uh... The viewer is gonna think this guy is biased.

    23. LF

      (laughs)

    24. GS

      Not true. I'm, I'm just telling the truth as it is. Yeah, so I- I feel linear algebra is, is, is just a nice part of math that people can get the idea of. They can understand something that's a little bit abstract, 'cause once you get to 10 or 100 dimensions, um... And very, very, very useful. That's what's happened in, in my lifetime is the, the, the importance of data which does come in matrix form, so it's really set up for algebra. It's not set up for differential equations. And, uh, uh, let me f- fairly add probability. The ideas of probability and statistics have become very, very important, have also jumped forward. So... And that's not... That's different from linear algebra, quite different. So now, we really have three major areas to me. Uh, calculus, uh, linear algebra, matrices, and probability statistics, and, uh, they all deserve a important place.

    25. LF

      Oh, yeah.

    26. GS

      And, and, and calculus is traditionally had a, had a, it's a lion's share of the time-

    27. LF

      A disproportionate share.

    28. GS

      ... a dis- thank you.

    29. LF

      Yeah.

    30. GS

      Disproportionate, that's a good word.

  14. 41:53 – 46:20

    A favorite matrix and the joy of teaching: second derivatives, assessments, and ‘getting it’

    1. LF

      I know it's hard to pick favorites, but what is your favorite matrix?

    2. GS

      What's my (laughs) favorite matrix?

    3. LF

      (laughs)

    4. GS

      Okay. So my favorite matrix is square, I admit it. It's a square bunch of numbers, and it has twos running down the main diagonal. T- and, uh, on the next diagonal, so think of top left to bottom right, twos down, down the middle of the matrix, and minus ones just above those twos, and minus ones just below those twos. And otherwise, all zero. So mostly zeros. Just three non-zero diagonals coming down.

    5. LF

      What is interesting about it?

    6. GS

      Well, all the different ways it comes up.You see it in engineering, you see it as analogous in calculus to second derivative. So, calculus learns about taking the derivative, the figuring out how much, how fast something's changing. But second derivative, now that's also important. That's how fast the change is changing, how fast the graph is bending. How, how fast it's, it's curving. Uh, and Ei- Einstein showed that that's fundamental to understand space. So, second derivatives should have a bigger place in calculus, second mice, matrices, which are like the linear algebra version of second derivatives, are neat in, in linear algebra. Yeah. Just everything comes out right with those guys. (laughs)

    7. LF

      (laughs)

    8. GS

      Yeah.

    9. LF

      Beautiful. What have you learned about the process of learning by having taught so many students math over the years?

    10. GS

      Ooh, that is hard. I'll have to admit here that I'm not, I'm not really a good teacher.

    11. LF

      (laughs)

    12. GS

      Because I don't get into the exam part. The exam's the part of my life that I don't like, and grading them, and giving the students A or B or whatever. I do it because it's, I'm supposed to do it. But, uh, but I tell the class at the beginning, I don't know if they believe me, probably they don't, I, I tell the class, "I'm here to teach you. I'm here to teach you math, and not to grade you." And, but they're thinking, "Okay, this guy is gonna, you know, when's he gonna, is he gonna give me an A minus? Is he gonna give me a B plus? What?"

    13. LF

      What, what have you learned about the process of learning?

    14. GS

      Of learning? Yeah. Well-

    15. LF

      So the students-

    16. GS

      ... maybe to be the, to give you a legitimate answer about learning, I should've paid more attention to the assessment, the evaluation part at the end. But I like the teaching part at the start, that's the sexy part-

    17. LF

      (laughs)

    18. GS

      ... to, to tell somebody for the first time about a matrix. Wow.

    19. LF

      But is there, are there moments, so you, you are teaching a concept, are there moments of learning that y- you just see in the students' eyes, you don't need to look at the grades-

    20. GS

      Yeah.

    21. LF

      ... but you see in their eyes that, that you hook them. That, you know, that you connect with them in a way where, you know what, they, they f- they fall in love with this-

    22. GS

      Yeah.

    23. LF

      ... with this beautiful world of mathematics.

    24. GS

      They see that it's got some beauty there.

    25. LF

      It g- it, it, see-

    26. GS

      Yeah.

    27. LF

      Or conversely-

    28. GS

      Yeah.

    29. LF

      ... that they give up at that point-

    30. GS

      Uh-huh.

  15. 46:20 – 49:47

    Advice for students and a life in math: passion, fun, and meaningful connection

    1. LF

      What advice do you have to a student just starting their journey in mathematics today? How do they get started? (laughs)

    2. GS

      (laughs) Oh, yeah, that's hard. Well, I hope you, you have a teacher, professor who, uh, is still enjoying what he's doing.

    3. LF

      Mm-hmm.

    4. GS

      What he's teaching. He's still looking for new ways to teach and to, and to understand math. Uh, 'cause that's the pleasure to, to, the moment when you see, "Oh, yeah, that works."

    5. LF

      So it's less about the material, you-

    6. GS

      Yeah.

    7. LF

      ... you study, it's more about the source of the teacher being full of passion for the subject.

    8. GS

      Yeah, more about the fun. Yeah.

    9. LF

      The fun.

    10. GS

      The, the, the moment of un- of getting it.

    11. LF

      But in terms of topics?

    12. GS

      Well-

    13. LF

      Linear algebra?

    14. GS

      Well, that's my, my topic. But, oh, there's beautiful things in geometry to understand. What's wonderful is that in the end, there, uh, there's a pattern, there, there's a, there are rules that, that, uh, that are followed in, uh, biology as there are in every field.

    15. LF

      You describe the life of a mathematician as, as, uh, 100% wonderful.

    16. GS

      (laughs)

    17. LF

      Uh, except for the grade stuff-

    18. GS

      Yeah.

    19. LF

      ... and having good grades.

    20. GS

      Except for grades.

    21. LF

      Yeah. When you look back at your life-

    22. GS

      Yeah.

    23. LF

      ... what memories bring you the most joy and pride?

    24. GS

      Well, that's a good question. I certainly feel good when I, maybe I'm giving a class in, in 18.06, that's MIT's linear algebra course that I started, so sort of there's a good feeling that, "Okay, I started this course. A lot of students take it, quite a few like it." Yeah. So I'm, I'm sort of happy when I feel I'm helping, helping make a connection between ideas and students. Between, uh, theory and the reader. Uh, yeah, it's, uh, I get a lot of very nice messages, uh, from, uh, people who've watched the videos, and it's, uh, inspiring. I just, and I'll maybe just take this chance to say thank you.

    25. LF

      Well, there's millions of students who you've taught, and I am grateful to be one of them, so Gilbert-

    26. GS

      Thank you.

    27. LF

      ... thank you so much. It's been an honor. Thank you for talking today.

    28. GS

      I- it was a pleasure. Thanks.

    29. LF

      Thank you for listening to this conversation with Gilbert Strang. And thank you to our presenting sponsor, Cash App. Download it, use code LEXPODCAST, you'll get $10, and $10 will go to FIRST, a STEM education nonprofit that inspires hundreds of thousands of young minds to learn and to dream of engineering our future. If you enjoy this podcast, subscribe on YouTube, give it five stars on Apple Podcasts, support on Patreon, or connect with me on Twitter. Finally, some closing words of advice from the great Richard Feynman, "Study hard what interests you the most, in the most undisciplined, irreverent, and original manner possible." Thank you for listening, and hope to see you next time.

Episode duration: 49:52

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