Lex Fridman Podcast - Gilbert Strang: Linear Algebra, Deep Learning, Teaching, and MIT OpenCourseWare

Episode Date: November 25, 2019

Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times. This conversation is part of the Artificial Intelligence podcast. If you would like to get more information about this podcast go to https://lexfridman.com/ai or connect with @lexfridman on Twitter, LinkedIn, Facebook, Medium, or YouTube where you can watch the video versions of these conversations. If you enjoy the podcast, please rate it 5 stars on Apple Podcasts or support it on Patreon. This episode is presented by Cash App. Download it, use code LexPodcast.  And it is supported by ZipRecruiter. Try it: http://ziprecruiter.com/lexpod Here's the outline of the episode. On some podcast players you should be able to click the timestamp to jump to that time. 00:00 - Introduction 03:45 - Math rockstar 05:10 - MIT OpenCourseWare 07:29 - Four Fundamental Subspaces of Linear Algebra 13:11 - Linear Algebra vs Calculus 15:03 - Singular value decomposition 19:47 - Why people like math 23:38 - Teaching by example 25:04 - Andrew Yang 26:46 - Society for Industrial and Applied Mathematics 29:21 - Deep learning 37:28 - Theory vs application 38:54 - Open problems in mathematics 39:00 - Linear algebra as a subfield of mathematics 41:52 - Favorite matrix 46:19 - Advice for students on their journey through math 47:37 - Looking back

The following is a conversation with Gilbert Strang. He's a professor of mathematics in MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT Open Coursework 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. There's at once calm, simple, and yet full of passion for the elegance inherent to mathematics. I remember doing the exercise 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 5 stars and Apple Podcasts, support it on Patreon, or simply connect with me on Twitter,

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I've personally seen inspire girls and boys to dream of engineering a better world. And now here's my conversation with Gilbert Strain. How does it feel to be one of the modern-day rock stars of mathematics? I don't feel like a rock star, that's kind of crazy for old math person, but it's true that the videos in linear algebra that I made way back in 2000, I think. I've been watched a lot. And well, partly the importance of linear algebra, which I'm sure you'll ask me, and give me a chance to say that linear algebra

as a subject is just surged and important. But also, it was a class that I taught a bunch of times, so I kind of got it organized and enjoyed doing it. It was just the videos were just the class. So they're on OpenCourseWare and on YouTube and translated. That's fun. But there's something about that chalkboard and the simplicity of the way you explain

the basic concepts in the beginning. To be honest, when I went to undergrad, you know, do linear algebra probably. Of course, did linear algebra. Yeah, yeah, of course, but I before going through the course at my university, I was going through open course where I was, you were my instructor. Oh, right. Yeah.

And that, I mean, we're 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, yeah, that's really powerful. So, how do you think the idea of putting lectures online would really MIT OpenCourseWare has innovated. That was a wonderful idea. I think the story that I've heard is the committee was appointed by the president, president Vest, at that time, a wonderful guy. And the idea of the committee was to figure out

how MIT could be like other universities, market, the work we were doing. And then they didn't see a way, and after a weekend, and they had an inspiration, came back to the present vest, and said, what if we just gave it away? And he decided that was, okay, good idea. So you know, that's a crazy idea. That's, if we think of a university as a thing that creates a product, isn't knowledge. Right. You know, the kind of educational knowledge isn't the product and giving that away.

Yeah. Are you surprised that it went through the result that he did it. Well, knowing a little bit President Vest, it was like him, I think. And it was really the right idea. MIT is kind of, it's known for being high-level technical things. And this is the best way we can say, we can show what MIT really is like. Because in my case, those 1806 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 don't know where that comes through. It's somehow a bit friendly with a class.

I like students. And then you're algebra, we get to give the subject most of the credit. It really has come forward and importance in these years. So let's talk about linear algebra a little bit, because it is such a, it's both a powerful and a beautiful subfield of mathematics. So what's your favorite specific topic in linear algebra or even math in general to give a lecture on to two people? I would convey to tell

a story to teach students. 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 four fundamental subsp, which are of course known before or long before my name for them, but can you go through them? Can you go through the first question? Sure I can. Yeah, so the first one to understand is, so the matrix, maybe I should say the matrix.

What is the matrix? What's a matrix? Well, so we have a rectangle of numbers. So it's got in columns, got a bunch of columns, and also got an M rows, let's say, and the relation between, so of course, the columns and the rows, it's the same numbers. So there's got to be connections there, but they're not simple. The columns might be longer than the rows, and they're both different, the numbers are mixed up. First space to think about is, take the columns.

So those are vectors, those are points in end dimensions. What's a vector? So a physicist would imagine a vector or might imagine a vector as an arrow in space or the point it ends at in space. For me, it's a column of numbers. So you often think of, this is very interesting in terms of linear algebra in terms of a vector. You think a little bit more abstract than how it's very commonly used, perhaps. You think this arbitrary spate, multidimensional, is what I'm saying.

Right away, I'm in high dimensions in the dreamland. Yeah, that's right, in the 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 arrow in 10 dimensional space, but whatever. You can add one bunch of 10 numbers to another bunch of 10 numbers, so you can add a vector to a vector. And 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. You know, 10 dimensions. Yeah. You know, there's this beautiful thing about math. If you

look at strength theory and all these theories, which are really fundamentally derived through math, but are very difficult to visualize. visualize. How do you think about the things like a 10-dimensional vector that we can't really visualize? And yet, math reveals some beauty underlying our world in that weird thing we can't visualize. How do you think about that difference? 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 it goes on to ten dimensions with no problem. I mean, if you're just seeing what happens if you add two vectors in 3D, you're just adding the 10 components. So I can't say that I have a picture, but yet I try to push the class to think of a flat surface in 10 dimensions.

So a plane in 10 dimensions. So that's one of the spaces. Take all the columns of the matrix, take all their combinations, 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 my imagination is just seeing like a piece of paper in 3D. But anyway, so that's one of the spaces, 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 So we got the column space, all combinations of the columns, and then we got the row space, all combinations of the rows. So those words are easy for

me to say, and I can't really draw them on a blackboard, but I try with my thick chalk. Everybody likes that railroad chalk. And me too, I wouldn't use anything else now. And then the other two spaces are perpendicular to those. So like if you have a plane in 3D, just a plane is just a flat surface in 3D, then perpendicular to that plane would be a line. So that would be the null space. So we've got a column space, a row space, and there are two perpendicular spaces.

So those four fit together in a beautiful picture of a matrix. Yeah. Yeah. It's sort of a fundamental. It's not a difficult idea. It comes pretty early in 1806, and it's sort of a fundamental. It's not a difficult idea. It comes, comes pretty early in 1806 and it's basic. Plains in these multi-dimensional spaces, how, how difficult of an idea is that to, to come to? Do you think if you, if you look back in time, I think mathematically it makes sense, but I don't know if it's intuitive for us to imagine just as we're talking about. It feels like calculus is easier to...

I see! ...into it. Well, calculus, I have to admit calculus came earlier. Earlier than linear algebra. So Newton and Leibniz were the great men to understand the key ideas of calculus. But linear algebra, to me, is like, okay, it's the starting point because it's all about flat things. Calculus has got all the complications of calculus come from the curves, the bending, the curved surfaces. Linear algebra, the surfaces are all flat nothing bands in the linear algebra. So it should have 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 that'll be calculus and then I say enough of it like okay get to get to the good stuff. And that's- Do you think linear algebra should come first? Well, it really, I'm okay with it not coming first, but it should. Yeah, it should.

It's simpler. Because everything is flat. Yeah, everything's flat. Well, of course, for that reason, calculus sort of sticks to one dimension, or eventually you do multi-variate, but that basically means two dimensions. Linerality, it take off into ten dimensions, no problem.

It just feels scary and dangerous to go beyond two dimensions, but that's all. If everything is flat, you can't go wrong. So what concept or theorem in linear algebra or in math, you find most beautiful. It gives you pause that leaves you in awe. Well, I'll stick with linear algebra here. I hope the viewer knows that really mathematics is amazing, amazing subject and deep, deep connections between ideas that didn't look connected, they turned out they were. But if we stick with linear algebra, so we have a matrix. That's like the basic thing, a rectangle of numbers.

And it might be a rectangle of data. You're probably going to ask me later about data science, where an often data comes in a matrix. You have maybe every column corresponds to a drug in every row corresponds to a patient. And if the patient reacted favorably to the drug, then you put up some positive number in there. Anyway, rectangle of numbers, matrix is basic. So the big problem is to understand all those numbers.

You got a big set of numbers. And what are the patterns? What's going on? And so one of the ways to break down that matrix into simple pieces is use as something called singular values. And that's come on as fundamental in the last, certainly in my lifetime. I can values, if you have viewers who've done engineering math or basic linear algebra, I can values were in there, but those are restricted to square matrices. And data comes in rectangular matrices.

So you've got to take that next step. I'm always pushing math faculty. Get on, do it, do it, singular values. So those are a way to make, to find the important pieces of the matrix, which add up to the whole matrix. So you're breaking a matrix into simple pieces and the first piece is the most important part

of the data, the second piece is the second most important part. And then often, so a data scientist will like, if a data scientist can find those first and second pieces, stop there. The rest of the data is probably round off, experimental error, maybe. So you're looking for the important part. Yeah. So what do you find beautiful about singular values?

Well, yeah. I didn't give the theorem. So here's the idea of singular values. Every matrix, every matrix, 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 just a diagonal matrix, otherwise all zeroes except on one diagonal, and then the third factor is another rotation.

So rotation, stretch, rotation is the breakup of any matrix. The structure of that, the ability that you can do that, what do you find appealing, what do you find beautiful about it? Well, geometrically, as I freely admit, the action of a matrix is not so easy to visualize, but everybody can visualize a rotation. Take two dimensional space and just turn it around the center. Take three dimensional spaces. So a pilot has to know about, well, what are the three? The yaw is one of them.

I've forgotten all the three turns that a pilot makes. Up to 10 dimensions, you've got 10 ways to turn. But you can visualize a rotation. Take this space and turn it, but you can visualize a rotation. Take this space and turn it, and you can visualize a stretch. So to break a matrix with all those numbers in it, into something you can visualize, rotate, stretch, rotate. It's pretty neat. That's pretty neat. Yeah, that's pretty powerful. I need to 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'm trying to understand why that is perhaps you you can help me give me clues. So it's not just the kinds of lectures that you give, but it's also just other folks like with number file, there's a channel where they just chat about things that are extremely complicated, actually. People nevertheless connect with them. What do you think that is?

It's wonderful, isn't it? I mean, I wasn't really aware of it Do so 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, no, 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, or, you know,

things are not obvious, but they're true. So it's really cheering to think that so many people really want to learn more about math. Yeah. In terms of truth, again, I'm sorry to slide into philosophy at times, but math does reveal pretty strongly what things are true. I mean, that's the whole point of proving things.

And yet, sort of our real world is messy and complicated. What do you think about the nature of truth that math reveals? Oh, wow. Because it is a source of comfort like you've mentioned. Yeah, that's right. Well, I have to say, I'm not much of a philosopher. I just like numbers, you know, I'd say, kid, I would, this was before you had to go in when you're in the end of filling your teeth, yeah, to kind of just take it. So what I did was think about math, you know, like take powers of two, two, four, eight, sixteen up until the time the tooth stopped hurting and the dentist said you're through. Or counting.

So that was a source of just piece almost. Yeah. What is it about math? You think that brings that? Yeah, what is that? Well, you know where you are. Simetry. It's certainty.

The fact that, you know, if you two, if you mobile it's 10 times you get a thousand and twenty-four period, everybody's going to get that. GC Math is a powerful tool or an art form. So it's both. That's really one of the neat things. You can be an artist and like math, you can be an engineer and use math. Which are you? Which to my?

What did you connect with most? Yeah, I'm in here. In here, between. I'm certainly not an artist type philosopher type person. Might sound that way this morning, but I'm not. Yeah, I really enjoy teaching engineers. They go for an answer. And yeah, so probably within the MIT math department, most people enjoy teaching students who

get abstract idea. I'm okay with, I'm good with engineers who are looking for a way to find answers. Yeah. Actually that's an interesting question. Do you think, do you think for teaching, and in general, but think about new concepts, do you think it's better to plug in the numbers or to think more abstractly? So looking at theorems and proving the theorems or actually building up a basic intuition of the theorem or the method the approach

and then just plugging in numbers and seeing it work. Yeah, well, certainly many of us like to see examples. First, we understand it might be a pretty abstract sounding example like a three-dimensional rotation. How are you going to understand a rotation in 3D or in tendi, or... But... And then some of us like to keep going with it to the point where you got numbers. Where you got ten angles, ten axes, ten angles. But the best, the great mathematicians, probably, I don't know if they do that,

because they... for them, for them, an example would be a highly abstract thing to the rest of it. Right, but nevertheless working with it in the space of examples. Yeah, exactly. It seems to. It seems to examples of structure. Our brains seem to connect with that.

Yeah. Yeah. So, I'm not sure if you're familiar with him, but Andrew Yang is the presidential candidate currently running with a math in all capital letters and his hats as a slogan. I see. It stands for Make America Think Card. Okay. I'll vote for him. So, uh, and his name rhymes with yours, Yang, Strang, so, but he also loves math and he comes from that world. But he also Looking at it makes me realize that math science and engineering are not really part of our politics

Political discourse about political like government in general. Yeah. What do you think? That is well, what are your thoughts, government in general. Yeah. What do you think that is? Well, what are your thoughts on that in general? Well, certainly somewhere in the system, we need people who are comfortable with numbers, comfortable with quantities, you know, if you say this leads to that, they see it, and it's undeniable. But isn't it strange to you that we have almost no, I mean, I'm pretty sure we have no elected officials in Congress or obviously the president that

either has an engineering degree or a math. Yeah, well, that's too bad. A few could make the connection. It would have to be people who understand engineering or science and at the same time can make speeches and lead. And inspire people. You were speaking of inspiration, the president of the society for industrial applied mathematics. Oh, yes. It's a major organization in math and applied math. What do you see as a role of that society, you know, in our public discourse?

Right. In public. Yeah. So what it was fun to be president at the time, a couple of years, two years around, around 2000. This hope that's president of a pretty small society, but nevertheless, it was a time when math was getting some more attention in Washington. But yeah, I got to give a little in Washington. But yeah, I got to give a little 10 minutes to a committee of the House of Representatives and I was talking about who I met. And then actually it was fun because one of the members of the House had been a student, it had been in my class. What do you think of that? As you say, a pretty rare, most members of the house have had a different training, different background, but there was one from New Hampshire who was my friend, really, by being in the class. Yeah. Those years were good. Then, of course, other things take over an importance in Washington and the math just at this point is not so visible. But for a little moment, it was.

There's some excitement, some concern about artificial intelligence in Washington now, about the future. Yeah. And I think at the core of that is math., about the future. Yeah. And I think at the core of that is math. Well, it is. Yeah. But maybe it's hidden, maybe it's wearing a different hat. Oh, wow. Well, artificial intelligence, and particularly,

kind of, I use the words deep learning. It's a deep learning as a particular approach to understanding data. Again, you've got a big whole lot of data. Where data is just swamping the computers of the world and to understand it, out of all those numbers to find what's important in climate and everything. And artificial intelligence is two words for one approach

to data. Deep learning is a specific approach there, which uses a lot of linear algebra, so I got into it. I thought, okay, I've got to learn about this. So maybe from your perspective, I may ask the question. Yeah, how do you think of on your own network? What is it? And you're on that one? Yeah, okay. So can I start with the idea about deep learning? What does that mean? Sure. What is deep learning? What is deep learning? Yeah. So, so we're trying to learn from all this data, we're trying to learn what's important, what's it telling us. So you've got data.

You've got some inputs for which you know the right outputs. The question is, can you see the pattern there? Can you figure out a way for a new input, which we haven't seen, to understand what the output will be from a new input, which we haven't seen, to get the 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 patterns, some rule that will take those inputs, which we know about, to the correct million outputs. And this idea of a neural net is part of the structure of our new way to create a rule. We're looking

for a rule that will take these training inputs to the known outputs. And then we're going to use that rule on new inputs that we don't know the output and see what comes. Linear algebra is a big part of defining that rule. That's right. Linear algebra is a big part. Not all the parts. People were leaning on matrices, that's good, still do. Linear is something special. It's all about straight lines and flat planes. And data isn't quite like that, you know. It's more complicated.

So you gotta introduce some complication. So you have to have some function that's not a straight line. And it turned out, non-linear non-linear And it turned out that it was enough to use the function that's one straight line and then a different one halfway So piecewise linear piecewise piece of one piece has one slope one piece the other piece has a second slope and has one slope, one piece, the other piece has the second slope. Yeah. And so that, getting that nonlinear, simple nonlinearity in blue, the problem open.

That little piece makes it sufficiently complicated to make things interesting. Exactly. Because you're going to use that piece over and over a million times. So, it has a fold in the graph, the graph, two pieces. But when you fold something a million times, you've got a pretty complicated function that's pretty realistic. So that's the thing about neural networks is they have a lot of these. A lot of these. Exactly. So, why do you think neural networks by using, so, formulating an objective function, they're not a plane. Yeah.

A function of the- Lots of folds. Lots of folds of the inputs, the outputs. Why do you think they work to be able to find a role that we don't know is optimal, but it just seems to be pretty good in a lot of cases. What's your intuition? Is it surprising to you as it is to many people? You have an intuition of why this works at all?

Well I'm beginning to have a better intuition. This idea of things that are piecewise linear, flat pieces, but with folds between them. Like think of a roof of a complicated, infinitely complicated house or something. That curve, it almost curved, but every piece is flat. That's been used by engineers. That idea has been used by engineers. Is used by engineers. Big time, something called the finite element method. If you want to design a bridge, design a building, design a airplane.

You're using this idea of piecewise flat as a good simple, computable approximation. But you have a sense that there's a lot of expressive power in this piecewise linear that's combined together. You use the right word. If you measure the expressivity, how complicated a thing can this piecewise flat guys express the answer is

very complicated. What do you think are the limits of such piecewise linear or just neural networks the expressivity of neural nodes? Well, you would have said a while ago that they're just computational limits. It's a problem beyond a certain size. A supercomputer isn't gonna do it. But that was keep getting more powerful. So that limit has been moved to allow more and more complicated surfaces. So in terms of just mapping from inputs to outputs,

looking at data, what do you think of, you know, in the context of neural networks in general, data is just tensor vectors, matrices, tensors. Right. How do you think about learning from data? How much of our world can be expressed in this way? How useful is this process? I guess that's another way to ask, what are the limits of this? 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 we're not

going to find a useful rule because there isn't one. So the extreme of having a rule is like knowing Newton's law, you know, if you hit a ball and moves. So that's where you had laws of physics, Newton and Einstein and other great people have found those laws and laws of the distribution of oil in an underground thing. I mean, that's so engineers, petroleum engineers, understand how oil will sit in an underground basin. So there were rules. Now, the new idea of artificial intelligence is, learn the rules rules instead of figuring out the rules by with help from Newton or Einstein. The computer is looking for the rules.

So that's another step. But if there are no rules at all that the computer could find, if it's totally random data, well, you've got nothing. You've got no science to discover. It's automated search for the underlying rules. Yeah, search for the rules, yeah, exactly. And there will be a lot of random parts. I'm not knocking random because that's there. There's a lot of randomness built in, but there's got to be some basic.

It's almost always signal, right? In most cases, there's got to be some signal. Yeah, if it's all in the way, then you're not going to get it anywhere. Well, this world around us does seem to be, does seem to always have a signal, some kind to be discovered. Right. That's it. So, what excites you more? Yeah, that's right. To be discovered. Right. That's it. So what

excites you more, the we just talked about a little bit of application, what excites you more theory or the application of mathematics? Well, for myself, I'm probably a theory person. I'm not, I'm speaking here pretty freely about applications, but I'm not the person who really, I'm not a physicist or a chemist or a neuroscientist. So for myself, I like the structure and this flat subspaces and the relation of matrices, columns to rows, that's my part in the spectrum. So there really, really science is a big spectrum of people from asking practical questions and answering them, using some math, then some math guys,

like myself, who are in the middle of it, and then the geniuses of math and physics and chemistry who are finding fundamental rules and doing really understanding nature. That's incredible. At its lowest simplest level, maybe just a quick and broad stroke from your perspective, what is linear algebra sit as a subfield of mathematics? What are the various subfields that you think about

in relation to linear algebra? So the big fields of math are algebra as a whole and problems like calculus and differential equations. So that's a second, quite different field. Then maybe geometry deserves to be understood as a different field. maybe geometry deserves to be under thought of as a different field to understand the geometry of high dimensional surfaces. So I think, am I allowed to say this here? I think this is where personal view comes in. I think, This is where personal view comes in. I think math or thinking about undergraduate math, what millions of students study, I think we overdo the calculus at the cost of the algebra at the cost of linear. See, of this title, calculus versus linear algebra.

That's right. That's right. And you say that linear algebra wins. So can you dig into that a little bit of why does linear algebra win? Right. Well, OK. The viewer is going to think this guy is biased.

Not true. I'm just telling the truth as it is. Yeah. So I feel linear algebra is just a nice part of math that people can get the idea of. They can understand something that's a little bit abstract because once you get to 10 or 100 dimensions, and very, very, very useful.

That's what's happened in my lifetime is 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 let me fairly add probability. The ideas of probability and statistics have become very, very important. I've 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, calculus, linear algebra, matrices, and probability statistics. And they all deserve an important place. And Calculus has traditionally had a lion's share of the time. And disproportionate share.

It is, but thank you. Disproportionate. That's a good word. Of the love and attention from the excited young minds. Yeah. I know it's hard to pick favorites, but what is your favorite matrix? What's my favorite matrix? Okay, so my favorite matrix is square, I've met it. It's a square bunch of numbers, and has two's running down the main diagonal. And on the next diagonal, so think of top left to bottom right,

two's down down the middle of the matrix, and minus ones just above those two's and minus ones just below those two's, and otherwise all zeroes. So mostly zeroes, just three non-zero diagonals coming down. What is interesting about it? 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, 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 fast it's it's curving. And Einstein showed that that's fundamental to understand space. So second derivatives should have a bigger place in calculus. Second, my matrices, which are like

the linear algebra version of second derivatives are neat in linear algebra. Yeah. Just everything comes out right with those guys. Beautiful. What did you learn about the process of learning by having taught so many students math over the years? Oh, that is hard. I'll have to admit here that I'm not, I'm not really a good teacher because I don't get into the exam part, the exam is 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 but I tell the class at the beginning. I don't know if they believe me. Probably they don't. I tell the class, I'm here to teach you.

I'm here to teach you math and not to grade you. But they're thinking, okay, this guy is going to, you know, when is he gonna give me an A minus? Is he gonna give me a B plus? What, what did you learn about the process of learning? Of learning. Yeah, well, maybe to give you a legitimate answer about learning, I should have 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 to tell somebody for the first time about a matrix. Wow. Is there are there moments? So you are teaching a concept. Are there moments of learning that you just see in the students eyes? You don't need to look at the grades. But you see in their eyes that you hook them, that you connect with them in a way where you know what, they fall in love with this beautiful world of math.

They see that it's got some beauty in there. Or conversely that they give up at that point is the opposite. The darker say that math, I'm just not good at math and a little walk away. Yeah, yeah. Maybe because of the approach in the past, they were discouraged, but don't be discouraged. It's too good to miss. Yeah, I, well, if I'm teaching a big class, do I know when I think maybe I do? Sort of, I mentioned at the very start the four fundamental subspaces and the structure of the fundamental theorem of linear algebra. The fundamental theorem of linear algebra. That is the relation of those four subspaces,

those four spaces. Yeah, so I think that I feel that the class gets it. When they want to see it. What advice do you have to a student just starting their journey mathematics today? How do they get started? Oh, no. That's hard.

Well, I hope you have a teacher, a professor who is still enjoying what he's doing, what he's teaching. He's still looking for new ways to teach and to understand math, because that's the pleasure, ways to teach and to understand math. Cause that's the pleasure to the moment when you see, oh yeah, that works. So it's us about the material you study.

It's more about the source of the teacher being full of passion. Yeah, more about the fun, the moment of getting it. But in terms of topics, the algebra. Well, that's my topic. But oh, there's beautiful things in geometry to understand. What's wonderful is that in the end, there's a pattern. There are rules that are followed in biology as there are in every field. You describe the life of a mathematician as 100 percent wonderful, except for the grade stuff, and the grades. Yeah, when you look back at your life

Yeah, what memories bring you the most joy and pride Well, that's a good question I certainly feel good when I maybe I'm giving a class in 1806 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 sort of happy when I feel I'm helping make a connection between ideas and students between theory and the reader.

Yeah, it's... I get a lot of very nice messages from people who've watched the videos and it's inspiring. I just... maybe take this chance to say thank you. Well, there's millions of students who you've taught. And I am grateful to be one of them. So good, but thank you so much. It's been an honor. Thank you for talking to it. It was a pleasure. Thanks. Thank you for listening to this conversation with Gilbert's

string. And thank you to our presenting sponsor, cash app, download it, use code, Lex podcast, 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, we have five stars in Apple Podcast, supporting on Patreon, or connect with me on Twitter. Finally, some closing words of advice from the great Richard Feynman. Study hard would interest you the most in the most undisciplined, irreverent, and original

manner possible. Thank you for listening, and hope to see you next time. you