Gilbert Strang: Linear Algebra, Deep Learning, Teaching, and MIT OpenCourseWare

[0] The following is a conversation with Gilbert Strang.

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

[2] His MIT open courseware lectures on linear algebra have been viewed millions of times.

[3] As an undergraduate student, I was one of those millions of students.

[4] There's something inspiring about the way he teaches.

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

[6] 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.

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[31] And now here's my conversation with Gilbert Strang.

[32] How does it feel to be one of the modern day rock stars of mathematics?

[33] I don't feel like a rock star.

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

[35] 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.

[36] 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 in importance.

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

[38] It was just, the videos were just the class.

[39] So they're on open courseware and on YouTube and translated.

[40] That's fun.

[41] But there's something about that chalkboard and the simplicity of the way you explained the basic concepts in the beginning.

[42] You know, to be honest, when I went to undergrad...

[43] You didn't do linear algebra, probably.

[44] Of course, there's lineology, but yeah, yeah, yeah, of course.

[45] But before going through the course at my university, I was going through an open course where I was, you were my instructor for linear algebra.

[46] And that, I mean, we're using your book.

[47] 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 really powerful.

[48] So how do you think the idea of, putting lectures online, what really MIT OpenCourseware has innovated?

[49] That was a wonderful idea.

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

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

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

[53] And he decided that was okay, good idea.

[54] You know, that's a crazy idea.

[55] If we think of a university is a thing that creates a product.

[56] Yes.

[57] Isn't knowledge?

[58] Right.

[59] The kind of educational knowledge isn't the product and giving that away Are you surprised that it went through?

[60] The result that he did it.

[61] Well, knowing a little bit present vest was like him, I think.

[62] And it was really the right idea.

[63] MIT is known for being high -level technical things.

[64] And this is the best way we can say, tell, we can show what MIT really is like.

[65] because in my case, those 1806 videos are just teaching the class.

[66] They were there in 26, 100.

[67] They're kind of fun to look at.

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

[69] Somehow I've been friendly with a class.

[70] I like students.

[71] And linear algebra, we've got to give the subject most of the credit.

[72] It really has come forward in importance in these years.

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

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

[75] Okay.

[76] Okay.

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

[78] Can you go through them?

[79] Can you go through the four subspaces?

[80] Sure, I can, yeah.

[81] So the first one to understand is, so the matrix.

[82] is maybe I should say the matrix.

[83] What is a matrix?

[84] What's a matrix?

[85] Well, so we have a like a rectangle of numbers.

[86] So it's got N columns, got a bunch of columns, and also got an M rows, let's say.

[87] And the relation between, so of course the columns and the rows, it's the same numbers.

[88] So there's got to be connections there, but they're not simple.

[89] The columns might be longer than the rows, and they're all different.

[90] The numbers are mixed up.

[91] First space to think about is take the columns.

[92] So those are vectors.

[93] Those are points in n dimensions.

[94] What's a vector?

[95] 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 in space.

[96] For me, it's a column of numbers.

[97] You often think of This is very interesting in terms of linear algebra, in terms of a vector.

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

[99] You think this arbitrary, multidimensional space.

[100] Right away, I'm in high dimensions.

[101] Dreamland.

[102] Yeah, that's right.

[103] In the lecture, I try to.

[104] 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 an 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 multiply a vector by 3, and that's, if you know how to do those, you've got linear algebra.

[105] You know, 10 dimensions, there's this beautiful thing about, math if we look at string theory and all these theories which are really fundamentally derived through math but are very difficult to visualize.

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

[107] Yeah.

[108] Do you, and yet math reveals some beauty underlying our world in that weird thing we can't visualize.

[109] How do you think about that difference?

[110] well probably i'm not a very geometric person so i probably thinking in three dimensions and the beauty of linear algebra is that is that it goes on to 10 dimensions with no problem i mean that if you're just seeing what happens if you add two vectors in 3d you then you can add them in 10d you're just adding the 10 components so so i i can't say that i have a picture but yes I try to push the class to think of a flat surface in ten dimensions.

[111] So a plane in ten dimensions.

[112] So that's one of the spaces.

[113] Take all the columns of the matrix, take all their combinations, so much of this column, so much of this one.

[114] Then if you put all those together, you get some kind of a flat surface that I call a vector space, space of vectors.

[115] And, And my imagination is just seeing like a piece of paper in 3D.

[116] But anyway, so that's one of the spaces.

[117] That's space number one, the column space of the matrix.

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

[119] So we got the column space, all combinations of the columns.

[120] And then we've got the row space, all combinations of the rows.

[121] 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 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 A plane is just a flat surface In 3D then perpendicular to that plane would be a line.

[122] So that would be the null space.

[123] So we've got two, we've got a column space, a row space, and there are two perpendicular spaces.

[124] So those four fit together in a beautiful picture of a matrix.

[125] Yeah, yeah.

[126] It's sort of a fundamental.

[127] It's not a difficult idea.

[128] It comes pretty early in 1806, and it's basic.

[129] Plains in these multidimensional spaces, how difficult of an idea is that to come to?

[130] Do you think if you look back in time, I think mathematically it makes sense, but I don't know if it's intuitive for us to imagine, just as what we're talking about.

[131] It feels like calculus is easier to intuit.

[132] I see.

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

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

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

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

[137] Linear algebra, the surfaces are all flat.

[138] Nothing bends in linear algebra.

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

[140] Calculus also comes first in high school classes, in college class.

[141] It'll be freshman math.

[142] It'll be calculus.

[143] And then I say, enough of it.

[144] Like, okay, get to the good stuff.

[145] Do you think linear algebra should come first?

[146] Well, it really, I'm okay with it not coming first, but it should.

[147] Yeah, it should.

[148] It's simpler.

[149] Because everything is flat.

[150] Yeah, everything's flat.

[151] Well, of course, for that reason, the calculus sort of sticks to one dimension or eventually you do multivariate, but that basically means two dimensions.

[152] Linear algebra, you take off into ten dimensions, no problem.

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

[154] That's all.

[155] If everything is flat, you can't go wrong.

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

[157] Well, I'll stick with linear algebra here.

[158] I hope that viewer knows that really mathematics is amazing, amazing subject and deep connections between ideas that didn't look connected.

[159] They turned out they were.

[160] But if we stick with linear algebra, so we have a matrix.

[161] That's like the basic thing, a rectangle of numbers.

[162] And it might be a rectangle of data.

[163] You're probably going to ask me later about data science, where often data comes in a matrix.

[164] You have maybe every column corresponds to a drug and every row corresponds to a patient.

[165] And if the patient reacted favorably to the drug, then you put up some positive number in there.

[166] Anyway, rectangle of numbers, a matrix is basic.

[167] So the big problem is to understand all those numbers.

[168] You've got a big set of numbers.

[169] And what are the patterns?

[170] What's going on?

[171] And so one of the ways to break down that matrix, into simple pieces uses something called singular values.

[172] And that's come on as fundamental in the last certainly in my lifetime.

[173] Igen values if you have viewers who've done engineering math or or basic linear algebra, eigen values were in there.

[174] But those are restricted to square matrices.

[175] And data comes in rectangular matrices.

[176] So you've got to take that next step.

[177] I'm always pushing math faculty.

[178] Get on.

[179] Do it, do it.

[180] Do it.

[181] Singular values.

[182] So those are a way to break, to make, to find the important pieces of the matrix, which add up to the whole matrix.

[183] So the first piece is the most important part of the data, the second piece is the second most important part.

[184] And then often, so, a data scientist will, like, if a data scientist can find those first and second pieces, stop there.

[185] The rest of the data is probably round off, you know, experimental error, maybe.

[186] So you're looking for the important part.

[187] Yeah.

[188] So, what do you find beautiful about singular values?

[189] Well, yeah, I didn't give the theorem.

[190] So here's the idea of singular values.

[191] Every matrix, every matrix, rectangular, square, whatever, can be written as a product of three very simple special matrices.

[192] So that's the theorem.

[193] Every matrix can be written as a rotation times a stretch, which is just a matrix, a diagonal matrix, Otherwise, all zeros except on the one diagonal.

[194] And then the third factor is another rotation.

[195] So rotation, stretch, rotation is the breakup of any matrix.

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

[197] What do you find beautiful about it?

[198] Well, geometrically, as I freely admit, the action of a matrix is not so easy to visualize.

[199] But everybody can visualize a rotation.

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

[201] Take three -dimensional space.

[202] So a pilot has to know about, well, what are the three?

[203] The yaw is one of them.

[204] I've forgotten all the three turns that a pilot makes.

[205] Up to ten dimensions, you've got ten ways to turn.

[206] But you can visualize a rotation.

[207] take the 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 is pretty neat it's pretty neat 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.

[208] I'm trying to understand why that is.

[209] Perhaps you can help give me clues.

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

[211] People nevertheless connect with them.

[212] What do you think that is?

[213] It's wonderful, isn't it?

[214] I mean, I wasn't really aware of it.

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

[216] 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 going to learn some more math.

[217] I get a lot of those.

[218] It's really encouraging.

[219] And I think what people like is that there's some order, you know, a lot of order, or, you know, things are not obvious, but they're true.

[220] So it's really cheering to think that so many people really want to learn more about math.

[221] Yeah.

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

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

[224] It is, yeah.

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

[226] It is.

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

[228] Oh, wow.

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

[230] Yeah, that's right.

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

[232] I just liked numbers, you know, as a kid.

[233] I would, this was before you had to go in when you had a Philly in your teeth, you had to kind of just take it.

[234] So 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 and the dentist said you were through.

[235] Or counting, yeah.

[236] So that was a source of just, source of peace almost.

[237] Yeah.

[238] What is it about math, do you think that brings that?

[239] Yeah.

[240] What is that?

[241] Well, you know where you are, yeah.

[242] It's symmetry.

[243] It's certainty.

[244] The fact that, you know, if you multiply two by itself ten times, you get a thousand and twenty -four period.

[245] Everybody's going to get that.

[246] Do you see math as a powerful tool or is an art form?

[247] So it's both.

[248] That's really one of the neat things.

[249] You can be an artist and like math.

[250] You can be a engineer and use math.

[251] Which are you?

[252] Which am I?

[253] What did you connect with most?

[254] Yeah, I'm somewhere between.

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

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

[257] Yeah, I really enjoy teaching engineers.

[258] Because they go for an answer.

[259] And, yeah, so probably within the MIT Math Department, most people enjoy teaching people teaching students who get the abstract idea i'm okay with with i'm good with engineers who are looking for a way to find answers yeah so actually that's an 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 or to think more abstractly so so looking at theorems and proving the theorems or actually building up a basic intuition of the theorem or the method of the approach and then just plugging in numbers and seeing it work.

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

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

[262] How are you going to understand a rotation in 3D or in 10D or but and then some of us like to keep going with it to the point where you got numbers where you got 10 angles 10 axes 10 angles but the best the great mathematicians probably I don't know if they do that because they they for them for them an example would be a highly abstract thing to the rest of it.

[263] Right, but nevertheless is working in the space of examples.

[264] Yeah, examples.

[265] It seems to...

[266] Examples of structure.

[267] Our brain seemed to connect with that.

[268] Yeah, yeah.

[269] So I'm not sure if you're familiar with him, but Andrew Yang is a presidential candidate currently running.

[270] Yeah.

[271] With math in all capital letters and his hats as a slogan.

[272] I see.

[273] Stans for Make America Think Hard.

[274] Okay.

[275] I'll vote for him.

[276] So, and his name rhymes with yours, Yang, Strang.

[277] But he also loves math, and he comes from that world.

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

[279] yeah why 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 if you say this leads to that they see it and it's undeniable but isn't that 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 is either has an engineering degree or a math.

[280] Yeah, well, that's too bad.

[281] A few could, a few who could make the connection.

[282] It would have to be people who understand engineering or science and at the same time can make speeches and lead.

[283] Yeah.

[284] Yeah, inspire people.

[285] You were, speaking of inspiration, the president of the Society for Industrial Applied Mathematics.

[286] Oh, yes.

[287] That's a major organization in math and applied math.

[288] What do you see as a role of that society, you know, in our public discourse?

[289] Right.

[290] Yeah.

[291] So, well, it was fun to be president at the time.

[292] A couple years, two years.

[293] Two years, around 2000.

[294] It was a president of a pretty small society, but nevertheless, it was a time when math was getting some more attention in Washington.

[295] But yeah, I got to give a little 10 minutes to committee of the House of Representatives talking about why Matt.

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

[297] What do you think of that?

[298] Yeah, as you say, pretty rare.

[299] 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.

[300] Yeah, so those years were good.

[301] Then, of course, other things take over an importance in Washington.

[302] and math, math just, at this point, is not so visible, but for a little moment it was.

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

[304] Yes, sure.

[305] About the future.

[306] Yeah.

[307] And I think at the core of that is math.

[308] Well, it is, yeah.

[309] Maybe it's hidden.

[310] Maybe it's wearing a different hat.

[311] Well, artificial intelligence, and particularly, can I use the words deep learning, If the deep learning is a particular approach to understanding data.

[312] Again, you've got a big, a 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.

[313] And artificial intelligence is two words for one approach to data.

[314] deep learning is a specific approach there, which uses a lot of linear algebra.

[315] So I got into it.

[316] I thought, okay, I've got to learn about this.

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

[318] How do you think of a neural network?

[319] What is it in your own network?

[320] Yeah, okay.

[321] So can I start with an idea about deep learning?

[322] What does that mean?

[323] Sure.

[324] What is deep learning?

[325] What is deep learning?

[326] Yeah.

[327] So we're trying to learn, from all this data, we're trying to learn what's important, what's it telling us.

[328] So you've got data.

[329] You've got some inputs for which you know the right outputs.

[330] The question is, can you see the pattern there?

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

[332] new input.

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

[334] 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.

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

[336] We'd look at it.

[337] We'd look at it.

[338] for a rule that will take these training inputs to the known outputs.

[339] And then we're going to use that rule on new inputs that we don't know the output and see what comes.

[340] Linear algebra is a big part of finding that rule.

[341] That's right.

[342] Linear algebra is a big part.

[343] Not all the part.

[344] People were leaning on matrices.

[345] That's good.

[346] Still do.

[347] Linear is something special.

[348] It's all about straight lines and flat planes and and data isn't quite like that you know it's it's more complicated so you got to introduce some complication so you have to have some function that's not a straight line and it turned out it non -linear non -linear not 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 piece of one piece has one slope one piece the other piece has the second slope and so that getting that nonlinear simple nonlinearity in blew the problem open that little piece makes it sufficiently complicated to make things interesting because you're going to use that piece over and over million times so you so you it has a it has a fold in the graph the graph two pieces and but but when you fold something a million times, you've got a pretty complicated function that's pretty realistic.

[349] So that's the thing about neural networks is they have a lot of these.

[350] A lot of them.

[351] That's right.

[352] So why do you think neural networks by using a, so formulating an objective function, very not a plain function.

[353] Yeah.

[354] A function of the, lots of folds.

[355] Lots of folds.

[356] the inputs, the outputs, why do you think they work to be able to find a rule that we don't know is optimal, but it's just seems to be pretty good in a lot of cases.

[357] What's your intuition?

[358] Is it surprising to you as it is to many people?

[359] Do you have an intuition of why this works at all?

[360] Well, I'm beginning to have a better intuition.

[361] This idea of things that are piecewise linear, flat pieces but but with folds between them like think of a roof of a complicated infinitely complicated house or something that curved it almost curved but every piece is flat that that's been used by engineers that idea has been used by engineers is used by engineer big time something called the finite element method if you want to if you want to design a bridge design a building design airplane.

[362] You're using this idea of piecewise flat as a good simple, computable approximation.

[363] But you have a sense that there's a lot of expressive power in this kind of piecewise linear functions combined together.

[364] You use the right word.

[365] If you measure the expressivity, how complicated the thing can can this piecewise flat guys express, the answer is very complicated, yeah.

[366] What do you think are the limits of such piecewise linear or just of neural networks, the expressivity of neural networks?

[367] Well, you would have said a while ago that they're just computational limits.

[368] It's a problem beyond a certain size, a supercomputer isn't going to do it.

[369] But those keep getting more powerful.

[370] So that limit has been moved to allow more and more complicated surfaces.

[371] 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.

[372] right 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 to asking 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.

[373] So the extreme of having a rule is like knowing Newton's law, you know, if you hit a ball and moves.

[374] So that's where you had laws of physics.

[375] Newton and Einstein and other great, great people have found those laws and laws of the distribution of oil in an underground thing.

[376] I mean, so engineers, petroleum engineers, understand how oil will sit in an underground basin.

[377] so there were rules now now the new idea of artificial intelligence is learn the rules instead of figuring out the rules 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 to discover.

[378] It's an automated search for the underlying rules.

[379] Yeah, search for the rules.

[380] Yeah, exactly.

[381] And there will be a lot of random parts, a lot of, I mean, I'm not knocking random, because that's there.

[382] There's a lot of randomness built in, but there's got to be some basic structure, right?

[383] There's got to be some signal, yeah.

[384] If it's all noise, then there's, you're not going to get anywhere.

[385] Well, this world around us does seem to be, this seems to always have a signal of some kind.

[386] Yeah, yeah, that's right.

[387] To be discovered.

[388] Right, that's it.

[389] So what excites you more?

[390] We just talked about a little bit of application.

[391] What excites you more, theory or the application of mathematics?

[392] Well, for myself, I'm probably a theory person.

[393] And 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.

[394] So for myself, I like the structure and the flat subspaces and the relation of matrices, columns to rows, that's my part in the spectrum.

[395] So 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.

[396] And then the geniuses of math and physics and chemistry.

[397] who are finding fundamental rules and then doing the really understanding nature.

[398] That's incredible.

[399] At its lowest, simplest level, maybe just a quick and broad strokes from your perspective, what does linear algebra sit as a subfield of mathematics?

[400] What are the various subfields that you're, that you think about in relation to linear algebra.

[401] So the big fields of math are algebra as a whole and problems like calculus and differential equations.

[402] So that's a second quite different field.

[403] Then maybe geometry deserves to be thought of as a different field to understand the geometry of high dimensional surfaces.

[404] So I think, am I allowed to say this here?

[405] I think, this is where personal view comes in.

[406] I think math, we're thinking about undergraduate math, what millions of students study.

[407] I think we overdo the calculus at the cost of the algebra, at the cost of linear.

[408] You have this talk titled Calculus versus linear algebra.

[409] That's right.

[410] And you say that linear algebra wins.

[411] So can you dig into that a little bit?

[412] Why does linear algebra win?

[413] Right.

[414] Well, okay.

[415] The viewer is going to think this guy is biased.

[416] Not true.

[417] I'm just telling the truth as it is.

[418] Yeah.

[419] So I feel linear algebra is just a nice part of math that people can get the idea of.

[420] They can understand something that's a little bit abstract because once you get to 10 or 100 dimensions.

[421] And very, very, very useful.

[422] That's what's happened in my lifetime is the importance of data, which does come in matrix form.

[423] So it's really set up for algebra.

[424] It's not set up for differential equation.

[425] And let me fairly add.

[426] The ideas of probability and statistics have become very, very important, have also jumped forward.

[427] So, and that's not, that's different from linear algebra, quite different.

[428] So now we really have three major areas to me, calculus, linear algebra, matrices, and probability statistics.

[429] and they all deserve a important place.

[430] And Calculus has traditionally had a lion's share of the time.

[431] Disproportionate share.

[432] It is, thank you, disproportionate.

[433] That's a good word.

[434] Of the love and attention from the excited young minds.

[435] Yeah.

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

[437] What's my favorite matrix?

[438] Okay.

[439] So my favorite matrix is square, I admit it.

[440] It's a square bunch of numbers, and it has twos running down the main diagonal.

[441] And on the next diagonal, so think of top left to bottom right, twos down the middle of the matrix, and minus ones just above those twos, and minus ones just below those twos, and otherwise all zeros.

[442] So mostly zeros.

[443] Just three non -zero diagonals coming down.

[444] What is interesting about it?

[445] Well, all the different ways it comes up.

[446] You see it in engineering, you see it as analogous in calculus to second derivative.

[447] So calculus learns about taking the derivative, figuring out how much, how fast something's changing.

[448] But second derivative, now that's also important.

[449] That's how fast to change.

[450] is changing, how fast the graph is bending, how fast it's curving.

[451] And Einstein showed that that's fundamental to understand space.

[452] So second derivatives should have a bigger place in calculus.

[453] Second, my matrices, which are like the linear algebra version of second derivatives, are neat.

[454] in linear algebra.

[455] Just everything comes out right with those guys.

[456] Beautiful.

[457] What did you learn about the process of learning by having taught so many students' math over the years?

[458] Ooh, that is hard.

[459] I'll have to admit here that I'm not really a good teacher because I don't get into the exam part.

[460] 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.

[461] I do it because I'm supposed to do it, but I tell the class at the beginning, I don't know if they believe me. Probably they don't.

[462] I tell the class, I'm here to teach you.

[463] I'm here to teach you math and not to grade you.

[464] But they're thinking, okay, this guy, is going to, you know, when's he going to give me an A minus?

[465] Is he going to give me a B plus?

[466] What?

[467] What have you learned about the process of learning?

[468] Of learning.

[469] 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.

[470] But I like the teaching part at the start.

[471] That's the sexy part to tell somebody for the first time about a mate.

[472] matrix wow but is there are there moments so you you are teaching a concept are there moments of learning that you just see in the student's eyes you don't need to look at the grades yeah 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 fall in love with this with this beautiful world of math or they see that it's got some beauty there.

[473] Or conversely, that they give up at that point is the opposite.

[474] The darkest say that math, I'm just not good at math.

[475] I don't want to walk away.

[476] Yeah, yeah.

[477] Maybe because of the approach in the past, they were discouraged, but don't be discouraged.

[478] It's too good to miss. Yeah, 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.

[479] The fundamental theorem of linear algebra.

[480] That is the relation of those four subspaces, those four spaces.

[481] Yeah.

[482] So I think that, I feel that the class gets it.

[483] When they see it.

[484] Yeah.

[485] What advice do you have to a student just, starting their journey mathematics today.

[486] How do they get started?

[487] No. Yeah, that's hard.

[488] Well, I hope you have a teacher, professor, who is still enjoying what he's doing, what he's teaching.

[489] He's still looking for new ways to teach and to understand math.

[490] Because that's the pleasure to the moment when you see.

[491] Oh, yeah, that works.

[492] So it's less about the material you study.

[493] It's more about the source of the teacher being full of passion for the time.

[494] Yeah, more about the fun, yeah, the moment of getting it.

[495] But in terms of topics, linear algebra?

[496] Well, that's my topic, but, oh, there's beautiful things in geometry to understand.

[497] And what's wonderful is that in the end, there's a pattern.

[498] There are rules that are followed in biology as there are in every field.

[499] You describe the life of a mathematician as 100 % wonderful, except for the grade stuff.

[500] Yeah, and the grades.

[501] Except for grades.

[502] Yeah.

[503] Yeah, when you look back at your life, what memories bring you the most joy and pride?

[504] Well, that's a good question.

[505] I certainly feel good when I, maybe I'm giving a class in 1806, that's MIT's linear algebra course that I started.

[506] So sort of is a good feeling that, okay, I started this course, a lot of students take it, quite if you like it.

[507] 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.

[508] Yeah, I get a lot of very nice messages from people who've watched the videos, and it's inspiring.

[509] I just, I'll maybe take this chance to say thank you.

[510] Well, there's millions of students who you've taught, and I am grateful to be one of them.

[511] So, Gilbert, thank you so much.

[512] It's been an honor.

[513] Thank you for talking today.

[514] It was a pleasure.

[515] Thanks.

[516] Thank you for listening to this conversation with Gilbert Strang.

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[523] Finally, some closing words of advice from the great Richard Feynman.

[524] Study hard would interest you the most in the most undisciplined, irreverent, and original manner possible.

[525] Thank you for listening and hope to see you next time.