Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare | Lex Fridman Podcast #52
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 opencourseware 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 exercises in his book introduction of 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 miracle 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 an apple podcast support on patreon or simply connect with me on Twitter Alex Friedman spelled Fri D ma N this podcast is supported by zip recruiter 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 the tools that we built ourselves but reinventing the wheel was painful so zip recruiters a tool that's already available for you it seeks to make hiring simple fast and smart for example codable co-founder Gretchen Abner used the recruiter to find a new game artist to join her education tech company by using zip recruiter 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 zip recruiter the smartest way to hire CY zip recruiter is effective for businesses of all sizes by signing up as I did for free a zip recruiter comm slash FlexPod that's zip recruiter calm slash Lex pod 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 bank transfer to become investable through cash up it takes seconds cash app also has a new investing feature you can buy a fraction of stock which to me is a really interesting concept so you can buy $1 worth no matter what the stock price is brokerage services are provided by cash app investing a subsidiary of square and member si PC 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 and 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 Lex podcast you get ten dollars in cash app will also donate ten dollars to the 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 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 it's partly the importance of linear algebra which way 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 any I'm enjoy doing it was just the videos were just the class so they're on OpenCourseWare and YouTube and translated that's fun but there's something about that chalkboard in 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 do linear algebra broadly of course this lineage I before going through the course at my university I was going through open courseware I was you were my instructor oh yeah you're 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 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 and the idea of the committee was to figure out how a mighty could make be like other universities market the market work we were doing and then they didn't see away and after a weekend and they had an inspiration came back to the president vest and said what if we just gave it away and he decided that was ok good idea so that's a crazy idea that's uh if we think of a university is a thing that creates a product yes isn't knowledge right the you know the kind of educational knowledge isn't the products and giving that away are you surprised that you went through it the result that he did it well knowing a little bit president vest it was like him I think and and it was really the right idea you know MIT as I kind of it's known for being high level technical things and and this is the best way we can say tell we can show what MIT really is like because the the in my case those 1806 videos are just teaching the class they were there in 26100 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 somehow I big friendly with a class I like students and and then your algebra the subject we got to give this subject most of the credit it 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 a subfield of mathematics so what's your favorite specific topic in linear algebra or even math in general to give a lecture on to convey to tell a story to teach students okay well on the teaching side so it's not deep mathematics at all but I I'm kind of proud of the idea of the four subspaces there are four fundamental subspaces which are of course known before long before my name for them but can you go through them can you go through the future I can yes so the first one to understand is so the matrix is maybe I should say the matrix what is the matrix what's a matrix well so we have a like a rectangle of numbers so it's got n columns got a bunch of columns and also got an M rows let's say and the relation between so of course the columns and the rows it's the same numbers so there's got to be connections there but they're not simple the they're much the columns might be longer than the rows and they've 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 what's the vector 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 for me it's a column of numbers does it you often think of this is very interesting in terms of linear algebra of a vector you think a little bit more abstract than the how it's very commonly used perhaps yeah you think this arbitrary Speight multi-dimensional right away I'm in high dimensions and in the room lands yeah that's right in the lecture I tried a so if you think of two vectors in ten 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 int n dimensional space but whatever you can a 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 you know ten dimensions yeah you know there's this beautiful thing about math if we look string theory and all these theories which are really fundamentally derived through math yeah but it very difficult to visualize it yeah how do you think about the things like a 10 dimensional vector that we can't really visualize yeah do you and and yet math reveals some beauty Oh underlying me yeah 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 is that it goes on to ten 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 ten D or you're just adding the ten components so so I I can't say that I have a picture but yet I try to push the class to think of a flat surface in ten dimensions so a plane in ten dimensions and so that's one of the spaces take all the columns of the matrix take all their combinations so on 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 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 then we've got the row space all combinations of the rows so those are 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 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 two we've got a column space a row space and they're two perpendicular spaces so those four fit together in the in a beautiful picture of a matrix yeah yeah it's sort of a fundamental it's not a difficult idea comes comes pretty early in 1806 and it's basic planes in these multi-dimensional spaces how how difficult of an idea is that to come to do you think if you if you look back in time yeah I think mathematically it makes sense but I don't know if it's intuitive for us to imagine just what we're talking about 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 this is a curved surfaces linear algebra the surfaces are all flat nothing bends in 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 I'll be calculus and then I say enough of it like okay get to get to the good stuff and that you think linear algebra should come first well it really yeah 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 of course for that reason you sort of sticks to one dimension or so or eventually you do multivariate but that basically means two dimensions linear algebra you take off into ten dimensions no problem it just feels scary and dangerous to go beyond two dimensions 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 and oh well I'll stick with linear algebra here I hope that viewer knows that really mathematics is amazing amazing subject and deep deep 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's like the basic thing a rectangle of numbers and might be a rectangle of data you're probably going to ask me later about data science where and often data comes in a matrix you have you know maybe every column corresponds to a to a drug in every row corresponds to a patient and and if the patient reacted favorably to the drug then you put up some positive number in there anyway rectangle of numbers a matrix is basic so the big problem is to understand all those numbers you got a big 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 uses something called singular values and that's come on as fundamental in the last and certainly in my lifetime I can values bro you if you have viewers who've done engineering math or or more basic linear algebra eigenvalues were in there but those are restricted to square matrices and data comes in rectangular matrices so you got to take that you got to take that next step I'm I'm always pushing math faculty get on do it don't do it do it singular values so those are a way to break to make to find these the important pieces of the matrix which add up to the whole matrix so 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 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 we're yeah 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 here's the idea of singular values every matrix every matrix rectangular square whatever you 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 matrix diagonal matrix otherwise all zeros except on the one diagonal and then a third and the third factor is another rotation so rotation stretch rotation is the breakup of a of any matrix the structure that the ability that you can do that what do you find appealing what do you find beautiful bodies well geometrically as I freely admit the mate action of a matrix it's 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 the three turns that a pilot makes up to ten dimensions you've got ten ways to turn but you can visualize a rotation take the space and turn it and you can visualize a stretch so to break a a matrix with all those numbers in it into something you can visualize rotate stretch rotate it's pretty neat 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 I I'm trying to understand why that is perhaps you can help yeah I mean give me clues so it's not just to let the kinds of lectures that you give but it's also just other folks were like with numberphile there's a channel where they just chat about things that are extremely complicated actually yeah people nevertheless connect with them you know what do you think that is what 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 liked it 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 or you know things are not obvious but they're true so it's really cheering to think that that so many people really want to learn more about math yeah in terms of truth again sorry to slide into philosophy at times yeah math does reveal pretty strongly what things are true yeah I mean it's the whole point of proving things is 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 think yeah I would you this was before you had you had to go in when you're in the other filling your teeth yeah I kind of just take it yeah so I what I did was think about math you know like take powers of 2 2 4 8 16 up until the time the two stopped hurting and the dentist said you're through or Counting yeah so so that was the source of just such a piece almost yeah what what what what is it about math you think that brings that yeah what is that well you know where you are yeah symmetry it's it's certainty the fact that you know if you'd to if you multiply 2 by itself 10 times you get a thousand 24 period that's everybody's gonna get that do you see math is a powerful tool or is an art form so it's both 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 engineer and use math which are you which am I what did you connect with most yeah I'm in here between I'm certainly not a artist type philosopher type person might sound that way this morning but I'm not yeah I I really enjoy teaching engineers because they they they go for an answer and yeah so of probably within the mountain 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 actually that's a interesting question do you think do you think for teaching and in general but thinking about new concepts do you think it's better to plug in the numbers or to think more abstractly so looking at theorems and proving the theorems or actually you know building up a basically tuition of the theorem or the method the approach and then just plugging in numbers and seeing it work you know well certainly many of us like to see examples first we understand it might be a pretty abstract sounding example like a three dimensional rotation how are you gonna how are you gonna understand a rotation in 3d or in 10 D 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 is 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 right but nevertheless working within the space of examples yeah example it seems to examples of structure our brain seemed to connect with that yeah yeah so uh I'm not sure if you're familiar with him but Andrew yang is the presidential candidate currently running yeah with the math in all capital letters and his hats as a slogan Isis stands for make America think hard okay I'll vote for it so and his name rhymes with yours yang strang so but he also loves math and then 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 right political discourse about political a government in general yeah what do you think that is well what are your thoughts on that in general well certainly somewhere in this 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 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 yeah that is either it has an engineering degree or a mess yeah well that's too bad you know a few could a few who could make the connection yeah it would have to be people who are at the door who understand engineering or science and at the same time can make speeches and and lead yeah inspire people yeah yeah you were speaking of inspiration the president of the Society for industrial applied mathematics oh yeah as a major organization in math and Padma what do you see as a role of that society you know in our public discourse right yeah so well it was fun to be president at the at the time of years year two years around around 2000 his hope as president of a pretty small society but nevertheless it was a time when math was getting some more attention in Washington but yeah I got to give a little 10 minutes to into committee of the House of Representatives talking about why mint and then actually it was fun because one of the members of the house he had been a student had been in my class what do you think of that yeah as you say a pretty rare most most members of the House have had a different training different background but there was one from New Hampshire who who was my friend really bye-bye being in the class yeah so that those years were good then of course other things take take over and importance in Washington and maths 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 yes about the future yeah and I think at the core of that is math well it is yeah yeah but maybe it's hidden maybe he's wearing a different hat but uh well artificial intelligence and and particularly can I use the words deep learning it's a deep learning is a particular approach to understanding data again you've got a big a whole lot of data where data is just swapping the computers of the world and and - and understand it out of all those numbers to find what's important you know in climate in everything and artificial intelligence is two words for 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 and I asked the this most basic question yeah how do you think of a neural network what is it and you're on the 1 yeah ok so can I start with a 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 day that we're trying to learn what's important what was some 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 understand what the output will be from that new input so we've got a million inputs with their out so we're trying to create some patterns some rule that'll take those inputs those million training inputs which we know about to the correct million outputs and this idea of a neural net is part of the structure of 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 going to use that rule on new inputs that we don't know the output and see what comes the linear algebra is a big part of defining a finding that rule 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 all about straight lines and flat planes and 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 not only doubt it on linear nonlinear nonlinear and it turned out that the it was enough to use the function that's one straight line and then a different one halfway that's so piecewise then he said one piece has one slope one piece the other piece has the second slope and so that introduced getting that nonlinear simple non-linearity in blue the problem open that little piece makes it sufficiently complicated to make things interesting because you're gonna use that piece over and over a million times so you so you it has a it has a fold in the in the graph the graph two pieces and but when you fold something a million times you've got 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 that's so why do you think neural networks by using a sort of formulating an objective function very not a plane yeah function holds lots of folds of 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 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 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 ideas been used by engineers is used by engineers big time something called the finite element method if you want to if you want to design a bridge design a building design airplane you're using this idea of piecewise flat as a good simple computable approximation to pay your you have a sense that that there's a lot of expressive power in this kind of piecewise linear yeah well that's combined together you use the right word if you measure the expressivity how how many how complicated a thing can can this piecewise flat guys express the answer is very complicated yeah what do you think are the limits of such piecewise linear or just neural networks its passivity of nool nose well you would have said a while ago that they're just computational limits it you you know you the problem beyond a certain size a supercomputer isn't going to do it but that does keep getting more powerful so that's that limit has been moved to allow more and more complicated surfaces so in terms of just mapping from inputs to the outputs looking at data yeah what do you think of you know in a context in your networks in general data is just tensor vectors matrices tensors 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 the 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 the 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 hit a ball and 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 the the the distribution of oil in a underground thing I mean that so so engineers petroleum engineers and understand how how oil will sit in a in an underground basin so there were rules now now the the new idea of artificial intelligence is learn the rules instead of 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 for 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 the automated search for the underlying rules yeah search for the rules yeah exactly yeah there will be a lot of random parts a lot I'm not knocking random because the that's there the the the there's a lot of randomness built in but there's got to be some basic it's almost always signature right in most there's got to be some signal yeah if it's all noise then there's there's you're not gonna get it anywhere well this world around us does seem to be this seem to always have a signal some kind yeah yeah they discovered right that's it so what excites you more the we just talked about a little bit of application what excites me 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 a person who really I'm not a physicist or a chemist or a neuroscientist so for myself I like this structure and this flat subspaces and and and the relation of matrices columns to rows that's my part in the spectrum so the really science is a big spectrum of people from asking practical practical questions and answering them using some math than some math guys like myself or in the middle of it and then the geniuses of math and physics and chemistry and who are finding fundamental rules and doing doing really understanding nature that's at its lowest simplest level maybe just a quick in broad strokes from your perspective what is uh where does the linear algebra sit as a subfield of mathematics what what are the various subfields a year okay a you think about in relation to linear algebra so the big fields of math or 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 under sort 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 and I think math for thinking about undergraduate math what millions of students study I think we overdo the calculus at at the cost of the algebra at the cost of linear dog titled calculus versus linear that's right and and you say that linear algebra wins so you can you want can you dig into that a little bit why does linear algebra win right well ok I'm the viewer is gonna 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 it's a little bit abstract because once you get to ten or a hundred dimensions 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 a differential equation and now let me fairly add probability they're 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 of matrices and probability statistics and they all deserve a important place and and calculus has traditionally had a had a lion's share of the time and disproportionate share yes but thank you proportionate that's a good work of the 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 admit it's a square bunch of numbers and it has twos running down the main diagonal and 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 zeros so mostly zeros just three nonzero 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 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 fast it's it's curving and my Feinstein 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 beautiful what did you learn about the process of learning by having taught so many students math over the years whoo 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 exams 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 and but they're thinking okay this guy it's gonna you know when's he gonna kids he's gonna give me an A - is he gonna give me a b-plus what what would have you learn about the process of learning of learning yeah well maybe to be elated 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 but 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 yeah you see in their eyes that that you hooked them that you know that you connect with them in a way where you know what they fall in love with this yes beautiful world amazing see that it's got some beauty it's yeah or conversely yeah that they give up at that point is the opposite the darkest a the math I'm just not good at math and alcohol yeah yeah maybe because of the approach in the past they were discouraged but don't be discouraged it's 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 the what advice do you have to a student just starting their journey mathematics today how do they get started Oh No yeah that's hard well I hope you have a teacher professor who is still enjoying what he's doing and what he's teaching they're still looking for new ways to teach and and to understand math because that's the pleasure to the moment when you see oh yeah that works so it's s about the material you yeah you study it's more about the source of the teacher being full of passion yeah more about the fun yeah there's a moment of of getting it but in terms of topics linear algebra well that's not my topic but oh there's beautiful things in geometry to understand what's wonderful is that in the end there there's a pattern there there's their rules that that that are followed in biology as there are in every field you describe imitation as as a hundred percent wonderful except for the great stuff yeah and the grades were great 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 in 1806 that's mi t--'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 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 not maybe it's 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 birth thank you so much has been an honor thank you for talking to it was a pleasure thanks thank you for listening to this conversation with Gilbert Strang and thank you to our presenting sponsor cash app downloaded used collects podcast you'll get ten dollars and ten dollars 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 had five stars in an apple podcast support on patreon I'll connect with me on Twitter finally some closing words of advice from the great Richard Fineman study hard would interest you the most in the most undisciplined irreverent an original manner possible thank you for listening and hope to see you next time you