Gilbert Strang: Linear Algebra vs Calculus

[@lexfridman]

Full episode with Gilbert Strang (Nov 2019): kzbin.info/www/bejne/ona9gZmjfKh4oZI New clips channel (Lex Clips): kzbin.info Once it reaches 20,000 subscribers, I'll start posting the clips there instead. (more links below) For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days. Podcast full episodes playlist: kzbin.info/aero/PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Podcasts clips playlist: kzbin.info/aero/PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 Podcast website: lexfridman.com/ai Podcast on Apple Podcasts (iTunes): apple.co/2lwqZIr Podcast on Spotify: spoti.fi/2nEwCF8 Podcast RSS: lexfridman.com/category/ai/feed/

[@Ilyaswashere]

Just saw his final lecture…he is a true legend!

[@pavelestrada7988]

It's so interesting to hear that Linear Álgebra should go first, me who I just started on programming just took Linear Álgebra first without knowing this.

[@I_hu85ghjo]

@@pavelestrada7988 do you find calculus easier than linear algebra? What should i learn first as a programmer ?

[@pavelestrada7988]

@@I_hu85ghjo I haven't learn Calculus yet, but I think You can go either way.

[@friedrichmyers]

@@I_hu85ghjo Learn the conventional way. Calculus is easier. In linear algebra you have to know how to see things.

[@eduardorivera508]

At my uni, Calc 2 was a requirement for Linear Algebra. When I took Linear, I remember thinking that it felt like it was too good to be true that it was this straight forward. Definitely makes sense to me to take Linear as a culminating class to a, say, college Algebra class, although, I was blown away with how the link was made to Calc 2 where you could use Linear Algebra to solve integrals. Shit was nuts.

[@liftheadleyweight]

We went over page rank today and used eigenvalues to a markov model and I was losing it

[@KAn-br6py]

@@liftheadleyweight feel like most people don't know HOW difficult linear algebra gets as soon as u go beyond learning about eigenvalues..... there's so much ideas to learn and study that as a student it is Overwhelming...

[@calum.macleod]

@@liftheadleyweight I'd love to know the context. Mixing graph centrality with timeseries is the most fun I get to have in life.

[@DetectiveConan990v3]

@@KAn-br6py lol thankfully my linear algebra course ends with eigenvalues/vectors

[@pikapuff123]

I often think that Linear Algebra should come before Calculus in math curriculum, but then I realize that my poor heart could not take it if they end up doing to the Linear Algebra curriculum what they do to Calculus. I remember my first course in linear algebra in college, we never learned anything beyond row reduction. There was no meaningful discussion of vector spaces, bases, factorization, decomposition. I can only image what would happen if we put high school teachers to the task of teaching it...

[@NoName-cm9tx]

Sadly I am facing it right now while in college 😂

[@user-qy6tu9ip9v]

@@NoName-cm9tx Does linear algebra have a lot of memorization? Would you also please give me advice for this course? I want to do well when I get into engineering.

[@divedeepintomaths5172]

@@user-qy6tu9ip9v Not at all, it requires practice and revision. When you study Linear algebra first time, it will be very difficult for you. When you study it second time, you will understand half of it and when you study it third time, you will love it.

[@maxwellsequation4887]

Do you expect them to teach all linear algebra to those dumbasses in high school?

[@HappyGoLuckyPanda]

Group theory should come first, especially finite group

[@odesangel]

Strang's OCW lectures were a huge help when I was studying linear algebra. Thank you, professor Strang.

[@speedspeed121]

I just made the connection that he was my textbook author. I literally watched the MIT videos while using his textbook for linear, but never made the connection. LOLOLOL

[@clydedsouzaauthor]

I don't understand 99% percent of the math and science discussed in your podcast but i listen to every one of them.

[@neoblackcyptron]

Prof strang is a legend. God bless him. I am ramping up on all his lectures before my return to university after 10 years as a software engineer to specialize in AI and robotics. AI is the new frontier. In want to taste it. His math lectures will help a lot.

[@mattsharkey8437]

AI will be the death of us

[@neoblackcyptron]

​@@mattsharkey8437 As a Christian Engineer (yes I know it is an oxymoron). I take solace in the fact during the second coming of Christ. We still see only the Anti-Christ (a demon) and humans being in charge, machines still serve men. So I am not that worried of AI taking over. At least at the moment AI is still in the nascent stage, we can do some nice things with Neural Networks, Reinforcement Learning mixed with Robotics. Within a year at school I've learnt quite a lot to create workable products to practically immediately assist humans. Sure the Phd's do the tough job, but for us engineers its all about taking the research and putting it to use practically.

[@mattsharkey8437]

@@neoblackcyptron not buying it. You're just one man lol

[@iosifpuha6114]

@@neoblackcyptron Christian Engineer is not an oxymoron at all

[@neoblackcyptron]

@@iosifpuha6114 According to popular view of "intellectuals" religious people are primitive and don't make good STEM Engineers or Scientists. Hence the statement.

[@waqqiali2960]

Gilbert Strang: Calculus is about curves while in linear algebra everything is flat. Me: Comments section is gonna be EPIC!

[@Darklegend_royal-metamorphosis]

Sadly no

[@mr.erikchun5863]

@@Darklegend_royal-metamorphosis the poor fool was hoping for likes

[@TheBrainn]

he’s right though.

[@RioMusicIndia]

If everything is flat, you can't go wrong. - Professor Gilbert Strang

[@Tabu11211]

This was the most inspiring video I've seen so far! I am currently trying to self study both.

[@sturdyspamsl2277]

How did it ago 1 year later ?

[@amann02]

Mastered it yet ? Or not

[@Tabu11211]

@sturdyspamsl2277 not great haha. I own the books I need now atleast haha. I ended up getting a job making financial astrology tools!

[@Tabu11211]

@@amann02 unfortunately not yet.

[@amann02]

@@Tabu11211 Cool! That is the kind of motivation i wanted ;(

[@TheLethalDomain]

In fairness, much of elementary and high school mathematics serves as a basis to introduce the simplest aspects, operations, and functions that are common in both linear algebra and calculus. It's also important to note how much they inform each other. Now... Strang of all people knows this and is of course banking on the field of math for which he is most legendary for, but what I say here I believe is a worthy consideration for those not in the know. Linear algebra generalizes much of what is learned in early mathematics. It's more complicated in the sense that not only does it require you to be at least decent at the rudiments of arithmetic and elementary algebra, but then it expands to generalized vector spaces where things like commutativity laws themselves can change and have different behaviors according to dimension or rank. Certain aspects of it, however, are even simpler than elementary algebra. It allows for organization of information/data in a way that would be tiring to deal with the traditional old way. It also gives us a powerful computation/calculation method for many other fields of math, especially in fields like computer science itself, tensor analysis, or even quantum theory. And indeed, notions of curvature can arise in more advanced algebras, especially in multilinear algebra, Lie algebras, and geometric algebras. All follow the underlying rules of linear algebra. Don't forget that vector calculus itself is an extension of linear algebra to differentiable manifolds.

[@YessimkhanSeidikarim]

Cheers! Such a good comment. Concise, information-packed and simply true.

[@grayson1946]

Now who could argue with that? 😂😂😂😂

[@michaelh4227]

Currently reading this guy's Linear Algebra textbook right now! It's interesting to hear from the author himself.

[@rhydes_]

His book on linear algebra was top tier. Straight to the point helpful for a first year undergraduate student.

[@atulsrmcem]

as a matter of fact, I am currently reading Dr. Strang's book on Calculus, and it's one of the best ones I came across.

[@Mathin3D]

GS: In Linear algebra everything is flat But the came, functional analysis!

[@ABHISHEKSINGH-nv1se]

Then came Functional analysis and made every thing curvy flatty😊

[@hrperformance]

I like that guy. He's done some really good lectures on MIT Open courses

[@NapoleonPicard]

I love Professor Strang. What a hero

[@BrickBreaker21]

Linear algebra has more to memorize. I think there is a lot of visualization. Not so much of higher spaces, but of matrix order, transpositions, etc. I would teach myself to visualize operations like cofactor expansion, matrix multiplication, and other operations.

[@yoyoezzijr]

yeah theres alot of stuff like inverse of matrices, transpose, basis, linear transformations, eigenvectors, diagonalizing, cramer’s rule, hermitian/unitary, etc. it can be easy to forget unlike calculus

[@zappist751]

@@yoyoezzijr I disagree. I felt like if you actually understand a lot of the concepts in linear algebra then memorization is not necessary. I haven't found the need to memorize much after having to proof a lot of the concepts, which were much easier to prove when compared to calculus.

[@omniyambot9876]

@@zappist751 I'm new to linear algebra. I like rigorous style of learning then going intuitive and analysis. I absolutely hate memorization. Would you give me any advice? I'm taking this next month I better be ready. I'm top student in calculus in my university so there's so much expectations:(. I know deeply that these are very hard subjects. Is a month enough for me to familiarize heavily?

[@zappist751]

@@omniyambot9876 Don't worry! I find linear algebra to be easier than calculus in terms of understanding the concepts intuitively. Once you really understand the specific topic, you won't really have to do much memorization. I think a month is definitely enough for you to familiarize yourself with linear algebra (UC Berkeley's linear algebra course Math W54 is a month throughout the summer so I think you can definitely do it).

[@Manu-se5tx]

@@omniyambot9876 search for 3blue1brown mini course on linear algebra, without going in depth it gives a clear visualization of what a lot of linear algebra concepts are in a visual way, by having a clear image of what things really are you won't have any problem with the theory

[@JohnKooz]

Just a wise mathematician and an AI podcaster talking about their preference of Calculus and Linear Algebra like they were having a dialectic about which vehicle performs the best! This exchange between Gilbert Strang and Lex Fridman really, honestly, delighted me! Thanks, Lex!

[@psibarpsi]

*Flat earthers, after watching this* : lEt'S bUrN cAlCuLuS tExTbOoKs. AlL hAiL lInEaR aLgEbRa!

[@feezy243]

😂😂😂

[@integralboi2900]

A flat earther would never watch a video containing math.

[@of8155]

@@integralboi2900 well said boi

[@Peter_1986]

My engineering physics program in Sweden had a course at the end of my first year that was called "Functions Of Several Variables" (or "Math 4") - which was basically the equivalent of an American "Calculus 3" course - and I remember that we sometimes used the Wronskian matrix from Linear Algebra for checking if a set of functions were independent of each other.

[@Tabu11211]

Cool!

[@Nisirat]

functions is literally calculus 1 introduction calculus 3 is vectors

[@_mark_3814]

@@Nisirat they were dealing with abstract vector spaces they are talking about linear independence of functions

[@YoungLink51423]

Totally forgot about the Wronskian til now lol good shit

[@Peter_1986]

@@YoungLink51423 I think that we used the Hessian matrix for some things as well. It was quite satisfying to use ideas from Linear Algebra here and there like that.

[@JaspreetSingh-zp2nm]

These two courses are must of every math major, here are few more undergraduate courses that I think are interesting to study: Real and complex analysis, Abstract algebra, topology, analytic and algebraic number theory, ring theory, probability and statistics, combinatorics, graph theory, Differential equations, functional analysis, commutative algebra, Logic and set theory & many more. P.S : I haven’t studied them all.

[@bohanxu6125]

Linear algebra are typically proof-based classes while calculus can be proof-based or formula-based. Early college level calculus can just be an easy-to-learn formula-based class (first to learn)... but linear algebra classes are almost always proof based regardless if it is matrix based or linear space based (second to learn)... proof based calculus like real analysis with limit, continuity, derivative, differential form are indeed much harder to learn (third to learn)

[@Cantor214]

I still have nightmares about the Spectral Theorem.

[@sarvasvarora]

"It (calculus) kinda came first, but it didn't" Wow 😂🙏

[@133839297]

"If everything is flat, it can't go wrong". - Gil Strang, 2019.

[@raneena5079]

I think linear algebra should be taught first and then calc i, ii, iii. This is because ideas like linear operators, vectors, and determinants from linear algebra are useful in calculus, especially calc iii.

[@mr.erikchun5863]

there is a proof based linear algebra that I dont think a high schooler would be able to comprehend jumping from algebra or trig

[@raneena5079]

@@mr.erikchun5863 Yeah, I think more application-based/procedural linear algebra should be taught first, maybe mixing in some proofs/reasoning. I think Strang's book and courses are a good example. But proof-based LA is definitely more for college.

[@lowlightevangelist9431]

I'm in over 30 hours of Strang's lectures on KZbin, now. Glad for this interview, and Strang's mathematical candour. Mind blown when Strang said calculus is, at most, two dimensional. Thank you fpr sharing your knowledge, Mr Strang. However, Strang teaches mathematics as magic, therefore he is not of God and cannot be trusted.

[@RD-jc2eu]

I interpreted his remark about calculus to be saying that, in most college courses, even "multivariate" calculus is usually only working with two dimensions (even though calculus can be applied to more than that).

[@lowlightevangelist9431]

@@RD-jc2eu yeah, calculus seems to be for plotting a two dimensional course, and accounting for curves. What you wrote seems to be accurate.

[@sabinlevesque1250]

College Math teacher from Canada here. I'm always joking with my colleagues that we need to go down decolonial path to free ourselves from the tyranny of calculus. Statistics and Linear Algebra should be seen earlier and more profoundly for most students.

[@TB-ni4ur]

Honestly for people who can make it to linear algebra, it is a breath of fresh air and almost magical. For people who can't get past calculus, linear algebra would simply go right over their heads and it would turn into a plug and chug exercise with no understanding of just how elegant it is.

[@MarkWitucke]

How do you not fall in love with Gilbert Strang as an educator? A man of wonders.

[@metalmilitia1890]

He aays linear Algebra is simpler, but I have yet to see it. For whatever reason, calculus has always been a more relaxed class for me

[@vasilykartashov8970]

You interviewed an absolute legend

[@urisimchoni3936]

I remember studying linear algebra, and you have all those abstract definition, and I guess the way to go about it is "...Uhhh... OK...." without trying to imagine what those things mean in real life, because they really don't have to. And if you "just play with the symbols" according to the rules, it is simpler than Calculus. I was not as anxious before Algebra exam as I was before a Calculus exam or a Physics exam because it seemed there's a straightforward solution to anything they might throw at you, without need for tricks. Of course, I don't know if High School students have the same capacity for abstract thinking (maybe they have more?)

[@tharunsankar4926]

PDEs are essentially a system of ODEs where the system becomes infinitely big. You can convert a calculus problem into a linear algebra problem

[@andrewsovik3858]

No PDEs are just a system of ODEs where the variables are independent-either type of system can be linearized

[@jc6380]

Beautiful conversation

[@zackcarl7861]

Calculus is harder actually, it's just that in linear algebra have more formulas

[@dw5chaosfan]

And then there's functional analysis: why not both?

[@patinho5589]

They are both easy to intuit.. once you start to intuit it all with the lessons .. having done a few of strang’s online lectures.. I got to intuit what he was talking about.

[@FirstLast-gk6lg]

idk who Gilbert Strang is, but his facial expression looks like a man who is perpetually having a painfully complex thought.

[@patinho5589]

He’s a lovely teacher person. MIT lecturer. His MIT lecture course on linear algebra is on KZbin

[@ishaanparikh485]

Most complicated math exists purely to convert problems into linear algebra problems

[@raphaelreichmannrolim25]

Interesting comment. Some of my work does just that. I have created the concept of an arithmetical space, to study problems of number theory using linear operators.

[@zfolwick]

This guys chalkboard game was so strong

[@Quidoute]

idk man, calculus always felt more relaxing to me than linear algebra, and calculus it self at its core, stands by the idea that despite a graph beeing a curve we can work with flat slopes to visualize it

[@dabossbabie3605]

Interestingly, we learned at my school basic linear algebra before calculus. In our precalc course, we went over dot and cross, inverses, even linear transformations/mapping, eigenvalues/vectors, and such. I think it was good to be exposed to, because if not I would've never even known Linear Algebra existed today

[@arbaran01]

Same, some linear algebra is part of my HS's precalc course

[@Lemurai]

After I finished my Chem E degree, I was so burnt out I got a job as a production manager because I didn’t want to see another equation 😂

[@The-South-African-Host]

Linear algebra is not as intuitive case calculus. It's very abstract hahaha

[@scottbrandon6244]

Gil just retired from MIT (in May 2023) after a 61 year teaching career. Total legend in the field of mathematics.

[@Youtube304s]

When i took math econ i was stunned at the fact that i didnt learn it even before regular algebra. Seemed to make much more intuitive sense to me.

[@69erthx1138]

The whole concept of mutual orthogonality, regardless of dimension. Gil Strang is a legend, Hope you're enjoying retirement professor.

[@Mulkek]

Wow! great interview 👌

[@BigFatSandwitch]

Flat surface vs Curved surface. My work involved more flat surfaces but my personal preference is curved surfaces

[@kevinjones9586]

Both linear algebra and calculus have such a unique beauty. Different, but equally as beautiful.

[@renanpereira5487]

Strang is legend!

[@jaimelima2420]

The 21st is century is the century of the revenge of Linear Algebra.

[@uncoded0]

"If everything's flat, you can't go wrong." 😂

[@OlivierBCO]

Linear algebra brings you to the infinite horizon of plane space while calculus brings you to the infinite horizon of infinitesimal curved space...

[@joefuentes2977]

Linear algebra is a prerequisite for non-mickey-mouse calculus (post calc 1&2)

[@timcrnkovic8991]

Prof Strang is right - linear combinations involve addition and multiplication only. No integration of crazy exponential and logarithmic functions. It’s no contest.

[@gabrielmora6144]

The first time I used a matrix to solve an integral I lost my mind.

[@OmkarKulkarni-wf7ug]

Nothing bends in linear algebra except your mind

[@alejrandom6592]

I remember watching this before I even knew what lin alg is

[@royalfinest]

Linear Algebra approaches no limits.

[@ruskolnikov7211]

I think they work together. To ask what is Calculus?, I think it’s good to have linear algebra to explain linear operators and linear maps. It’s good to be able to picture abstract vector spaces and be able to view an integral as a very long dot product.

[@kichigan1]

True, Algebra is mostly 'flat'. But it became incredibly useful in economics, engineering, construction, aviation, etc. The program Excel is Algebra. It's a tool to minimize unnecessary algorithms. It's also 'fun', because it can solve age, sort, area, exponential, electrical, weight, speed, etc. problems.

[@austinharding9734]

i always found calculus to be easier than algebra n general

[@PeteZam]

thats what all my college math teachers used to say. the calculus is easy, its all the algebra that is often needed that is the hard part. and in the higher level classes that was often true for me. the calculus part was only one or two easy steps. its was all the manipulation needed to get it into a form that you wanted that was the hard part, which was usually algebra or trig or both.

[@VinyZikss]

in my uni, it's the complete opposite. In our first semester the percentage of students failing calculus is much much higher than linear algebra

[@adi29raj]

If everything is flat, you can never go wrong So true!!!

[@quirkyquester]

Professor Strang has done it!

[@ibgarrido]

Linear algebra and (multi variable) calculus work together and fine. They have something in common: vector spaces.

[@justanormalyoutubeuser3868]

In Italy, single variable calculus and linear algebra are taught at the same time in math, physics and engineering courses. However only calculus is taught in high school.

[@nope7200]

this guy is a legend

[@faeancestor]

heard about that

[@DanielDTUBWeinberger]

The multiple dimensions make perfect sense to me, it's the actual stuff we do and learn for a lot of portions. Calculus has been a lot easier for me, and it feels like linear algebra should be easy, but I'm struggling. My medical problems should make both hard, so maybe I am just better at working in 2 dimensions.

[@wesleyblack8302]

this man and herbert gross are why calculus is a hobby of mine.

[@MCLooyverse]

Linear algebra coming first would make sense, since calculus primarily studies two/a family of linear operators on nice real-to-real functions.

[@walidslimani]

THIS IS THE POWER OF MATH ,i love it and after that comes the little ones ( phyisics , biology ... ) applied stuff

[@deEvolutionary1729]

In my country, linear algebra does come first. I don't know why you'd have it the other way around.

[@speedspeed121]

I just made the connection that he was my textbook author. I literally watched the MIT videos while using his textbook for linear, but never made the connection

[@sketchfam8037]

from the moment i met calculus i knew there couldn't possibly be anything worse

[@jonetyson]

Well, linear algebra at the simple level of Strang's class is simpler, but I strongly disagree that it's simpler at a higher level, say at that of Bhatia's book Matrix Analysis. The idea that "everything is flat" is already suspect in Strang's course, which spends a fair amount of time on the matrix function A -> A^-1. Anyone who has studied the function x -> 1/x in linear algebra knows this isn't flat, but the matrix inverse is just a generalization. At a higher level, many other nonlinear functions of matrices come into play, and inequalities involving matrix functions are in general much much harder to prove than inequalities involving scalar functions.

[@valizeth4073]

Looking at linear algebra from a more abstract view, such as through abstract algebra, you can definitely see some strong similarities between the two, some things even being.. essentially the same, just different ways of expressing them.

[@736939]

He's rights, Linear algebra applied in ML/DL, Electronics and Computer Graphics. But Calculus is rare (in our digital present).

[@antonkochnev4185]

Fun fact: the matrix on the preview is singular

[@jonsnow8550]

I used to watch these in front of my friends to appear smart, then I watched him on joe rogan, now I've convinced myself I totally get it.

[@venkateshg3171]

A hero to all students!

[@Lateratesfaye]

My girlfriend used to say that... "If everything is flat, you can't go wrong!"

[@nadavshemesh1231]

"If everything is flat you cant go wrong haha", Oh but I can.. I can go very wrong

[@damienthorne861]

He is an awesome professor

[@atrumluminarium]

1:21 that's not what she said

[@temjenlongkumer4213]

Then their crossover in multivariable calculus

[@markhughes7927]

Algebra is millennia older than calculus - but maybe not in placing in contemporary curricula. Newton himself of ‘fluxions/calculus’ fame is reported to have preferred geometric solutions to a problem over algebraic solutions - and a dim memory comes up with the name of Wallis as an algebraist whose work was a component in Newton’s great synthesis - thus anterior.

[@epicmarschmallow5049]

Linear algebra is not "algebra" i.e. the manipulation of symbols like x and x. It's the study of vector spaces and the linear transformations between them, with these ideas mostly originating in the mid 19th century (the absolute earliest you can argue is the time of Descartes).

[@markhughes7927]

@@epicmarschmallow5049 Thank you - I have become aware since making the comment that he is talking about something of a different order! - had not heard of it before.

[@aleksanderaksenov1363]

Agreed I admit that linear algabra and lagara itsel is a basis of mathematics

[@phnml8440]

The thing about maths is that the more i study maths the more i understand that mathematicians as such really only completely understand linear algebra and apply it to all sorts of things. Of course this is not really true and an overexaggeration since like integration is a really simple example for something that has hardly to do with linear algebra. But all of differential analysis stands upon the idea of the „best linear approximation“ which is a linear Transformation and thus linear algebra. This really only becomes clear when you talk higher dimensional diff. analysis when you talk about Jacobi and Hesse Matricies and finding extreme values of multivariable functions. Also another example would be numerics or manifolds in topolgy.

[@ivankrasimirovralev]

Maybe it's like that in the US. But for me I had Linear Algebra and Analytical Geometry in one class and Calculus 1 in the other class. This was my 1st semester of university.

[@johnnyq4260]

Linear algebra did come first for me at 13, then calculus later at 15.

[@spacewad8745]

now that's a teacher!

[@DreamCodeLove]

Gilbert likes Curves not Flats ... all of us agree with you Sir

[@Lklibertad]

linear algebra or multi linear algebra are basis for awesome things, Michael penn has a video on that (linear algebra)

[@inarmi7169]

In singapore we generaly do linear algebra before calculus (middle and high school levels)

[@KINGSLPK]

The host knows about mathematics???!!! I Never knew that, good to know!!!!

[@steveperryman8102]

What? You mean Lex isn't asking his guest about (non-existent) aliens for once? Shock, horror !

[@brahmlewandowski4155]

im taking both at the same time rn😂😭

[@karu6111]

Linear algebra comes first in my experience, but calculus bamboozled me and my mates. We were taught it SO BADLY, I don't think we ever learn anything from it. Our lecturer had us memorize formulas and that's it. None of the visualizations, none. We passed that course but boy, it was shit.

[@haoyuezhang3137]

这个老师讲得特别好，他拯救了我的线代期末