48 ms·

Too Much Calculus – Gilbert Strang (2001) [pdf]

214 points by nkurz 12y ago

evrim 12y ago
heh, try teaching galois for a year, you'll understand why calculus is so popular.
- banachtarski 12y ago
  it's a beautiful branch of mathematics though.
  - evrim 12y ago
    everybody thinks they know algebra if they study linear algebra. this is worse than not knowing about calculus i guess. can't image teaching point-set topology, teaching uryshons proof to undergrads, gee.
SixSigma 12y ago
I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"
- benburton 12y ago
  His lectures are wonderful! In college I skipped my regular Linear Algebra lectures, and used the time slot in my schedule to watch his MIT lectures instead. I ended up scoring one of the highest grades in the class.
  - SixSigma 12y ago
    Finding out the Fourier transform was a way to multiply matrices was a mind blower :)
    - jordigh 12y ago
      I suppose you mean this? http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/positive-definite-matrices-and-applications/complex-matrices-fast-fourier-transform-fft/MIT18_06SCF11_Ses3.2sum.pdf http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr... I was a bit confused by your statement. You mean that the FFT can be done by matrix multiplication? I tried to interpret your statement the other way around, that you can somehow transform two matrices before you multiply them, trying to fit it into the convolution theorem, but I couldn't make sense of it that way.
      - pbsd 12y ago
        I was puzzled as well. There is an analogous of the FFT method of multiplying polynomials and integers to the matrix multiplication case [1], but it is not as simple and probably does not belong in an introduction to linear algebra. [1] http://arxiv.org/abs/math/0307321v2 http://arxiv.org/abs/math/0307321v2
- baldfat 12y ago
  As someone who did work at a University who also loved classes I would bring friends who were visiting to classes and every time they would say something like this, "I hated Philosophy as a college student, but somehow now I loved every second of that class." They were always surprised by that experience.
- tptacek 12y ago
  Those lectures are fantastic. Pro-tip: watch them at 2.5x speed (you'll keep up just fine, and you can always pause to play with the concepts in Sage or whatever). Bonus: when you switch him back to normal speed, he sounds stoned out of his mind.
  - YAYERKA 12y ago
    This is a great tip. I've been using it for a few years now to watch various lectures on youtube--especially mathematics. Professors lecturing mathematics tend to speak slowly since they don't want to say anything erroneous. Speeding them up is usually amazing. For youtube; open the javascript console in your browser and type '$('video').playbackRate = 2.5;'. I've found that each lecturer usually has their own magic number regarding speed--after watching someone for a few hours and varying the speed you can usually find it.
- joeyyang 12y ago
  This OCW was my lifeline during my college Linear Algebra course. Learned almost all the course material from 18.06 videos. Awesome, awesome stuff.
- blt 12y ago
  Incredible. I learned LA in school but it didn't click 100% until I watched Strang's lectures. I think they are ideal for someone who's already reasonably comfortable with matrices and vectors, but wants to see that higher connection and beauty. His lecture deriving the determinant from a few simple properties is magical to me. His emphasis on the fundamental subspaces of a matrix, and the geometric interpretation over the equation-solving one, are ideal for anyone who thinks visually/spatially. That was the first online lecture series I ever watched, and I've been hooked ever since. I am overflowing with gratitude that so many professors at the world's best schools have put their lectures online. It is so good for a student like me, who went to a liberal arts school that didn't offer many advanced math or CS courses, and learns better from lectures than books. Strang's Linear Algebra course was one of the first way back in 2002. You're right, he is a wonderful teacher.
  - SixSigma 12y ago
    Well, I was going to say "you should also check out Walter Lewin's Physics lectures, they are equally stunning" But it seems MIT took them down http://www.reddit.com/r/Physics/comments/2ope15/mit_indefinitely_removes_online_physics_lectures/ http://www.reddit.com/r/Physics/comments/2ope15/mit_indefini... This borders on criminal and is certainly a tragedy. I'm quite struck and emotional about it.
wwweston 12y ago
Possibly understated... not only is there the three course sequence + diff eq, there's usually also a two-sequence real analysis course (more or less "careful calculus") and probably either PDEs or complex analysis to boot. I took the discrete emphasis when I studied and I think I actually ended up with fewer discrete classes (5, IIRC) than calculus classes (6).
- titanomachy 12y ago
  Although it's not explicitly stated in the article, I think Strang's remarks are actually directed at non-math programs in STEM (I assume you are at least a minor in math because of 3 analysis courses). Math majors will get plenty of algebra either way, but it's the engineers and scientists who miss out from overemphasising calc over algebra.
oleks 12y ago
Perhaps amusingly, the Computer Science education at the University of Copenhagen includes no mandatory Calculus course, but it does include a Linear Algebra course. Even so, students have a hard time seeing what they'll use Linear Algebra for, before later on in their education.
- marktangotango 12y ago
  I think this observation highlights the problem with Linear Algebra, the applications aren't encountered until upper level courses in other fields, and then they teach what you need to know for the course. I took two semesters of Linear Algebra in college, the first course was an overview for physicists and engineers and was little more than the mechanics of matrix operations, and find eigen vectors and eigen values. The second course focused on Linear Algebra as an abstract algebra, with near zero application. Subsequently, reading this paper[1] and the way the author used Linear Algebra blew my mind, why wasn't I taught about THAT kind of Linear Algebra? I suppose my examples illustrates: great teach Linear Algebra, but what facets to concentrate on? Abstract or concrete applications? [1] http://www.geometrictools.com/Documentation/IntersectionSphereCone.pdf http://www.geometrictools.com/Documentation/IntersectionSphe...
  - cableshaft 12y ago
    I took a 3d graphics programming course where we built our own raytracer for our final project. We learned the math for a few different intersections in order to make the raytracer. So if they offered it, that would have been the course to take.
- ajuc 12y ago
  Make them code simple wolfenstain 3d clone with opengl, directx or even unity3d. It was a revelation for me when I've tried to get into game programming, and half of the math taught in primary and secondary school became immediately useful, also linear algebra (which was only started at the final year of high school). Made linear algebra courses on the university a lot more interesting. There's a difference if you look at a class "what I need to pass an exam", and "what I could use in my next game". Also tinkering with 3d graphics makes thinking in vectors, planes, matrices, dot and cross products intuitive after a few weeks. BTW I wonder hwo many kids learnt basic linear algebra from Denthor's Asphyxia tutorials :)
ekidd 12y ago
As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's free textbook: http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ (GFDL license, LaTeX source available) (I get nothing out of recommending this open source textbook other than the joy of sharing a book I loved.)
- jimhefferon 12y ago
  Hey, thanks!
  - jsuskin 12y ago
    That was really cool.
- jseliger 12y ago
  until I read Jim Hefferon's free textbook I still prefer reading textbooks in paper, and one interesting note: it looks like a couple of companies and individuals are offering paper copies on Amazon, but none of them appear to be Henderson himself. And the companies or individual offering paper copies still charge $20 – $30!
  - coopaq 12y ago
    http://joshua.smcvt.edu/linearalgebra/book.pdf http://joshua.smcvt.edu/linearalgebra/book.pdf
  - krussell 12y ago
    I just noticed under the "Download Linear Algebra" section on http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ , he gives a link to a paperback version of Linear Algebra on Amazon (ISBN-13: 978-0989897563).
  - 12y ago
nicholasdrake 12y ago
i actually have taught some linear algebra concepts on youtube and have gotten several hundred thousand views and some positive feedback. i think the key is trying to figure out a framework of intuition to help students through all the math but i liked the idea of four key equations we are trying to solve as a framework as well
xamuel 12y ago
(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on very neat topics for these students (mathematics of voting, etc.) so I'd say math curriculum has already been successfully reformed for them. As for the average STEM student, in my experience, most freshmen wouldn't be ready for linear algebra (unless by "linear algebra" you mean "matrix arithmetic"). They have enough trouble understanding functions in Calculus I: linear algebra's "functions are basically matrix multiplication" feature would NOT help. Another thing to keep in mind is calculus has had much more impact on history/society than linear algebra. The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance. Pragmatically, calculus is the most suitable mathematics for the type of testing-oriented, uniformizable/transferable pedagogy that a university really (in cold practical terms) must depend on for first-year students. It would be extremely difficult to force the kind of uniformity onto linear algebra that would be necessary if you seriously want to have millions of transfer students moving around between thousands of universities every quarter, gossiping about which linear algebra teacher gives the easiest grades, etc.
- davidwihl 12y ago
  I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification for its continuation.
  - weland 12y ago
    > I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study). We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.
cageface 12y ago
I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.
- andrepd 12y ago
  If you were looking to become a programmer, maybe you shouldn't have majored in chemstry, where strong Calculus foundations make perfect sense :)
  - samatman 12y ago
    Chemist here, who also programs. Inorganic chemists do calculus all the time. Organic chemists do arithmetic (stoichiometry) and some discrete analysis (instrumentation, the computers do the heavy lifting), orgo is primarily a linguistic discipline, i.e. we are expected to parse sentences like "benzophenone in THF was refluxed in the presence of Pd black for 2 hours" and to know, without being told, that there's a workup before the next step in the reaction.
    - cageface 12y ago
      Exactly. I was an organic chemist. Never used a lick of Calculus. I didn't get into programming until I'd already been working as a chemist for about five years.
- relaytheurgency 12y ago
  "I learned more than enough welding when apprenticing as a metalsmith but now as a chef use none of it."
  - cageface 12y ago
    At least at that time the comp sci people I knew had to take about the same number of math classes the chem people did. Maybe that's changed now.
kowdermeister 12y ago
The Khan Academy lectures are a pretty way to start: https://www.youtube.com/playlist?list=PL835D2252574274C5 https://www.youtube.com/playlist?list=PL835D2252574274C5
clebio 12y ago
What is the context of this short essay? Why did he feel the need to write it, at that time?
ludicast 12y ago
I agree. As far as calculus goes, I am more enamored with books like Spivak's (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...) that take a proof-centric approach to teach calculus from first principles. Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment: https://www.edx.org/course/linear-algebra-foundations-frontiers-utaustinx-ut-5-02x https://www.edx.org/course/linear-algebra-foundations-fronti... (from UT Austin) https://www.edx.org/course/applications-linear-algebra-part-1-davidsonx-d003x-1 https://www.edx.org/course/applications-linear-algebra-part-... (from Davidson) http://coursera.org/course/matrix http://coursera.org/course/matrix (from Brown) The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index.html http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index... . And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.
- pakled_engineer 12y ago
  My calc I course in university was applied calculus without a text. I had to go back and redo single variable by reading Spivak (and Polya's How to Solve It) to figure out the proofs in Concrete Math by D. Knuth, et. al.
jkbyc 12y ago
My linear algebra professor insisted that we, as computer science graduates, know and can reproduce most if not all the proofs of the theorems presented in the class. He didn't have such demands for math majors. His point was that we as computer scientists / software engineers / programmers will always have to understand and construct all the minutiae of the programs we write (otherwise they won't work). In contrast, mathematicians could use a lot of it "just" as tools. I also had to go through a lot of heavy calculus courses which were very proof-centric and built from first principles. Rarely we skipped a proof of a theorem and we were expected to be able to reproduce and understand all of them at exams. I hated it often back then. However, these days I can appreciate the rigor it has taught me and I would wish I could send some of the programmers I met to similar courses so that they would learn to think clearly and precisely and be able to spot unmet or missing assumptions. Our teachers and the creators of the curriculum were not stupid or naive. They knew well how much of hands on calculus we would use in practice vs. algebra or linear algebra. One of their main reasons for including calculus was to teach us rigor. I know that once someone overheard two teachers talking - the calculus teacher was asking someone from a CS department what to teach the CS graduates and the reply was that he can teach us whatever he wants, he just should make sure to teach us to think. I think they were quite successful at it even if it wasn't a very pleasant experience for most. That also reminds me of this very relevant and true comic: http://savethepenguins.net/image/237344573 http://savethepenguins.net/image/237344573 ("Wtf, man. I just wanted to learn how to program video games.")
jmilloy 12y ago
> We need to present the mathematics that is most useful to the most students. I wish it weren't this way! Mathematics is a wide and fascinating field, and I think more students could find interest if our schooling reflected that. But moving some of the detail and drudgery that is the calculus sequence to subject matter that requires it, in order to make way for linear algebra, is a great start.
- jeffreyrogers 12y ago
  I agree completely, but calculus touches so much of advanced math that at some point you have to learn it in depth (assuming you actually want to learn advanced mathematics). I'd argue that linear algebra is a more interesting subject for most people, but calculus contains more important ideas that are crucial if you intend to continue studying mathematics.
  - jmilloy 12y ago
    I have a major in Mathematics, and I truly didn't need calculus after the multivariable/vector calculus course I had to take my Freshman fall. Why did I have to do so much if it wasn't going to be necessary until graduate school at the earliest, or perhaps never depending on specialization? And why was it a prerequisite for my other math courses (in algebra, topology, real analysis, number theory, and combinatorics)?
    - peterfirefly 12y ago
      How was topology motivated without talking about generalizations of limits? (Honest question, no snark.)
      - jmilloy 12y ago
        It's a good question. I think my first topology course, like in many intro analysis courses, shared concepts from calculus without requiring any previous expertise from our calculus courses. A rigorous approach to real analysis or point-set topology will be very set-theoretic, and you build the general definitions of continuity, compactness, etc from the ground up. So, I would have used concepts from my analysis and topology courses in calculus, if only it had been taught in the other order. I guess it feels like limits are a rigorous foundation for calculus, but they are not calculus. In any case, I didn't need anything from differential or integral calculus (or that entire required course developing the theoretical basis for Stokes...), which is what I mean to say.
ivan_ah 12y ago
Anyone interested in getting started with linear algebra can check out this printable tutorial: Linear algebra in four pages: http://minireference.com/blog/linear-algebra-tutorial/ http://minireference.com/blog/linear-algebra-tutorial/ The page also links to another short tutorial which explains more of the geometric intuition behind linear transformations. Neither tutorial can serve as a full course, but reading them will give you a good overview of what linear algebra is about.
hzhou321 12y ago
I wish Gilbert Strang would teach calculus and write a book as he did for linear algebra.
- dfan 12y ago
  You are in luck! Strang, Calculus, 2nd edition: http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang/dp/0980232740 http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang...
  - bewuethr 12y ago
    And the first edition is freely available at Open Courseware: http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
    - hzhou321 12y ago
      I wish every day is like this!
jeffreyrogers 12y ago
I think Strang has some good points, and his advice is true for the majority of students who won't go on to use substantial amounts of mathematics or need to learn much math in the future. However, if you do want to learn more math beyond calculus and linear algebra, then a thorough understanding of calculus is absolutely a prerequisite and you honestly probably haven't learned it well enough taking the typical sequence of classes. For example, I'm currently studying probability theory at a fairly rigorous level (building up from measure theory) and my relative lack of skill with calculus is the main stumbling block. And I was recently talking with one of my professors who said that the biggest obstacle he sees students facing in trying to understand advanced mathematics isn't the difficulty of the math itself, but in their lack of mastery of the fundamental calculations that are necessary to follow the ideas and to come up with them on your own. Fluency with calculus is an enormous part of this. That said, I agree with the point, which is that for most students linear algebra is more useful. Actually, I might go a bit farther and say that most students don't need to learn much math at all, at least in the way it is taught. Math is pretty poorly taught for a subject that is actually quite interesting once you start to understand it better (part of the problem is that in order to get to the interesting parts you need to first master all the boring stuff), and most people won't use any nontrivial math in their future lives.
- cbgb 12y ago
  This may be true for mathematics in the analytical tradition (things like Topology, Measure Theory, Real/Complex Analysis), but Linear Algebra is far more important in, unsurprisingly, algebraic disciplines. These include Number Theory, Field Theory, and Mathematical Logic. While studying mathematics in college, once I had finished my Real Analysis requirement, I jumped headfirst into the algebraic side of things and never found myself using any sort of calculus. Even in my Topology course, we focused much more on using techniques from Real Analysis than specifically calculus topics (the former just being a generalization of the latter). Probability theory is somewhat deceptive in its classification, since much of it "feels" a lot like a discrete mathematics course; however, much of the concepts, like you say, are underpinned by measure-theoretic principles, which is heavily analytic. It makes sense that calculus would come in handy in a much deeper study of probability theory.
laurentoget 12y ago
14 years later, the popularity of machine learning and "big data", which requires quite a bit of linear algebra, makes this only truer.
- tel 12y ago
  It also requires quite a bit of calculus though!
  - laurentoget 12y ago
    Calculus is indeed a required tool if you want to design new machine learning algorithm. But without linear algebra, good luck implementing an algorithm or even using a library to apply it on a concrete case.
    - tel 12y ago
      I don't see why this is an either/or decision? Without a firm understanding of probability then good luck interpreting or fitting a model. Without a firm understanding of linear algebra good luck implementing or using an algorithm. And hell, if you have a streaming algorithm you probably no longer need linear algebra but to grasp either the probabilistic or geometric interpretations you'll probably need calculus.
j2kun 12y ago
To reinforce this point: here's just a fraction of what you can do with linear algebra [1] [1]: https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-what-you-can-do-with-linear-algebra-633383d4153f https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-wha...
- aet 12y ago
  To be fair, almost all that stuff uses calculus too.
tel 12y ago
You can probably do worse than to cut mathematics up into "algebra" (not just linear) and "analysis" (calculus). They're almost completely diametric points of view by the bulk modern mathematical canon [0]. There's a fair point to say that one shouldn't learn so much calculus as to lose your linear algebra, but the opposite is just as true. There are a lot of comments in this thread along the lines of saying that linear algebra is far more useful than calculus in CS. This statement feel not exactly unfair but certainly like it's a reactionary measure. Whenever you have a notion of change you have some kind of calculus underlying it. Even if you end up having to view that notion of change algebraically. For instance, there's a famous functional programming result that there is a well-defined "derivative of an algebraic data type" which allows you to talk about changes in data "over time". I'd also feel very bad for anyone who tried to learn too much probability without studying quite a bit of calculus first. [0] Sure there's discrete calculus and algebra underlies the mechanism of most calculus computation. Likewise there are continuous groups and you can't avoid how topology mixes the two concepts.
deleted 12y ago
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forkandwait 12y ago
Another perspective: I am trying to teach myself physics from Morin's books, and for the first time I see how well structured was my lower division calc sequence d for this application.
graycat 12y ago
> Too Much Calculus Not for me! Heck, I never took freshman calculus or pre-calculus and, instead read a book and started on sophomore calculus then continued with, right, ordinary differential equations, advanced calculus for applications, and advanced calculus, topology, modern analysis, measure theory, functional analysis, exterior algebra, etc. Not enough calculus for me! Strang, of course, is emphasizing linear algebra. Great! It's terrific stuff. I had an abstract algebra course that started on vector spaces and matrix theory. Then I got famous book from Princeton on multi-variable calculus and read the first chapters on linear algebra, e.g., learned Gram-Schmidt. Then I got a linear algebra book and read a few chapters and applied them to some problems in classical mechanics. Then I wrote my math honors paper in group representation theory, right, more linear algebra. The out of school, I wanted to know linear algebra better and read a good book cover to cover, carefully. Then I got Halmos, Finite Dimensional Vector Spaces and read it cover to cover. Wrote Halmos about his proof of the Hamelton-Cayley theorem and got back a nice answwer! Then I got another famous text on multi-variable calculus and read it cover to cover, including the parts on exterior algebra. Then I did that again from Spivak's Calculus on Manifolds -- right, a lot of both calculus and linear algebra. Then I got Forsythe and Moler, Computer Solution of Linear Algebraic Systems, read it cover to cover, directed a project in interval arithmetic for linear algebra, took a course in numerical analysis with a lot of linear algebra for solution of partial differential equations, e.g., iterative solutions (Gauss-Seidel) and wrote corresponding code. I also used linear algebra in work with the fast Fourier transform, multi-variate statistics, and more. For more on linear algebra, I read the first part, linear algebra, of von Neumann, Quantum Mechanics. From a world class guy in linear algebra, I took an advanced course in linear algebra -- by then the course was beneath me and I led the class by a wide margin. Favorite theorem -- the polar decomposition. Read Coddington's book on differential equations -- gorgeous book. The combination of linear algebra and calculus, with other math, continued. At one time a little calculus did a lot: The initial value problem for the little first order, linear ordinary differential equation y'(t) = k y(t) ( b - y(t) ) saved FedEx, that is, kept it from going out of business. So, useful calculus! So, sure, I like linear algebra and like the combination! I definitely do not see that I had too much calculus! Here's what I saw, and still fume about: All I could get in chemistry was one too simple course in high school and one too simple course in college. All I could get in physics was one too simple course. I college, by the time I got to Maxwell's equations, I didn't know enough calculus to do really well. In college I never got math enough to do well with quantum mechanics. But in grades 9-12 and then in the first two years of college, I was force fed like a goose with six years of English literature, with each year yet another play by Shakespeare. I agree that there was a good writer in England in the 1600s -- Newton! So it was also Chaucer, Milton, Wordsworth, ..., Dickens, the great natural order (all corresponding theorems and proofs omitted!), etc. With good information about people? Nope. Then there was history: It never got to the 20th century. It never touched on technology, economics, or any other causes. There was plenty of time in history courses; it's just that the courses didn't have much content. If my startup works, then I'll get back to more in calculus, functional analysis, stochastic processes, and mathematical physics. "More calculus, Ma!".
lolwhat 12y ago
I remember taking Calculus 1, 2, 3 but only having 1 linear algebra course and some engineering math. Although I did well in them, as a developer, I wish I payed more attention to linear algebra or at least wished that my uni offered more courses related to it, like discrete math and maybe even statistics. Good thing I still have my old books!
cpitman 12y ago
Statistics are wildly important for both business and understanding the world around you, and without solid calculus skills you cannot truly work with statistics. Its not something that comes up every day at work, but it is still pretty often. For example, I was on a large team developing a complicated distributed system. The performance requirements were uncertain (like 500000 clients +- 200000, sending 10 +- 30 messages per second, etc). The system itself was in early development, hadn't been tuned, and we had only rough performance numbers for it. Given only that, I needed to purchase the correct amount of hardware for the test lab, keeping in mind that the lead time for hardware was several months. I needed both statistics and some basic calculus to build a model based on what we knew, calculating variances for each intermediate and output, and then getting 50%, 90%, 99% confidence level estimates. In the end, we got pretty close to a bullseye. On the other hand, I learned the important lesson to hide all the math from the client, lest they get awfully confused.
xxcode 12y ago
I took Gilbert Strang's Mathematical Methods for Engineers course at MIT. Can't seem to find the link for it, but I would recommend the online version for anyone who wants to learn linear algebra.
EdwardCoffin 12y ago
I don't disagree that linear algebra isn't given enough emphasis. However, the first thing I thought of when I read this was something attributed to Richard Hamming, who is known for the transcript of his talk that pops up on here frequently, You and your Research [1]. I got this from a review of his book on Amazon [2]: > Hamming, in discussion, concluded his life thinking "the best tool to teach thinking was to teach the calculus." I'm just thinking here that Strang is saying Linear Algebra is more useful for _applying_, where Hamming would say Calculus is more useful for reasons that go beyond applications (not that it doesn't also have applications). [1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html http://www.cs.virginia.edu/~robins/YouAndYourResearch.html [2] http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm?ie=UTF8&ASIN=9056995014 http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm...
hexagonc 12y ago
I had the fortune of taking 18.06 (Linear Algebra) from Gilbert Strang around the time of this essay. And like other commenters, I will agree that he was a very good teacher. I could tell that he held a great emotional attachment to matrices; he'd often anthropomorphize them with genders and refer to them as "little guys".
xlm1717 12y ago
I think taking less calculus and more linear algebra is a mistake. There is a really good book in two volumes that I feel shows how methods of linear algebra can be extended using calculus to solve a wide range of problems: A Course in Mathematics for Students of Physics: http://www.goodreads.com/book/show/2685577-a-course-in-mathematics-for-students-of-physics http://www.goodreads.com/book/show/2685577-a-course-in-mathe... Volume 2 is more important. It starts demonstrating how methods of Linear Algebra can be used to solve circuit problem, then generalizes this to higher dimensions and from discrete to continuous spaces using the exterior derivative. Seeing this demonstrated gave me a much clearer, more intuitive understanding for both calculus and linear algebra.
tdfx 12y ago
I've never understood why Calculus is taught as a separate course. It seems so detached from practicality. I've always thought students should take perhaps 4-6 semesters of Physics -- and introduce derivatives, integrals, etc. as needed when they can be demonstrated to more easily solve the physics problems. If the physics content was more spread out, there would be ample time to introduce the necessary math concepts, instead of stuffing them into a separate course.
- AnimalMuppet 12y ago
  I remember, in my high school calculus class, the teacher said that there were six or seven applications of derivatives, and more like 40 applications of integrals. It's way more than just physics.
jules 12y ago
I don't think you can cleanly separate the topics. Most applications of linear algebra come from calculus since after all you use calculus to linearize something. In case it was already linear it's probably a linear differential equation -> calculus again. Complex analysis is also closely related to linear algebra via geometric algebra. Rather than teaching them separately and emphasizing linear algebra more, it would make more sense to approach calculus + linear algebra as an integrated subject.
mjh2539 12y ago
The way this was typeset...it's making my head hurt.

214 points by nkurz 12y ago

evrim 12y ago

heh, try teaching galois for a year, you'll understand why calculus is so popular.

banachtarski 12y ago
it's a beautiful branch of mathematics though.
- evrim 12y ago
  everybody thinks they know algebra if they study linear algebra. this is worse than not knowing about calculus i guess. can't image teaching point-set topology, teaching uryshons proof to undergrads, gee.

SixSigma 12y ago

I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"

benburton 12y ago
His lectures are wonderful! In college I skipped my regular Linear Algebra lectures, and used the time slot in my schedule to watch his MIT lectures instead. I ended up scoring one of the highest grades in the class.
- SixSigma 12y ago
  Finding out the Fourier transform was a way to multiply matrices was a mind blower :)
  - jordigh 12y ago
    I suppose you mean this? http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/positive-definite-matrices-and-applications/complex-matrices-fast-fourier-transform-fft/MIT18_06SCF11_Ses3.2sum.pdf http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr... I was a bit confused by your statement. You mean that the FFT can be done by matrix multiplication? I tried to interpret your statement the other way around, that you can somehow transform two matrices before you multiply them, trying to fit it into the convolution theorem, but I couldn't make sense of it that way.
    - pbsd 12y ago
      I was puzzled as well. There is an analogous of the FFT method of multiplying polynomials and integers to the matrix multiplication case [1], but it is not as simple and probably does not belong in an introduction to linear algebra. [1] http://arxiv.org/abs/math/0307321v2 http://arxiv.org/abs/math/0307321v2
baldfat 12y ago
As someone who did work at a University who also loved classes I would bring friends who were visiting to classes and every time they would say something like this, "I hated Philosophy as a college student, but somehow now I loved every second of that class." They were always surprised by that experience.
tptacek 12y ago
Those lectures are fantastic. Pro-tip: watch them at 2.5x speed (you'll keep up just fine, and you can always pause to play with the concepts in Sage or whatever). Bonus: when you switch him back to normal speed, he sounds stoned out of his mind.
- YAYERKA 12y ago
  This is a great tip. I've been using it for a few years now to watch various lectures on youtube--especially mathematics. Professors lecturing mathematics tend to speak slowly since they don't want to say anything erroneous. Speeding them up is usually amazing. For youtube; open the javascript console in your browser and type '$('video').playbackRate = 2.5;'. I've found that each lecturer usually has their own magic number regarding speed--after watching someone for a few hours and varying the speed you can usually find it.
joeyyang 12y ago
This OCW was my lifeline during my college Linear Algebra course. Learned almost all the course material from 18.06 videos. Awesome, awesome stuff.
blt 12y ago
Incredible. I learned LA in school but it didn't click 100% until I watched Strang's lectures. I think they are ideal for someone who's already reasonably comfortable with matrices and vectors, but wants to see that higher connection and beauty. His lecture deriving the determinant from a few simple properties is magical to me. His emphasis on the fundamental subspaces of a matrix, and the geometric interpretation over the equation-solving one, are ideal for anyone who thinks visually/spatially. That was the first online lecture series I ever watched, and I've been hooked ever since. I am overflowing with gratitude that so many professors at the world's best schools have put their lectures online. It is so good for a student like me, who went to a liberal arts school that didn't offer many advanced math or CS courses, and learns better from lectures than books. Strang's Linear Algebra course was one of the first way back in 2002. You're right, he is a wonderful teacher.
- SixSigma 12y ago
  Well, I was going to say "you should also check out Walter Lewin's Physics lectures, they are equally stunning" But it seems MIT took them down http://www.reddit.com/r/Physics/comments/2ope15/mit_indefinitely_removes_online_physics_lectures/ http://www.reddit.com/r/Physics/comments/2ope15/mit_indefini... This borders on criminal and is certainly a tragedy. I'm quite struck and emotional about it.

wwweston 12y ago

Possibly understated... not only is there the three course sequence + diff eq, there's usually also a two-sequence real analysis course (more or less "careful calculus") and probably either PDEs or complex analysis to boot. I took the discrete emphasis when I studied and I think I actually ended up with fewer discrete classes (5, IIRC) than calculus classes (6).

titanomachy 12y ago
Although it's not explicitly stated in the article, I think Strang's remarks are actually directed at non-math programs in STEM (I assume you are at least a minor in math because of 3 analysis courses). Math majors will get plenty of algebra either way, but it's the engineers and scientists who miss out from overemphasising calc over algebra.

oleks 12y ago

Perhaps amusingly, the Computer Science education at the University of Copenhagen includes no mandatory Calculus course, but it does include a Linear Algebra course. Even so, students have a hard time seeing what they'll use Linear Algebra for, before later on in their education.

marktangotango 12y ago
I think this observation highlights the problem with Linear Algebra, the applications aren't encountered until upper level courses in other fields, and then they teach what you need to know for the course. I took two semesters of Linear Algebra in college, the first course was an overview for physicists and engineers and was little more than the mechanics of matrix operations, and find eigen vectors and eigen values. The second course focused on Linear Algebra as an abstract algebra, with near zero application. Subsequently, reading this paper[1] and the way the author used Linear Algebra blew my mind, why wasn't I taught about THAT kind of Linear Algebra? I suppose my examples illustrates: great teach Linear Algebra, but what facets to concentrate on? Abstract or concrete applications? [1] http://www.geometrictools.com/Documentation/IntersectionSphereCone.pdf http://www.geometrictools.com/Documentation/IntersectionSphe...
- cableshaft 12y ago
  I took a 3d graphics programming course where we built our own raytracer for our final project. We learned the math for a few different intersections in order to make the raytracer. So if they offered it, that would have been the course to take.
ajuc 12y ago
Make them code simple wolfenstain 3d clone with opengl, directx or even unity3d. It was a revelation for me when I've tried to get into game programming, and half of the math taught in primary and secondary school became immediately useful, also linear algebra (which was only started at the final year of high school). Made linear algebra courses on the university a lot more interesting. There's a difference if you look at a class "what I need to pass an exam", and "what I could use in my next game". Also tinkering with 3d graphics makes thinking in vectors, planes, matrices, dot and cross products intuitive after a few weeks. BTW I wonder hwo many kids learnt basic linear algebra from Denthor's Asphyxia tutorials :)

ekidd 12y ago

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's free textbook: http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ (GFDL license, LaTeX source available) (I get nothing out of recommending this open source textbook other than the joy of sharing a book I loved.)

jimhefferon 12y ago
Hey, thanks!
- jsuskin 12y ago
  That was really cool.
jseliger 12y ago
until I read Jim Hefferon's free textbook I still prefer reading textbooks in paper, and one interesting note: it looks like a couple of companies and individuals are offering paper copies on Amazon, but none of them appear to be Henderson himself. And the companies or individual offering paper copies still charge $20 – $30!
- coopaq 12y ago
  http://joshua.smcvt.edu/linearalgebra/book.pdf http://joshua.smcvt.edu/linearalgebra/book.pdf
- krussell 12y ago
  I just noticed under the "Download Linear Algebra" section on http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ , he gives a link to a paperback version of Linear Algebra on Amazon (ISBN-13: 978-0989897563).
- 12y ago

nicholasdrake 12y ago

i actually have taught some linear algebra concepts on youtube and have gotten several hundred thousand views and some positive feedback. i think the key is trying to figure out a framework of intuition to help students through all the math but i liked the idea of four key equations we are trying to solve as a framework as well

xamuel 12y ago

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on very neat topics for these students (mathematics of voting, etc.) so I'd say math curriculum has already been successfully reformed for them. As for the average STEM student, in my experience, most freshmen wouldn't be ready for linear algebra (unless by "linear algebra" you mean "matrix arithmetic"). They have enough trouble understanding functions in Calculus I: linear algebra's "functions are basically matrix multiplication" feature would NOT help. Another thing to keep in mind is calculus has had much more impact on history/society than linear algebra. The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance. Pragmatically, calculus is the most suitable mathematics for the type of testing-oriented, uniformizable/transferable pedagogy that a university really (in cold practical terms) must depend on for first-year students. It would be extremely difficult to force the kind of uniformity onto linear algebra that would be necessary if you seriously want to have millions of transfer students moving around between thousands of universities every quarter, gossiping about which linear algebra teacher gives the easiest grades, etc.

davidwihl 12y ago
I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification for its continuation.
- weland 12y ago
  > I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study). We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.

cageface 12y ago

I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.

andrepd 12y ago
If you were looking to become a programmer, maybe you shouldn't have majored in chemstry, where strong Calculus foundations make perfect sense :)
- samatman 12y ago
  Chemist here, who also programs. Inorganic chemists do calculus all the time. Organic chemists do arithmetic (stoichiometry) and some discrete analysis (instrumentation, the computers do the heavy lifting), orgo is primarily a linguistic discipline, i.e. we are expected to parse sentences like "benzophenone in THF was refluxed in the presence of Pd black for 2 hours" and to know, without being told, that there's a workup before the next step in the reaction.
  - cageface 12y ago
    Exactly. I was an organic chemist. Never used a lick of Calculus. I didn't get into programming until I'd already been working as a chemist for about five years.
relaytheurgency 12y ago
"I learned more than enough welding when apprenticing as a metalsmith but now as a chef use none of it."
- cageface 12y ago
  At least at that time the comp sci people I knew had to take about the same number of math classes the chem people did. Maybe that's changed now.

kowdermeister 12y ago

The Khan Academy lectures are a pretty way to start: https://www.youtube.com/playlist?list=PL835D2252574274C5 https://www.youtube.com/playlist?list=PL835D2252574274C5

clebio 12y ago

What is the context of this short essay? Why did he feel the need to write it, at that time?

ludicast 12y ago

I agree. As far as calculus goes, I am more enamored with books like Spivak's (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...) that take a proof-centric approach to teach calculus from first principles. Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment: https://www.edx.org/course/linear-algebra-foundations-frontiers-utaustinx-ut-5-02x https://www.edx.org/course/linear-algebra-foundations-fronti... (from UT Austin) https://www.edx.org/course/applications-linear-algebra-part-1-davidsonx-d003x-1 https://www.edx.org/course/applications-linear-algebra-part-... (from Davidson) http://coursera.org/course/matrix http://coursera.org/course/matrix (from Brown) The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index.html http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index... . And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.

pakled_engineer 12y ago
My calc I course in university was applied calculus without a text. I had to go back and redo single variable by reading Spivak (and Polya's How to Solve It) to figure out the proofs in Concrete Math by D. Knuth, et. al.

jkbyc 12y ago

My linear algebra professor insisted that we, as computer science graduates, know and can reproduce most if not all the proofs of the theorems presented in the class. He didn't have such demands for math majors. His point was that we as computer scientists / software engineers / programmers will always have to understand and construct all the minutiae of the programs we write (otherwise they won't work). In contrast, mathematicians could use a lot of it "just" as tools. I also had to go through a lot of heavy calculus courses which were very proof-centric and built from first principles. Rarely we skipped a proof of a theorem and we were expected to be able to reproduce and understand all of them at exams. I hated it often back then. However, these days I can appreciate the rigor it has taught me and I would wish I could send some of the programmers I met to similar courses so that they would learn to think clearly and precisely and be able to spot unmet or missing assumptions. Our teachers and the creators of the curriculum were not stupid or naive. They knew well how much of hands on calculus we would use in practice vs. algebra or linear algebra. One of their main reasons for including calculus was to teach us rigor. I know that once someone overheard two teachers talking - the calculus teacher was asking someone from a CS department what to teach the CS graduates and the reply was that he can teach us whatever he wants, he just should make sure to teach us to think. I think they were quite successful at it even if it wasn't a very pleasant experience for most. That also reminds me of this very relevant and true comic: http://savethepenguins.net/image/237344573 http://savethepenguins.net/image/237344573 ("Wtf, man. I just wanted to learn how to program video games.")

jmilloy 12y ago

> We need to present the mathematics that is most useful to the most students. I wish it weren't this way! Mathematics is a wide and fascinating field, and I think more students could find interest if our schooling reflected that. But moving some of the detail and drudgery that is the calculus sequence to subject matter that requires it, in order to make way for linear algebra, is a great start.

jeffreyrogers 12y ago
I agree completely, but calculus touches so much of advanced math that at some point you have to learn it in depth (assuming you actually want to learn advanced mathematics). I'd argue that linear algebra is a more interesting subject for most people, but calculus contains more important ideas that are crucial if you intend to continue studying mathematics.
- jmilloy 12y ago
  I have a major in Mathematics, and I truly didn't need calculus after the multivariable/vector calculus course I had to take my Freshman fall. Why did I have to do so much if it wasn't going to be necessary until graduate school at the earliest, or perhaps never depending on specialization? And why was it a prerequisite for my other math courses (in algebra, topology, real analysis, number theory, and combinatorics)?
  - peterfirefly 12y ago
    How was topology motivated without talking about generalizations of limits? (Honest question, no snark.)
    - jmilloy 12y ago
      It's a good question. I think my first topology course, like in many intro analysis courses, shared concepts from calculus without requiring any previous expertise from our calculus courses. A rigorous approach to real analysis or point-set topology will be very set-theoretic, and you build the general definitions of continuity, compactness, etc from the ground up. So, I would have used concepts from my analysis and topology courses in calculus, if only it had been taught in the other order. I guess it feels like limits are a rigorous foundation for calculus, but they are not calculus. In any case, I didn't need anything from differential or integral calculus (or that entire required course developing the theoretical basis for Stokes...), which is what I mean to say.

ivan_ah 12y ago

Anyone interested in getting started with linear algebra can check out this printable tutorial: Linear algebra in four pages: http://minireference.com/blog/linear-algebra-tutorial/ http://minireference.com/blog/linear-algebra-tutorial/ The page also links to another short tutorial which explains more of the geometric intuition behind linear transformations. Neither tutorial can serve as a full course, but reading them will give you a good overview of what linear algebra is about.

hzhou321 12y ago

I wish Gilbert Strang would teach calculus and write a book as he did for linear algebra.

dfan 12y ago
You are in luck! Strang, Calculus, 2nd edition: http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang/dp/0980232740 http://www.amazon.com/Calculus-Second-Edition-Gilbert-Strang...
- bewuethr 12y ago
  And the first edition is freely available at Open Courseware: http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
  - hzhou321 12y ago
    I wish every day is like this!

jeffreyrogers 12y ago

I think Strang has some good points, and his advice is true for the majority of students who won't go on to use substantial amounts of mathematics or need to learn much math in the future. However, if you do want to learn more math beyond calculus and linear algebra, then a thorough understanding of calculus is absolutely a prerequisite and you honestly probably haven't learned it well enough taking the typical sequence of classes. For example, I'm currently studying probability theory at a fairly rigorous level (building up from measure theory) and my relative lack of skill with calculus is the main stumbling block. And I was recently talking with one of my professors who said that the biggest obstacle he sees students facing in trying to understand advanced mathematics isn't the difficulty of the math itself, but in their lack of mastery of the fundamental calculations that are necessary to follow the ideas and to come up with them on your own. Fluency with calculus is an enormous part of this. That said, I agree with the point, which is that for most students linear algebra is more useful. Actually, I might go a bit farther and say that most students don't need to learn much math at all, at least in the way it is taught. Math is pretty poorly taught for a subject that is actually quite interesting once you start to understand it better (part of the problem is that in order to get to the interesting parts you need to first master all the boring stuff), and most people won't use any nontrivial math in their future lives.

cbgb 12y ago
This may be true for mathematics in the analytical tradition (things like Topology, Measure Theory, Real/Complex Analysis), but Linear Algebra is far more important in, unsurprisingly, algebraic disciplines. These include Number Theory, Field Theory, and Mathematical Logic. While studying mathematics in college, once I had finished my Real Analysis requirement, I jumped headfirst into the algebraic side of things and never found myself using any sort of calculus. Even in my Topology course, we focused much more on using techniques from Real Analysis than specifically calculus topics (the former just being a generalization of the latter). Probability theory is somewhat deceptive in its classification, since much of it "feels" a lot like a discrete mathematics course; however, much of the concepts, like you say, are underpinned by measure-theoretic principles, which is heavily analytic. It makes sense that calculus would come in handy in a much deeper study of probability theory.

laurentoget 12y ago

14 years later, the popularity of machine learning and "big data", which requires quite a bit of linear algebra, makes this only truer.

tel 12y ago
It also requires quite a bit of calculus though!
- laurentoget 12y ago
  Calculus is indeed a required tool if you want to design new machine learning algorithm. But without linear algebra, good luck implementing an algorithm or even using a library to apply it on a concrete case.
  - tel 12y ago
    I don't see why this is an either/or decision? Without a firm understanding of probability then good luck interpreting or fitting a model. Without a firm understanding of linear algebra good luck implementing or using an algorithm. And hell, if you have a streaming algorithm you probably no longer need linear algebra but to grasp either the probabilistic or geometric interpretations you'll probably need calculus.

j2kun 12y ago

To reinforce this point: here's just a fraction of what you can do with linear algebra [1] [1]: https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-what-you-can-do-with-linear-algebra-633383d4153f https://medium.com/@jeremyjkun/here-s-just-a-fraction-of-wha...

aet 12y ago
To be fair, almost all that stuff uses calculus too.

tel 12y ago

You can probably do worse than to cut mathematics up into "algebra" (not just linear) and "analysis" (calculus). They're almost completely diametric points of view by the bulk modern mathematical canon [0]. There's a fair point to say that one shouldn't learn so much calculus as to lose your linear algebra, but the opposite is just as true. There are a lot of comments in this thread along the lines of saying that linear algebra is far more useful than calculus in CS. This statement feel not exactly unfair but certainly like it's a reactionary measure. Whenever you have a notion of change you have some kind of calculus underlying it. Even if you end up having to view that notion of change algebraically. For instance, there's a famous functional programming result that there is a well-defined "derivative of an algebraic data type" which allows you to talk about changes in data "over time". I'd also feel very bad for anyone who tried to learn too much probability without studying quite a bit of calculus first. [0] Sure there's discrete calculus and algebra underlies the mechanism of most calculus computation. Likewise there are continuous groups and you can't avoid how topology mixes the two concepts.

deleted 12y ago

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forkandwait 12y ago

Another perspective: I am trying to teach myself physics from Morin's books, and for the first time I see how well structured was my lower division calc sequence d for this application.

graycat 12y ago

> Too Much Calculus Not for me! Heck, I never took freshman calculus or pre-calculus and, instead read a book and started on sophomore calculus then continued with, right, ordinary differential equations, advanced calculus for applications, and advanced calculus, topology, modern analysis, measure theory, functional analysis, exterior algebra, etc. Not enough calculus for me! Strang, of course, is emphasizing linear algebra. Great! It's terrific stuff. I had an abstract algebra course that started on vector spaces and matrix theory. Then I got famous book from Princeton on multi-variable calculus and read the first chapters on linear algebra, e.g., learned Gram-Schmidt. Then I got a linear algebra book and read a few chapters and applied them to some problems in classical mechanics. Then I wrote my math honors paper in group representation theory, right, more linear algebra. The out of school, I wanted to know linear algebra better and read a good book cover to cover, carefully. Then I got Halmos, Finite Dimensional Vector Spaces and read it cover to cover. Wrote Halmos about his proof of the Hamelton-Cayley theorem and got back a nice answwer! Then I got another famous text on multi-variable calculus and read it cover to cover, including the parts on exterior algebra. Then I did that again from Spivak's Calculus on Manifolds -- right, a lot of both calculus and linear algebra. Then I got Forsythe and Moler, Computer Solution of Linear Algebraic Systems, read it cover to cover, directed a project in interval arithmetic for linear algebra, took a course in numerical analysis with a lot of linear algebra for solution of partial differential equations, e.g., iterative solutions (Gauss-Seidel) and wrote corresponding code. I also used linear algebra in work with the fast Fourier transform, multi-variate statistics, and more. For more on linear algebra, I read the first part, linear algebra, of von Neumann, Quantum Mechanics. From a world class guy in linear algebra, I took an advanced course in linear algebra -- by then the course was beneath me and I led the class by a wide margin. Favorite theorem -- the polar decomposition. Read Coddington's book on differential equations -- gorgeous book. The combination of linear algebra and calculus, with other math, continued. At one time a little calculus did a lot: The initial value problem for the little first order, linear ordinary differential equation y'(t) = k y(t) ( b - y(t) ) saved FedEx, that is, kept it from going out of business. So, useful calculus! So, sure, I like linear algebra and like the combination! I definitely do not see that I had too much calculus! Here's what I saw, and still fume about: All I could get in chemistry was one too simple course in high school and one too simple course in college. All I could get in physics was one too simple course. I college, by the time I got to Maxwell's equations, I didn't know enough calculus to do really well. In college I never got math enough to do well with quantum mechanics. But in grades 9-12 and then in the first two years of college, I was force fed like a goose with six years of English literature, with each year yet another play by Shakespeare. I agree that there was a good writer in England in the 1600s -- Newton! So it was also Chaucer, Milton, Wordsworth, ..., Dickens, the great natural order (all corresponding theorems and proofs omitted!), etc. With good information about people? Nope. Then there was history: It never got to the 20th century. It never touched on technology, economics, or any other causes. There was plenty of time in history courses; it's just that the courses didn't have much content. If my startup works, then I'll get back to more in calculus, functional analysis, stochastic processes, and mathematical physics. "More calculus, Ma!".

lolwhat 12y ago

I remember taking Calculus 1, 2, 3 but only having 1 linear algebra course and some engineering math. Although I did well in them, as a developer, I wish I payed more attention to linear algebra or at least wished that my uni offered more courses related to it, like discrete math and maybe even statistics. Good thing I still have my old books!

cpitman 12y ago

Statistics are wildly important for both business and understanding the world around you, and without solid calculus skills you cannot truly work with statistics. Its not something that comes up every day at work, but it is still pretty often. For example, I was on a large team developing a complicated distributed system. The performance requirements were uncertain (like 500000 clients +- 200000, sending 10 +- 30 messages per second, etc). The system itself was in early development, hadn't been tuned, and we had only rough performance numbers for it. Given only that, I needed to purchase the correct amount of hardware for the test lab, keeping in mind that the lead time for hardware was several months. I needed both statistics and some basic calculus to build a model based on what we knew, calculating variances for each intermediate and output, and then getting 50%, 90%, 99% confidence level estimates. In the end, we got pretty close to a bullseye. On the other hand, I learned the important lesson to hide all the math from the client, lest they get awfully confused.

xxcode 12y ago

I took Gilbert Strang's Mathematical Methods for Engineers course at MIT. Can't seem to find the link for it, but I would recommend the online version for anyone who wants to learn linear algebra.

EdwardCoffin 12y ago

I don't disagree that linear algebra isn't given enough emphasis. However, the first thing I thought of when I read this was something attributed to Richard Hamming, who is known for the transcript of his talk that pops up on here frequently, You and your Research [1]. I got this from a review of his book on Amazon [2]: > Hamming, in discussion, concluded his life thinking "the best tool to teach thinking was to teach the calculus." I'm just thinking here that Strang is saying Linear Algebra is more useful for _applying_, where Hamming would say Calculus is more useful for reasons that go beyond applications (not that it doesn't also have applications). [1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html http://www.cs.virginia.edu/~robins/YouAndYourResearch.html [2] http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm?ie=UTF8&ASIN=9056995014 http://www.amazon.com/review/RWPF0EL3E7GJZ/ref=cm_cr_pr_perm...

hexagonc 12y ago

I had the fortune of taking 18.06 (Linear Algebra) from Gilbert Strang around the time of this essay. And like other commenters, I will agree that he was a very good teacher. I could tell that he held a great emotional attachment to matrices; he'd often anthropomorphize them with genders and refer to them as "little guys".

xlm1717 12y ago

I think taking less calculus and more linear algebra is a mistake. There is a really good book in two volumes that I feel shows how methods of linear algebra can be extended using calculus to solve a wide range of problems: A Course in Mathematics for Students of Physics: http://www.goodreads.com/book/show/2685577-a-course-in-mathematics-for-students-of-physics http://www.goodreads.com/book/show/2685577-a-course-in-mathe... Volume 2 is more important. It starts demonstrating how methods of Linear Algebra can be used to solve circuit problem, then generalizes this to higher dimensions and from discrete to continuous spaces using the exterior derivative. Seeing this demonstrated gave me a much clearer, more intuitive understanding for both calculus and linear algebra.

tdfx 12y ago

I've never understood why Calculus is taught as a separate course. It seems so detached from practicality. I've always thought students should take perhaps 4-6 semesters of Physics -- and introduce derivatives, integrals, etc. as needed when they can be demonstrated to more easily solve the physics problems. If the physics content was more spread out, there would be ample time to introduce the necessary math concepts, instead of stuffing them into a separate course.

AnimalMuppet 12y ago
I remember, in my high school calculus class, the teacher said that there were six or seven applications of derivatives, and more like 40 applications of integrals. It's way more than just physics.

jules 12y ago

I don't think you can cleanly separate the topics. Most applications of linear algebra come from calculus since after all you use calculus to linearize something. In case it was already linear it's probably a linear differential equation -> calculus again. Complex analysis is also closely related to linear algebra via geometric algebra. Rather than teaching them separately and emphasizing linear algebra more, it would make more sense to approach calculus + linear algebra as an integrated subject.

mjh2539 12y ago

The way this was typeset...it's making my head hurt.