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Comment by ekidd | Hacker News Reader

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/ (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.)

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I agree although I work in robotics / computer vision.

What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.

Do you have a recommendation for a good statistics textbook, for someone who is mathematically literate, but doesn't know almost any statistics?

This book is great and designed for exactly this type of audience: http://www.stat.cmu.edu/~larry/all-of-statistics/

http://smile.amazon.com/Introduction-Bayesian-Statistics-2nd...

Was extremely gentle and mind blowing.

But statistics and probability is really just integration. It's either that or stamp collection.

And optimization is really just differentiation (+ some geometry).

I'd say the problem is too little calculus and not too much.

Don't know why you're being downvoted. This is totally true.

Most of my downvotes so far, both here and on reddit, have been earned by writing things that are true but unpopular.

The weird thing is that I can't predict which things are going to be sufficiently unpopular amongst people impolite enough to downvote instead of coming up with a good counterargument (or even better: fact). I had /never/ expected something like this to get a single downvote (and I was down at -3).

I agree as well. Not sure why you're being downvoted. Maybe I don't know enough about linear algebra to judge but everything ML that I've ever worked in was spoken in calculus..

Totally agree, calculus is a prerequisite for understanding statistics, unless you're just plugging in numbers for an A-B test - and you don't really need a class to do that.

Linear Algebra is very useful too, but it's not really fair to say one is more valuable than another. It just depends on what kinds of applications you end up working on. Universities should just teach both.

Many interesting things require a combination of more than one area of math. I don't understand how you can do much within robotics or vision without using all three of calculus, linear algebra, and statistics :)

I'll go even further and state that without an understanding of calculus, linear algebra, and statistics, an engineer or even a data scientist is little more than a glorified script kiddy.

Statistic teaches you application and helps to develop correct intuition. Can's speak for everyone, but my college algebra courses were 100% abstract stuff detached from reality. All professors had the attitude that once you learn the theory you could apply it to anything. Problem is, it's hard to remember theory if you don't map to something tangible, and it's hard to apply it to reality when you're just used to crunching equations.

Statistics is, classically, more developed in the realm of "applied math" than "pure math" while linear algebra is, again classically, more a "pure math" discipline. The difference in rhetoric you cite falls out of this exactly.

In practice, both have wonderful presence in math both applied and pure. Linear algebra is one form of abstract algebra which is easily one of the most important mathematical disciplines. It's also the characterization and method of manipulating large arrays of numbers in interesting and practical ways. Statistics is the mechanism of interpreting live data under knowledge of uncertainty toward generalization. It's also probably well characterized as a particularly well-behavior substudy of measure theory... the pure mathematical notion of generalized "substance"

my college algebra courses

Sorry, I meant my college calculus courses, although incidentally the statement applies to both.

I think colleges would be much better off teaching introductory math as "applied". I got significantly more out of statistics, because that class taught us how to spot problems where that branch of mathematics is applicable. I really liked how Udacity used real-life orbit calculations from various space missions as assignments. It gives the whole thing some context.

Hey, thanks!

That was really cool.

Thanks for the link to the book.

I had a similar experience where my college linear algebra class was not good, taught by a grad student who seemed to have no enthusiasm for the material or for teaching, plus his accent was hard to understand at times. It was also the first exposure to proofs in the math curriculum so from the beginning I remember being overwhelmed by those and totally losing sight of the forest for the trees.

I sat through the whole semester thinking it was a pretty useless subject. My senior year I happened to take a machine learning and a quantum mechanics class at the same time and had a big ah-ha moment when I realized how big a role linear algebra plays in such a wide range of subjects. I went back through my old textbook and notes at that point and that new perspective made a huge difference, it was like it was completely different material and it was a lot easier to understand.

As an aside, I get really confused and disheartened when I hear that someone's first exposure to proof was in college in a linear algebra class.

I had a somewhat old-fashioned education for an American, and learned proof via Euclid when I was 12. And again, but poorly, when I was 16, but that's another story. Proof is supposed to be learned a) thoroughly b) early and c) as a part of geometry. What's the point in even studying Euclid if you don't learn the method of proof?

BLESS YOU. I have a computer vision exam on Tuesday. Naturally, the prof is awful at explaining math and only seems to care about the theory part.

Thank you for the link. I had the same experience in uni. It's a general issue that plagues mathematics education, I think: theory should come after some basic understanding of applications, not before.

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!

http://joshua.smcvt.edu/linearalgebra/book.pdf

I just noticed under the "Download Linear Algebra" section on http://joshua.smcvt.edu/linearalgebra/ , he gives a link to a paperback version of Linear Algebra on Amazon (ISBN-13: 978-0989897563).

> And the companies or individual offering paper copies still charge $20 – $30!

$20 for the official version (see the book's web page for a link). That is, essentially, the price of printing and binding.

Linear Algebra is also important in Finance too. Stats are useful in almost every field, as well as for developing a probability-based state of mind.

I wonder why the imbalance exists. Too much focus on preparing people for careers in Engineering? Or preparing them for more abstract concepts like Analysis and Analytic Geometry?

> Too much focus on preparing people for careers in Engineering? Or preparing them for more abstract concepts like Analysis and Analytic Geometry?

Yes. Computer Science curricula tend to focus overmuch on preparing students to work as software engineers, and Mathematics curricula tend to focus overmuch on analysis (to the point that once you get past Calc 1+2+3, Linear Algebra, Differential Equations, Mathematical Analysis/Epsilon-Delta Calculus-with-proofs, and Real Analysis, things like Algebra and Topology are senior-year electives).

Yes - it's too bad they keep the fun stuff for the end. I've been rediscovering the fun of number theory via Project Euler.

For some coding puzzles I was given during interviews, I found that sometimes linear algebra solutions can be used. I submitted such a solution once to a company for a take home problem, they rejected me but it was still an interesting experience.

As a programmer, I've used calculus and linear algebra exactly never in my professional (20+ years) career. It just doesn't really come up doing web apps, sysadmin or dba work.

A lot of this is whether or not you are looking for it, honestly. It is easy to stay away from it, though it is also easy to find if you look for it.

I work on games. I use linear algebra daily and calculus frequently, but far less often. Both are extremely relevant to game development although you can get by without calculus and only light linear algebra if you limit your scope or specialize (in something other than graphics or simulation).

In my 20-odd years, I'd agree---uses of calc or linear algebra have been vanishingly rare, although my current project is changing that a bit. Not because I didn't have opportunities to use it, but because I didn't seek out those opportunities.

On the other hand, I'm prepared to defend to the death the statement that programming is applied formal logic. (Or possibly abstract algebra; that is likely more appropriate for some more modern things. On the other hand, I grew up in the land of Dijkstra, so....)

Mechanics don't use calculus, even many engineers don't, but those working in R&D, especially towards the R side, do.

ekidd - thanks for the link!

Also, I know it's not free (the coursera course is free) but it's still a great read and seriously applicable http://codingthematrix.com/

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