Where to start learning Linear Algebra?

Question

I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.

I'm told that Linear Algebra is the best place to start. Where should I go?

Answer 1

You are right: Linear Algebra is not just the "best" place to start. It's THE place to start.

Among all the books cited in Wikipedia - Linear Algebra, I would recommend:

• Strang, Gilbert, Linear Algebra and Its Applications (4th ed.)

Strang's book has at least two reasons for being recommended. First, it's extremely easy and short. Second, it's the book they use at MIT for the extremely good video Linear Algebra course you'll find in the link of Unreasonable Sin.

For a view towards applications (though maybe not necessarily your applications) and still elementary:

• B. Noble & J.W. Daniel: Applied Linear Algebra, Prentice-Hall, 1977

Linear algebra has two sides: one more "theoretical", the other one more "applied". Strang's book is just elementary, but perhaps "theoretical". Noble-Daniel is definitively "applied". The distinction from the two points of view relies in the emphasis they put on "abstract" vector spaces vs specific ones such as $\mathbb{R}^n$ or $\mathbb{C}^n$, or on matrices vs linear maps.

Maybe because of my penchant towards "pure" maths, I must admit that sometimes I find matrices somewhat annoying. They are funny, specific, whereas linear maps can look more "abstract" and "ethereal". But, for instance: I can't stand the proof that the matrix product is associative, whereas the corresponding associativity for the composition of (linear or non linear) maps is true..., well, just because it can't help to be true the first moment you write it down.

Anyway, at a more advanced level in the "theoretical" side you can use:

• Greub, Werner H., Linear Algebra, Graduate Texts in Mathematics (4th ed.), Springer

• Halmos, Paul R., Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer

• Shilov, Georgi E., Linear algebra, Dover Publications

In the "applied" (?) side, a book that I love and you'll appreciate if you want to study, for instance, the exponential of a matrix is Gantmacher.

And, at any time, you'll need to do a lot of exercises. Lipschutz's is second to none in this:

• Lipschutz, Seymour, 3,000 Solved Problems in Linear Algebra, McGraw-Hill

Enjoy! :-)

Answer 2

MIT has a complete online course for linear algebra, complete with video lectures, lecture notes, and assignments.

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2005/

I'm not sure how much of the course is applicable to what you'll be doing in game development, but it's a solid foundation.

Answer 3

I'll add another title (it's a bit on the theoretical side, but still at the introductory level, very readable and definitely worth):

• Serge Lang, Linear algebra 3rd. ed., Springer

Answer 4

In my opinion, Schaum's Linear Algebra is very straight forward, has a nice balance between pure and applied, and has the advantage that it is cheap. Definitely, if you are auto-didactic, it will be easier to work through.

Answer 5

Can't comment, so this will have to be a post. As well as seconding Gilbert Strang's lively lectures on the open MIT site, his book, and Artin's, which are all excellent and all already recommended, I think you could also try Paul Dawkins' online math notes:

http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx

which are a fair bit shorter and, given your ultimate aim, the online OpenGL tutorials at NeHe are also probably useful (because you will see lots of examples of the application of linear algebra in the area of your interest):

http://nehe.gamedev.net/

Answer 6

Another excellent book, not mentioned here is Algebra by G. Birkhoff and S. Mac Lane.

NOT A Survey of Modern Algebra.

You can't go wrong with this one.

Answer 7

For a different take, try Practical Linear Algebra, A Geometry Toolbox by Gerald Farin & Dianne Hansford.

Answer 8

I'm very surprised no one's yet listed Sheldon Axler's Linear Algebra Done Right - unlike Strang and Lang, which are really great books, Linear Algebra Done Right has a lot of "common sense", and is great for someone who wants to understand what the point of it all is, as it carefully reorders the standard curriculum a bit to help someone understand what it's all about.

With a lot of the standard curriculum, you can get stuck in proofs and eigenvalues and kernels, before you ever appreciate the intuition and applications of what it's all about. This is great if you're a typical pure math type who deals with abstraction easily, but given the asker's description, I don't think that a rigorous pure math course is what he/she's asking for.

For the very practical view, yet also not at all sacrificing depth, I don't think you can do better than Linear Algebra Done Right - and if you are thirsty for more, after you've tried it, Lang and Strang are both great texts.

Answer 9

I would suggest

$1.)$ Linear Algebra , Kenneth Hoffman and Ray Kunze, Prentice Hall

$2.)$ Linear Algebra by Serge Lang , Springer

Answer 10

No-one seems to have mentioned Cliffs Notes Linear Algebra. If I just wanted to get up to speed quickly so as to have a broad picture and then decide how to proceed further, this is it - it may even be enough for your immediate needs. To go further you must decide whether you want.

1. something very practical in which theorems and proofs, whilst there, can be ignored or avoided, allowing you to concentrate on the "How to Do" aspects; or

2. something more theoretical (but still at undergrad level) telling you a bit about "The Why".

For ($1$) The two books by Schaum (Lipschutz) - one the actual text and the other the 3000 worked examples (latter already mentioned) - are solid and you can select what you need. Also, they are much cheaper than most. Stanley I Grossman's Elementary Linear Algebra and Strang's applications book, not his other, more theoretical book, are in this category too.

For ($2$) you cannot beat Axler despite what some have said - and he is still fairly practical. Lang's book on LinAlg is short, easy to read and has everything you will need. Of course, Strang's more theoretical book strikes a nice balance.

Finally, don't overlook the Web - it has some really solid stuff not only on Wiki but on many other sites. In fact, once you've got to grips with Cliffs Notes and therefore have an idea of the main topics you can browse the Web and get masses of info on vector spaces, matrices, linear transformations (very important for gaming, I would think), linear equations etc etc. Oh yes, what about books on computer graphics? I don't have any titles (look up Amazon) but the books I've seen were full of stuff about manipulating vectors, for example, when "playing" with graphics images.