# Where to start learning Linear Algebra?

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?

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My professors suggested "Algebra" by Micheal Artin...It's been really useful to me so far! Unfortunately I can't give you any info about the edition or whatever because I only know about the italian ones. –  Andy Sep 9 '10 at 18:00
@Andy For learning LINEAR ALGEBRA?!? Artin's great if you already know some linear algebra and you're ready for a serious introduction to abstract algebra. But boy,that would be a really tough slog for a beginner,even a talented one.A much gentler book with a similar slant and which presents linear algebra from jump is E.B.Vinberg's "A Course In Algebra". I think both of you will find that book much easier and equally informative. –  Mathemagician1234 Aug 17 '11 at 7:24
@Mathemagician: yes, the book was the first in the list of suggested books in my first year geometry class (which was basically onlyu linear algebra). I didn't find it that hard, and it was an invaluable source for my abstract algebra (as you said) and another linear algebra course I took. –  Andy Aug 17 '11 at 17:02
The gamedev.net math and physics forums are usually a pretty good place to hang out for what you're doing. They have a more practical and applied bent, and most of the people there come from a programming background such as yourself, though there are math and computer science phds that post there as well. You can do little projects on your own, ask questions, and pick up things as you need it. gamedev.net/forum/20-math-and-physics –  Nick Alger Aug 8 '12 at 5:06

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 recomended. 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 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! :-)

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Would you recommend? Introduction to Linear Algebra, Fourth Edition [Hardcover]? –  Sergio Tapia Sep 9 '10 at 17:56
Which "Introduction to Linear Algebra"? –  Agustí Roig Sep 9 '10 at 18:04
@Sergio If you mean Friedberg,Insel,etc.-YES,ABSOLUTELY,SERGIO. i think that's the best,most balanced linear algebra book around. It hits just the right balance between the pure theory and the applications of the subject-both of which to me are of equal importance. –  Mathemagician1234 Aug 17 '11 at 7:28

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.

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I'm very surprised noone'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.

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Honestly, Axler's book, in my opinion, lacked any sense of "common sense" when presented to students with no formal analytical algebraic background. Even now, when I read back through that book, I see things and say, "why would you present that!?" or "why would you leave that exercise for the reader!? It is fundamentally important to understanding the chapter!!!" In my personal experience, the pedagogy linked to the book was also quite poor; it took me a few years to truly understand any of the concepts therein. –  Arkamis Aug 8 '12 at 1:55

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
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I'm a fan of Linear Algebra by Linear Algebra Kenneth M Hoffman and Ray Kunze. It shows somewhat how Linear Algebra and Algebra relate...

But if you only intend to use one book it's not suitable.

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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/

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I would suggest

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

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

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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.

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I wouldn't use it as my only textbook unless I had no choice,Scott-but it's definitely a very good book to have when learning LA,for all the reasons you gave. –  Mathemagician1234 Aug 17 '11 at 7:29

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.

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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.

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I'd suggest organizing your answer into paragraphs. –  Jonah Sinick Nov 1 '12 at 14:54

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

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I am surprised that no one offered a book by David Poole.

It is a text we used at University of Edinburgh and it's comprehension and depth is enough to satisfy everyone. It starts from the basics, how to add vectors, dot product and other high-school stuff.

Also it has applications of linear algebra, such as binary vectors, ISBN code representation and everything is provided in the very first chapters.

After that, the whole thing gets tougher and even more interesting.

Another great aspect of this book is that it does not make you wonder where the equation was pulled out. It is explained quite thoroughly.

And another great asset of this book is exercises. There are a ton of them, ranging from simple ones to quite tough.

The author provides a study plan in the introduction for Computer Science guys and it is very good grounding. I walked through the book and I am happy to have learned a lot.

"Linear Algebra. A Modern Introduction" by David Poole. It's third edition already.

It is quite pricey when you search on USA stores but I bought it from UK ebay for 45 bucks. So do a thorough search before buying.

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