What are the prerequisites for learning Linear Algebra?

[AkshayV]

I'm a high school student and mostly I love maths as a hobby and try to learn advanced topics. I'm in 11th grade and have finished most of high school algebra and calculus and want to learn Linear and Abstract Algebra. So, what are the prerequisites? And which one should I do before -Linear or Abstract? And what are some books that might be of help? (Please don't say Michael Artin!)

 

[Fredrik]

I would say that you only need familiarity with real and complex numbers, and an understanding of the concept of functions...but those things may be explained in the book too.

Everyone studies linear algebra first, and abstract algebra at some later time (if they study it at all). I recommend that you do the same. There is nothing that really prevents you from doing it the other way round, but linear algebra is a bit easier and much more useful both for physicists and mathematicians.

Regarding books, I only know that I like Axler, and dislike Anton a lot. However, micromass seems to have given it a lot more thought, so you might want to check out his https://www.physicsforums.com/blog.php?b=3206 [Broken] about it. (Apparently the book to read is the one by Friedberg, Insel & Spence).

 

[lavinia]

I'm a high school student and mostly I love maths as a hobby and try to learn advanced topics. I'm in 11th grade and have finished most of high school algebra and calculus and want to learn Linear and Abstract Algebra. So, what are the prerequisites? And which one should I do before -Linear or Abstract? And what are some books that might be of help? (Please don't say Michael Artin!)

You seem to know enough already although I agree with Fredrik that you should know about the complex numbers and also something about the theory of polynomials over the complex numbers.

Abstract algebra does not require knowing linear algebra first. Group theory is key to physics so if you are a physics student you need to know something about it. Certain theorems in linear algebra are best understood for the abstract algebra standpoint but it is not necessary to go there at the start.

 

[Deveno]

strictly speaking, the prerequisites for linear algebra are pretty slim. if you can add, multiply, subtract and divide, you can solve systems of linear equations, which occupies a big chunk of linear algebra (at least the beginning part).

so what you want to do, is play around for matices a good bit. get used to them, because they can fool you. your ordinary intutions about arithmetic don't always apply. get an introductory book, which talks about matrices up-front.

but don't kid yourself, this is not linear algebra...it's more like linear arithmetic. you need to know the alphabet before you can write. the most useful examples of vector concepts come from the euclidean plane, and euclidean 3-space, and it's a good idea to get comfortable with these. i don't have a specific book, but it should have a title like: "introduction to linear algebra". artin is way too sophisticated for a first look.

if you've not done much multivariate calculus, now is a good time to do that, too. the early parts of linear algebra, and functions of vectors, inform each other like a pair of newlyweds. calculus of several variables goes part of the way towards explaing why we need these vector and matrix things. i like trotter, crowell and williamson's "calculus of vector functions" it's a bit dated, but it has enough to it for a first look (don't just plunge ahead into something like rudin, you'll get discouraged).

but these are just the appetizers, the real meat of linear algebra lies deeper. to really have an appreciation of it, you'll want at least a smattering of group theory, and a little knowledge about rings and fields will be helpful. pinter and herstein are two authors that come up a lot, pinter is a lot easier on the newbie.

after that, it's time for "real linear algebra", with a real linear algebra text. i like hoffman/kunze, and I'm sure there are others here who know what I'm talking about. yes, the typsetting is horrible, but that's not why it's good.

if you rush it, you can learn most of linear algebra in around 6-8 months. but don't. learn the basics of computation first, then learn more about "the rules of structure of abstract math thingies" (which is what abstract algebra is, in spades). there's a LOT of abstract math thingies, but groups (especially abelian groups) and fields are what you want to know the most about (as far as linear algebra is concerned). a vector space is what you get, when a field and an abelian group get married, and decide to start a family.

 

[micromass]

Fredrik makes excellent points. However, from a pure math point-of-view, it makes a bit more sense to study abstract algebra first and then linear algebra. The reason is that you can then use the abstract algebra lingo in linear algebra.

But of course, all university students study linear first and then abstract. For the reasons that linear is much more applicable, and that linear is also a bit easier and more intuitive than abstract algebra.

So it doesn't really matter if you study linear or abstract first. If you do begin with abstract, then try "a book on abstract algebra" by Pinter. It needs no previous knowledge of linear algebra and it is suitable for a high-school student.

In my blog, I've been a bit too hard on Axler. But the only reason for that is that I dislike his treatment of determinants. Whether you like it or not, determinants are a big part of linear algebra. If I want to calculate the eigenvalues of a matrix, then I immediately start off by calculating a determinant.

Apart from that, Axler is a wonderful book!

 

[mathwonk]

paul shields has a very nice little book, Elementary linear algebra, that I highly recommend. If that is too easy, you can move up, but it is a good place to start, and will convince you how easy the subject can be. And it is not mickey mouse - he was a professor at stanford when he wrote it, and also wrote Calculus of elementary functions for the SMSG group.

here are some used copies starting under $4:

http://www.abebooks.com/servlet/SearchResults?an=shields&tn=elementary+linear+algebra&x=58&y=14

In my opinion Axler is not the place for a high school student to begin. I think it is more useful as a second course.

 

[Deveno]

Fredrik makes excellent points. However, from a pure math point-of-view, it makes a bit more sense to study abstract algebra first and then linear algebra. The reason is that you can then use the abstract algebra lingo in linear algebra.

But of course, all university students study linear first and then abstract. For the reasons that linear is much more applicable, and that linear is also a bit easier and more intuitive than abstract algebra.

So it doesn't really matter if you study linear or abstract first. If you do begin with abstract, then try "a book on abstract algebra" by Pinter. It needs no previous knowledge of linear algebra and it is suitable for a high-school student.

In my blog, I've been a bit too hard on Axler. But the only reason for that is that I dislike his treatment of determinants. Whether you like it or not, determinants are a big part of linear algebra. If I want to calculate the eigenvalues of a matrix, then I immediately start off by calculating a determinant.

Apart from that, Axler is a wonderful book!

you know, math has this whole chicken-and-the-egg thing going on. do we try to present the general truth, and illustrate with examples? or do we start with the familiar, and then generalize? i am in favor of a hybrid approach. i believe a good explorer should survey the territory, and learn a bit about the strange customs of a new land, but not pull up stakes and move there until she's done her homework.

focusing too much on the abstract is perhaps a bit too efficient. it can give the impression linear algebra is HARD, when in fact, it's one of the more friendly beasts in the mathematical menagerie. you say linear transformation, i say matrix, erm, i mean potahto.