Learning Algebra: Questions and Advice

[brainy kevin]

I have a couple of questions regarding learning algebra, specifically abstract and linear algebra. I already know somewhat about both.

Which one should I really concentrate on learning first? I want to truly understand both of them. (I'm interested in mathematics for mathematics's sake)

Can you recommend any good introductory textbooks on both of them?

I've gotten through the first couple of chapters in my linear algebra book, but what comes after eigenvalues and eigenvectors? For some odd reason, this seems to be the stopping point for most introductory linear algebra books.

After I get down the concepts of linear algebra, I want to re-learn it, only in a less intuitive sense, but in a more abstract algebra-ish way. I would prefer them to be defined using algebraic structures instead of with applications. However, I only want to learn it like this after I understand it through an intuitive sense.

Thank you for your help!

 

[jbunniii]

I have a couple of questions regarding learning algebra, specifically abstract and linear algebra. I already know somewhat about both.

Which one should I really concentrate on learning first? I want to truly understand both of them. (I'm interested in mathematics for mathematics's sake)

Can you recommend any good introductory textbooks on both of them?

I've gotten through the first couple of chapters in my linear algebra book, but what comes after eigenvalues and eigenvectors? For some odd reason, this seems to be the stopping point for most introductory linear algebra books.

In principle you could learn either one first. In practice it's useful to know some linear algebra before learning abstract algebra in part because it provides some good concrete examples of groups and in part because you need some basic theorems about vector spaces in order to do field/Galois theory. (You don't need eigenvectors for that, though.)

What textbooks are "introductory" depends on your background. It sounds like you have some exposure to basic "matrix-oriented" linear algebra; if so, then a good starting point with a more theoretical orientation is Axler's "Linear Algebra Done Right." It's a very clear, readable introduction and you should probably start here unless you're already at that level.

Another book with more substantial coverage is Hoffman and Kunze's "Linear Algebra," but although I've tried, I never did warm to the style of that book. Roman's "Advanced Linear Algebra" is a good higher-level treatment of abstract linear algebra which you could (with a bit of a stretch) jump to straight after Axler. Another direction that may be of interest, and an excellent book in its own right, is Horn and Johnson's "Matrix Analysis."

For abstract algebra, it depends on your overall mathematical maturity. I had a bit of a nontraditional background and was already pretty comfortable with abstract, pure mathematics (mostly on the analysis side) before learning algebra, and as a result I found a lot of the introductory books really, really boring. Given that background, I found Herstein's "Topics in Algebra" and Dummit and Foote's "Abstract Algebra" to be a good starting point.

Herstein's book is beautifully written and assumes just enough mathematical maturity that you don't feel like you're being talked down to when you read it. (And its "starred" exercises are really challenging!) It has a very nice treatment of group theory but is a bit thin on rings and fields.

On the other hand, Dummit and Foote may well have all the algebra you'll ever need or want to know, depending on how far you want to go. It's an unusual book in that it doesn't assume any algebra background and yet it contains enough material to span both undergraduate and graduate-level algebra as taught at most universities. Depending on how far you go, this book may well have all the algebra you'll ever need or want.

Another nice book at a similar level to Dummit and Foote is Rotman's "Advanced Modern Algebra." I find its style to be more readable (it "flows" more smoothly) than Dummit and Foote, but that's really a matter of taste.

 

[HallsofIvy]

I agree with pretty much everything jbunniii said. I used Hoffman and Kunze as a text when I was an undergraduate and liked it (maybe it helped that Kunze was the teacher!). I would also recommend Halmos' "Finite Dimensional Vector Spaces".

But I would certainly recommend studying Linear Algebra before Abstract Algebra. It has enough grounding in concrete problems together with abstract thinking to for a good transition from "Calculus- like" courses to Abstract Algebra