# Can you learn linear algebra with an abstract algebra book?

I am trying to learn both linear and abstract algebra. I already took a basic Matrix Theory course using Anton's linear algebra book (not very rigorous) and I want to self study the rest of the subject. We got to basic eigen values and eigen spaces. I want to relearn linear algebra rigorously, and I am considering Hoffman and Kunze. I have a good background in rigor, and am reading Rudin's Principles of Mathematical Analysis, so I think I would be able to handle Hoffman and Kunze.

However, I also want to learn abstract algebra. My question is, do abstract algebra books, say like Dummit and Foote cover everything in a linear algebra book like Hoffman and Kunze? Or would I have to read them simultaneously?

• the sequence is 1) Group theory, 2)Ring and Field theory, 3) Vector Spaces theory, there, you would be doing linear algebra. – janmarqz Jun 27 '15 at 18:30

No there is not enough Linear algebra in an Abstract algebra book. But I have a really good suggestion for you. But that will eat up at least two months of your time but will get you crystal clear understanding of Algebras.

Give at least 20 days to the book (if you have never been introduced to this subject before, then may be at least 30 days) Volume 1: Groups.

Then give 20 - 30 days atleast to Volume 2: Rings

Then Finally you can start with Linear algebra- (which is nothing but study of modules over fields). Read at whatever pace you like, as it is a gem of a book Volume 3: Modules

Then if you feel like, you can finish up the series with Volume 4:Field Theory.

I hope it helps.

You should try Hoffman and Kunze and see how it goes.

If you want to start with an abstract algebra book, try Herstein's Topics in Algebra, which contains linear algebra at the end.

• – lhf Jun 27 '15 at 19:07

I would say no. The real issue is that "Linear Algebra" is a field that really is a mashup of different ideas: Solving systems of linear equations, matrix theory, the theory of vector spaces, the study of linear transformations...

Most abstract algebra books will cover some details about what a vector space is from an algebraic point of view, point out how matrix multiplication (by a vector) provides homomorphisms between vector spaces, and multiplication between invertible matrices provides a source of examples of groups.

With very few exceptions, they won't go through any of the details to actually do much linear algebra (they will probably completely ignore the whole "solving systems of linear equations" point of view; Dummit and Foote for example completely leaves out Gaussian elimination, echelon forms, and methods for finding a basis for a subspace, all of which are super fundamental. They simply expect you to know it already).