# Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra

Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.

But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.

This brings me to my question (provided there are any)

Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?

Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help!

• Hoffman's linear algebra works fine, I think. Mar 20, 2015 at 17:57
• "Linear Algebra Done Right" by Sheldon Axler is a great text. Mar 20, 2015 at 17:57
• "Algebra" by Hungerford has a quality chapter on Linear Algebra which follows the Module chapter. Depending on how much motivation and "teaching" you are looking for (it's fairly succinct), that might be a good option. Mar 20, 2015 at 21:11
• What is "the perspective of general algebra"? Jun 6, 2018 at 16:05

## 3 Answers

Linear Algebra by Hoffman and Kunze is usually seen as a harder book for those with no background in abstract algebra. You'll probably feel comfortable with it. Definitely give it a try.

Topics in Algebra by Herstein contains this material as well, perhaps in a more condensed form.

• I was reading it without any knowledge of linear algebra or abstract algebra whatsoever. I stopped a few times because of some circumstances, but overall, it can be understood with hard work and perseverance. It's the best book there is. Mar 20, 2015 at 18:04

I've been enjoying some more abstract linear algebra texts lately, both freely available online.

There's Linear Algebra via Exterior Products, by Sergei Winitzki, available here.

In addition, there's Linear Algebra Done Right by Sheldon Axler. A google search pulls up a pdf on the first page, but it's not from an .edu. Since I'm not sure if it's legitimate, and I'll refrain from spreading the link.

They may not be exactly what you're looking for, but they're both a fresh perspective on the material (and you can freely and quickly look at either to see if they're suitable). Neither focuses very heavily on matrices, but instead on linear transformations. Winitzki introduces dual vector spaces and tensor products within a chapter or two, for example.

• The official web site for Linear Algebra Done Right is linear.axler.net. It contains sample chapters.
– lhf
Mar 20, 2015 at 19:22

Advanced Linear Algebra by Roman certainly develops the key ideas of linear algebra in an abstract setting. The preliminaries section contains a quick introduction to some concepts and definitions from abstract algebra, but the book is more appropriate for someone, like yourself, who has already studied some abstract algebra. I'd also recommend having an introductory linear algebra book on hand for additional examples.