I know there are many questions related to linear algebra, but the textbook I'm using is not that widely used as other books, I guess.

My university uses 'Finite-Dimensional Linear Algebra' by Mark S.Gockenbach. This book is actually quite good but has several defects.

1. When you learn several new concepts, it's useful to derive many useful consequences from them. But this book doesn't mention some of them. They are not critical to the development of linear algebraic structure, but very useful when you have to quickly determine certain properties. I had to spend tons of time discovering(and proving) all those useful gadgets. In some cases I think I actually have done better than the author with the logical framework.

2. All I need is the fundamental mathematical foundation, not application. He puts those applications too often, which is distracting. I can do those applications in other classes such as numerical analysis.

3. (minor) The font is too big! And the book is not strong, I mean the book got torn apart due to its weight, especially the front cover... within a week! Embarassing.

I've searched this book on this forum and Amazon but couldn't find useful information. I'm now studying abstract algebra, analysis, algebraic combinatorics and so on. So I'm thinking buying a more challenging book would be great. But I've only learned the first half of this book (up to Jordan Canonical form: the book didn't do a great job here) so should I just buy another book just as difficult as this book? If so, what would be adequate at my level? I always love challenge so if possible I want to buy a harder one!

Thanks as always.

  • $\begingroup$ Yes, you should change your book. However, I have learned linear algebra in French, so I have no books to recommend (unless you know french!). Those books with references to physics, economics and chemistry are not good if you're willing to go further with working in abstract algebra where you often see vector spaces over arbitrary fields for instance. $\endgroup$ – Patrick Da Silva Sep 13 '14 at 12:09
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    $\begingroup$ Personally I found Linear Algebra by Hoffman and Kunze to be a good theoretic book. That being said, they do leave out some material I think should be in a good linear algebra book, such as some of the category theory (the notion of "natural" in the book is informal at best), proof that every vector space has a basis, and some more development of module theory. But it's certainly good for a first introduction $\endgroup$ – Hayden Sep 13 '14 at 12:13
  • $\begingroup$ @PatrickDaSilva I can't read French :( that's sad. This book does not offer references to other fields. The application is rather about errors and calculations: rather realistic things. It still harms the mathematical structure the author wants to talk about though. I'm not a computer science major!! $\endgroup$ – Taxxi Sep 13 '14 at 12:18
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    $\begingroup$ @Hayden Thanks for the recommendation. I've heard about the book being good quite a lot so I was also thinking about it. $\endgroup$ – Taxxi Sep 13 '14 at 12:20
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    $\begingroup$ @Hayden : Category theory is not necessary in a first course in linear algebra... to really appreciate category theory one should at least know some group theory or ring theory. Otherwise it's just useless super abstract language. What's the use of knowing that the determinant is a natural transformation if you're never gonna compute determinants over different fields simultaneously? $\endgroup$ – Patrick Da Silva Sep 13 '14 at 12:20

Many years ago I learned Linear Algebra from an Italian textbook that I can't, for obvious reasons, suggest to you. When I grew up as a mathematician, I collected many books in linear algebra. One of my favorite textbooks remains that written by Serge Lang. I used it for a course and I was rather satisfied by its quality. It provides the reader with a strong theoretical background, it doesn't focus on "practical" applications, and it covers almost every topic in basic linear algebra.

Another powerful but difficult book is Greub's Linear algebra. It is a very rigorous book under fairly general algebraic assumptions. It does not deal, as far as I remember, with matrix calculus and "undergraduate" topics, but many proofs carry over to infinite-dimensional vector spaces.

  • $\begingroup$ Thanks a lot for your recommendation! I'll have a look at those. I've searched Amazon and there were two books for Lang: Linear Algebra, Introduction to Linear Algebra. I guess the first one is what you're referring to? Also, there were two books for Greub as well but the content doesn't seem to have changed a lot :) $\endgroup$ – Taxxi Sep 13 '14 at 12:56
  • $\begingroup$ I've checked both books and both seem to be wonderful! I can afford Greub's one cheaper than Lang's one. But I'm a little bit scared by the front cover saying 'graduate'. If I can, I'm thinking of grabbing both of them. $\endgroup$ – Taxxi Sep 13 '14 at 13:52

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