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I have been using Axler's "Linear Algebra Done Right." In fact I have recommended it here often.

I was wondering if there is a text at that level or higher that uses "kernel" rather than "null space"? And that does not go so far out of it's way to avoid matrices.

Thanks.

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Is "kernel" vs "null space" an issue so important that it would veto the choice of a textbook? Why? Just read all occurrences of "null space" as "kernel". – Robert Israel Nov 22 '12 at 22:39
Artin's Algebra is very matrix oriented. On the other hand, it is not just a linear algebra textbook. Also, I agree with @RobertIsrael's comment above. – Rankeya Nov 22 '12 at 22:41
You're absolutely right, and having studied some algebra after "Axler," I now do that. – Andrew Nov 22 '12 at 22:43
If it has "Matrix Theory" in the title you may be better off. If not all material is covered, you can switch back to Axler or several other books. – Will Jagy Nov 22 '12 at 22:43
The entire point of Axler's book is to do things in a particular way. If you would like a book that does it the traditional way (i.e. the "wrong way), there are lots of them. Or do you mean you want a book to use as an alternative to Axler's book, but which is still not done in the traditional way? – Carl Mummert Nov 22 '12 at 23:43
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6 Answers

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There is A Terse Introduction of Linear Algebra, which rapidly overview the subject matter of a typical first course in an elegant way. Particularly, it does prefer the kernels over the null space. A free and legal draft of this book is available at here.

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If Linear Algebra Done Right doesn't work, then try Linear Algebra Done Wrong, by Sergei Treil. This seems to meet both of your requirements.

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Thanks very much. Gee, wish I could give you more upvotes. With regards, Andrew – Andrew Nov 23 '12 at 3:03
up vote 4 down vote

If you want a higher level textbook, allow me to suggest "Advanced Linear Algebra" by Steven Roman.

The text is primarily concerned with abstract vector spaces, but it does treat matrices in detail and uses them when it is natural to do so. It will probably also cover all the linear algebra you need in an undergraduate degree. You can read samples of it here:

http://books.google.com/books?id=FV_s8W58D4UC&hl

http://www.amazon.com/Advanced-Linear-Algebra-Graduate-Mathematics/dp/0387728287

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I loved working through this book. The writing is amazingly clear and motivated. Unlike Axler, his proof of the spectral theorem does not involve matrices at all. He establishes the structure theorem for modules before covering eigenvectors and canonical forms. Roman doesn't "avoid" matrices, but Axler definitely uses more matrices than Roman does. – wj32 Nov 22 '12 at 23:33
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I'm a fan of the book by Hoffman and Kunze. It's a standard, concise, proof-based text.

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Seconded. This is a really great book. – nigelvr Jan 23 at 15:12
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Try Lang's Linear Algebra, which is both concrete and abstract.

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up vote 1 down vote
  1. [GSM], Dym, Linear Algebra in Action
  2. [GTM], Roman, Advanced Linear Algebra
  3. Lax, Linear Algebra and its Applications
  4. [UTM], Halmos, Finite-Dimensional Vector Spaces
  5. [UTX], Curtis, Abstract Linear Algebra

These are both highly reputed and depth with high quality.

For simple computations and basic concepts, it could be obtained by any textbook or even lecture notes adequately.

For theory, it could be much more benefited to carefully select highly deep books.

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