Alternative to Axler's "Linear Algebra Done Right" 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.
 A: 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.
A: *

*[GSM], Dym, Linear Algebra in Action

*[GTM], Roman, Advanced Linear Algebra

*Lax, Linear Algebra and its Applications

*[UTM], Halmos, Finite-Dimensional Vector Spaces

*[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.
A: Out of fairness, I thought I would post this for anyone interested. I recently bought a copy of Axler's 3rd edition. It is substantially better than the 2nd. It includes sections on quotient spaces and dual spaces.
With regard to my prior remarks in the question: the treatment of matrices is much more robust with very explicit notation showing how matrices function as linear transformations.
Initially, I thought the graphics in the formatting were a little extreme, but I have come to appreciate them as the pages are more visually accessible.
Finally the problem sets are enhanced (although I don't know what I will do with 500+ problems) yet many of them are challenging and illustrate applications of the concepts. And their organization at the end of each subsection rather than at the end of each chapter reinforces the material.
A: 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.
A: Try Lang's Linear Algebra, which is both concrete and abstract.
A: 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
A: I'm a fan of the book by Hoffman and Kunze. It's a standard, concise, proof-based text. 
A: S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach)
