Axler's book is fantastic. But sadly Springer altered the typesetting on the 3rd edition. A really classic and clear LaTeX layout got turned into something much less clear. This freaked me out. Look inside and compare:

* Second edition: https://www.amazon.com/dp/0387982582

* Third edition: https://www.amazon.com/dp/3319307657

I wonder whether widespread adoption of his book pushed editors to make it look flashier and watered down. The contents are the same though.

I always have to mention my all-time favorite introductory book on this subject:

Liner Algebra Done Right (http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...)

Its breakthrough is its focus on the basic algebraic properties of vector spaces and linear maps between them. It de-emphasizes matrix computations and especially determinants (they are covered, but only insomuch as they are necessary).

In my experience, the result of a typical linear algebra course is most students don't fully understand the determinant and more importantly they don't understand the proofs of major theorems which involve long manipulations of the determinant. They also don't understand the more algebraic side of the subject because they aren't given a chance to--it's not covered in much detail. The result is they don't understand the subject overall much at all.

This book is based on the observation that the abstract algebra involved in linear algebra is actually remarkably easy, much more so than arcane determinant manipulations.

This is really basic. Noone(who uses the subject) should need a cheat sheet for this... Also there are really good books out there and linalg is a must nowdays. As for textbook: http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat... As for reference: http://www.amazon.com/Matrix-Analysis-Roger-Horn/dp/05213863...

This is my favorite introductory textbook on linear algebra which goes to great lengths to avoid matrix and determinant computations:

http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...