The resources for linear algebra abound. Listed below are the references that are or have sat on my desk shelf. Interlibrary loan is a wonderful resource.
Strang, Gilbert, Introduction to Linear Algebra Fourth Edition, Wellesley Cambridge Press, 2009.
This is the main textbook for the first portion of the course. It takes a very computational and smooth approach. Getting students comfortable with computations is important, for a couple of reasons. First, they are forced to use the objects under discussion even if the student's interest is theoretical. Second, they are able to see tangible results to their work. Linear algebra is often the first encounter with proof writing, and can be a little intimidating. I've found that computations enhance a student's repertoire of examples, which is helpful when trying to prove a statement.
Strang, Gilbert, Linear Algebra and Its Applications, Fourth Edition, Brooks Cole, 2006.
Unfortunately the cost of this book has become astronomically high. If you can find used copies, it is well worth the effort. He covers some advanced topics more throughly than what is found in his introductory book. Topics I tend to cover are:
Fuhrmann, Paul, A Polynomial Approach to Linear Algebra, Second Edition, Universitext, Springer, 2012.
I thought this was a very different approach, when I first encountered the book. Teaching from the book served to confirm this. I used this during a Winter trimester after a student expressed interest in learning more about polynomials. The student had a background in writing proofs and using mathematical structures through a summer program at MIT. We worked through aspects of this book, and neither one of us regretted it. We both agreed that it was a challenging read. Fuhrmann begins with the notion of rings and modules and uses $\mathbb{F}[z]$-modules extensively. The book is geared toward those with an interest in functional analysis and interpolation problems. He ends the book with a discussion regarding Rational Hardy Spaces. The style is terse, requires maturity, and an interest in the subject matter.
Treil, Sergei, Linear Algebra Done Wrong,Linear Algebra Done Wrong
I find this useful for a couple of reasons. He writes in a more mathematically sophisticated way. It is a good contrast to what is done in Strang's book. My students have all said they appreciate the style after an introduction through Strang. Topics not covered in Strang's book are:
Other Books: Here are some other books that I have found useful:
Papers: I've used papers in some form or another to get students familiar with reading technical matheamtics and working with abstraction. Here are some papers that my students have read. Expository papers are available, but one does have to put the effort into finding them. It's easy to walk off the deep-end and assign something that a student doesn't have the background to read.
Axler, Sheldon, Down with Determinatns
His point-of-view is that determinants are esoteric objects which are difficult to understand. There is little reasoning behind the why or the way they are computed. So, students go through these computations mindlessly and the thread of what they should be learning gets interrupted. I'm not sure that I disagree with this sentiment, but proving the statements without determinants takes a comfort level with abstraction that students are often not ready for, in a first course. This is the first paper I have students read as an introduction to abstraction, at this level, and getting them familiar with reading more technical literature.
Austin, David, We Recommend a Singular Value Decomposition, Featured Column
It can be found here. I've found this to be an excellent supplement to the discussion given in Strang.