The text Linear Algebra is free for downloading, It covers the material of a first undergraduate Linear Algebra course. You can use it either as a main text, as a supplement to another text, or for independent study.
When I started teaching the subject I found three kinds of texts. There were applied mathematics books that avoid proofs and covered the linear algebra only as needed for their applications. There were advanced books that assumed students could understand their elegant proofs and also understand how to answer the homework questions having seen only one or two examples. And, there were books that spent a good part of the semester doing elementary things such as multiplying matrices and computing determinants, only to suddenly change level to working with definitions and proofs.
Each of these three types was a problem in my classroom. The applications were interesting but I wanted to focus on the linear algebra. The advanced books were beautiful but my students were not ready for them. And, the level-switching books resulted in a great deal of grief.
I took a level-switching book as an undergraduate, so I understood the struggle my students had with this. At the start of the semester they thought that these were like calculus books, where material labelled `proof' should be skipped in favor of the computational examples. Then, when the level switched, no amount of discussion on my part could convince students to switch with it, and the semester ended unhappily.
That is, while I wish I could say that my students now perform at the level of the advanced books, I cannot. However, as a teacher I can work steadily to bring them up to it over the course of our undergraduate program. This means stepping back from focussing on rote computations in favor of focussing on an understanding of the mathematics. It means proving things and having students understand, e.g., that matrix multiplication is the application of a linear function. But it means also avoiding an approach that is too advanced for the students: the presentation must emphasize motivation, must have many illustrative examples, and must include exercises with many of the medium-difficult questions that are a challenge to a learner without being overwhelming. And, it means communicating to my students that the change of focus is what we are up to, right from the start.
I know of only two reviews. One is at a site for free texts. The other is a blog, I believe. (Also, someone posted a link to this page on the math sub-reddit, so there are a number of comments.)
The files are current as of 2008-Jul-24.
For a quick look, I suggest the second chapter. The first chapter is necessarily computational but the second chapter shows more clearly what the book works on: bridging between lower-division mathematics with its reliance on explicitly-given algorithms, and upper division college mathematics with its emphasis on concepts and proof.
The text has been class-tested here and elsewhere and I'm delighted to have it on student's desks, doing what it is for. Running off double-sided copies and comb-binding them costs students around $20.00 (we include the Preface, Table of Contents, Appendix, and Index to make about 450 pages. With the Answers to Exercises at about 215 pages the total is about $25.00.
License.
This text is Free.
Use it under either the
GNU Free Documentation License
or the 
Creative Commons Attribution-ShareAlike 2.5 License, at your discretion.
To bookstores: in particular, instructors have permission
to make copies of this material, either electronic or paper,
and sell those copies to students.
Many schools use this text in this way.
If you have further questions, please feel free to contact me.
I'd be glad for any comments. My email is jim at joshua.smcvt.edu.
In addition to the book, the answers to the exercises, and the source, you can get some materials that were contributed by others. If you in turn have something that you are able to share please write me. I have gotten comments ranging from a suggestion about a better way to prove a result or make an example, to an entire Topic. In particular, I would welcome used exams or problem sets, especially if you can contribute the TeX or LaTeX source, so that a user could cut or paste. (Some instructors have expressed reservations about using a text where the answers to the exercises are downloadable, so more exercises would be helpful.) Naturally, all contributions will be gratefully acknowledged, or anonymous if you like.
If you find the book useful, drop me a note in the mail.
In particular, if you use it in a class, a letter on letterhead
to the address below would be great.
Jim Hefferon
Mathematics, Saint Michael's College
Winooski Park
Colchester, VT USA, 05439
Some people don't believe me when I tell them that I'm working in here.
This site Joshua is located in the Mathematics Department of Saint Michael's College in Colchester, Vermont USA.
Open Source software is a great idea. This project would not have gotten done without it.
(Credit for the logo to Matt Ericson.)