There are two introductionary books on linear algebra I would recommend. "Linear Algebra Done Right" by Axler gives you a _very_ nice and rigorous way to the foundations (this text requires a small bit of mathematical maturity, so be aware).
After reading Axler's book you should go and grab a copy of Strang's "Introduction to Linear Algebra" to view the applied side of linear algebra (this is the book mentioned in the article).
Strang's book is a terrible choice for first textbook of linear algebra (for the reasons btilly laid out), but it presents very imortant view on linear algebra, provided that you have mastered the foundations enough to compensate for the lack of rigour.
Really why by a book when you can just grab a PDF. I used Linear Algebra Done Wrong for my class this semester, and found it a pretty good book for being free. check it out: www.math.brown.edu/~treil/papers/LADW/LADW.pdf
Skimming the preface, this book appears to be something like the approach btilly describes. Thanks for the link. Intend to download to my iPod Touch to compete for my rare moments of free time with Elements of Statistical Learning, and the other free Math and CS books I'm sure will be on there soon.
After reading Axler's book you should go and grab a copy of Strang's "Introduction to Linear Algebra" to view the applied side of linear algebra (this is the book mentioned in the article).
Strang's book is a terrible choice for first textbook of linear algebra (for the reasons btilly laid out), but it presents very imortant view on linear algebra, provided that you have mastered the foundations enough to compensate for the lack of rigour.