by Peter D. Lax
One way of seeing Linear Algebra is: Linear Algebra is finite dimensional daughter of Functional Analysis. In my opinion, this is the spirit of this book. The Fields medalist P. Lax presents Linear Algebra in a modern way, which for those who are familiar with functional analysis, looks the firsts steps to introduce the much more general field of functional analysis. For the mathematicians, this book is a must-have!
Rodrigo Ribeiro 