This book consists of the expanded notes from an upper level linear algebra course given some years ago by the author. Each section, or lecture, covers about a week's worth of material and includes a full set of exercises of interest. It should feel like a very readable series of lectures. The notes cover all the basics of linear algebra but from a mature point of view. The author starts by briefly discussing fields and uses those axioms to define and explain vector spaces. Then he carefully explores the relationship between linear transformations and matrices. Determinants are introduced as volume functions and as a way to determine whether vectors are linearly independent. Also included is a full chapter on bilinear forms and a brief chapter on infinite dimensional spaces.
The book is very well written, with numerous examples and exercises. It includes proofs and techniques that the author has developed over the years to make the material easier to understand and to compute.