This book serves as an advanced companion to an initial linear algebra text with a computational focus. It encompasses conventional topics such as vector spaces, linear transformations, and bases, and it delves into non-standard subjects typically encountered in abstract algebra courses such as structure theorems for linear transformations, tensor products of vector spaces, and exterior algebras. In addition, it features chapters devoted to bilinear and sesquilinear forms, inner product spaces, and the spectral theorem.
However, the primary emphasis lies in demonstrating how results from an introductory linear algebra course naturally emerge when examined in a broader context. For instance, the seemingly unintuitive definition of the determinant becomes clear when explored in the context of exterior algebras.
Upon completing this book and working through the exercises, readers will possess a solid foundation in advanced linear algebra. They will also gain insight into the concepts from their initial linear algebra course, equipping them for further study in any field requiring a deep understanding of linear algebra.