This book is two textbooks in one. The first half is a straightforward (although very theoretical) introduction to linear algebra and matrices at the lower-division undergraduate level, while the second half is a much more in-depth look at the theory of linear algebra at the upper-division undergraduate level. The two courses are not intended to be presented back-to-back, but separated by a course in linear abstract algebra. The present volume is a 2014 Dover reprint of the 1992 Oxford edition, with two new pages of errata and notes.

The first half, although theoretical, is not abstract. Most work is done in **R**^{2} and **R**^{3}, and the book never ventures beyond finite-dimensional spaces. It uses some of the language of abstract algebra, including quotient spaces and a few commutative diagrams. It is proof-oriented, with few worked examples, and the exercises are mostly proofs with a little drill included. The presentation starts in terms of linear mappings, and only after these are established does it begin using matrix language. The second half is a collection of special topics in the deeper theory of linear algebra. This again is oriented toward proofs.

The linear algebra market has shifted since this book was first published, and it’s not clear what course today could use the first half of the book. It omits all applications, and it omits such now-common topics as Gaussian elimination, special factorizations such as LU and QR, Singular Value Decomposition, and any numerical considerations. The second half is still useful, and contains hard-to-find items such as a thorough development of the theory of determinants and a development of rational and Jordan canonical forms that does not require much algebra background.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.