This book is a revised and up-dated fourth edition of a textbook designed for upper division courses in linear algebra. It includes the basic results on vector spaces over fields, determinants, the theory of a single linear tranformation, and inner product spaces. While it does not presuppose an ealier course, many connections between linear algebrta and calculus are worked into the discussion, making it best suited for students who have completed the calulus sequence.A special feature of the book is the inclusion of sections devoted to applications of linear algebra, which can either be part of a course, or used for independent study. The topics covered in these secions are the geometric interpretation of systems of linear equations, the classification of finite symmetry groups in two and three dimensions, the exponential of a matrix and its application to solving systems of first order linear differential equations with constant coefficients, and Hurwitz's theorem on the composition of quadratic forms. This revised fourth edition contains a new section on analytic methods in matrix theory, with applications to Markov chains in probability theory.Proofs of all the main theorems are included, and are presented on an equal footing with methods for solving numerical problems. Worked examples are included in almost every section, to bring out the meaning of the theorems, and to illustrate techniques for solving problems. Many numerical exercises are included, which use all the ideas, and develop computational skills. There are also exercises of a theoretical nature, which provide opportunities for students to discover interesting theings for themselves.