[For the first edition (1992) see Zbl 0754.15002.]
In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. In particular, the presentation of tensor products and the umbral calculus are greatly expanded. More over, there are two new chapters: Chapter 15: Positive solutions to linear systems: Convexity and separation; Chapter 17: QR decomposition, singular values, and pseudo inverses.
As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. In almost all chapters, the exercises are rewritten and expanded. The book starts with a chapter on preliminaries, and then it is divided into two parts. Part I covers the basic theory of linear algebra, including vector spaces and linear transformations, quotient spaces, and isomorphism theorems. Also, it contains a good treatment of the theory of modules, in particular over principal ideal domains. It emphasizes on the difference between modules and vector spaces. The cyclic decomposition theorem is first established for finitely generated modules, and then it is adapted for vector spaces, and eventually is used to obtain the rational canonical form. Furthermore, a nice treatment of eigenvalues and eigenvectors, the Jordan canonical form, triangularizability and diagonalizability, real and complex inner product spaces, best approximation, the Riesz representation theorem, and a comprehensive structure theory of normal operators and the spectral theorem are given.
Part II covers selected topics such as: Metric vector spaces (i.e. vector spaces on which a bilinear forms is defined), metric spaces; Hilbert spaces (including basic theory, approximation problem, Fourier expansion, characterization of Hilbert spaces, and the Riesz representationtheorem); tensor products (without too much getting involved in category theory); positive solutions to linear systems; affine geometry; operator factorizations (QR and singular-value decompositions) and the umbral calculus.
Over all, I found the book a very useful one. The only extra thing I wish it had is an index of notations. It is a suitable choice as a graduate text or a reference book.