See Henry Ricardo's review of the second edition. In addition to many local corrections and improvements, the new edition has a new chapter on associative algebras. Included in that chapter are theorems characterizing finite-dimensional division algebras over R and over a finite field. The bibliography has also been enlarged with over a hundred new items.
* Vector Spaces * Linear Transformations * The Isomorphism Theorems * Modules I: Basic Properties * Modules II: Free and Noetherian Modules * Modules over a Principal Ideal Domain * The Structure of a Linear Operator * Eigenvalues and Eigenvectors * Real and Complex Inner Product Spaces * Structure Theory for Normal Operators * Metric Vector Spaces: The Theory of Bilinear Forms * Metric Spaces * Hilbert Spaces * Tensor Products * Positive Solutions to Linear Systems: Convexity and Separation * Affine Geometry * Operator Factorizations: QR and Singular Value * The Umbral Calculus * References * Index