Linear Algebra is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorems for linear maps, including eigenvectors and eigenvalues, quadric and hermitian forms, diagonalization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants, and linear maps. However, the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.
I Vector Spaces.- II Matrices.- III Linear Mappings.- IV Linear Maps and Matrices.- V Scalar Products and Orthogonality.- VI Determinants.- VII Symmetric, Hermitian, and Unitary Operators.- VIII Eigenvectors and Eigenvalues.- IX Polynomials and Matrices.- X Triangulation of Matrices and Linear Maps.- XI Polynomials and Primary Decomposition.- XII Convex Sets.- Appendix I Complex Numbers.- Appendix II Iwasawa Decomposition and Others.
