This engaging textbook for advanced undergraduate students and beginning graduates covers the core subjects in linear algebra. The author motivates the concepts by drawing clear links to applications and other important areas, such as differential topology and quantum mechanics. The book places particular emphasis on integrating ideas from analysis wherever appropriate. For example, the notion of determinant is shown to appear from calculating the index of a vector field which leads to a self-contained proof of the Fundamental Theorem of Algebra, and the Cayley-Hamilton theorem is established by recognizing the fact that the set of complex matrices of distinct eigenvalues is dense. The material is supplemented by a rich collection of over 350 mostly proof-oriented exercises, suitable for students from a wide variety of backgrounds. Selected solutions are provided at the back of the book, making it suitable for self-study as well as for use as a course text.
Yisong Yang is Professor of Mathematics at the Polytechnic School of Engineering, New York University. His areas of research are partial differential equations and mathematical physics. He is a Fellow of the American Mathematical Society and the author of Solitons in Field Theory and Nonlinear Analysis (2001).
Preface; Notation and convention; 1. Vector spaces; 2. Linear mappings; 3. Determinants; 4. Scalar products; 5. Real quadratic forms and self-adjoint mappings; 6. Complex quadratic forms and self-adjoint mappings; 7. Jordan decomposition; 8. Selected topics; 9. Excursion: quantum mechanics in a nutshell; Solutions to selected problems; Bibliographic notes; References; Index.