A Concise Text on Advanced Linear Algebra

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.

Table of Contents

Chapter 1 Vector spaces
Chapter 2 Linear mappings
Chapter 3 Determinants
Chapter 4 Scalar products
Chapter 5 Real quadratic forms and self-adjoint mappings
Chapter 6 Complex quadratic forms and self-adjoint mappings
Chapter 7 Jordan decomposition
Chapter 8 Selected topics
Chapter 9 Excursion: Quantum mechanics in a nutshell