Synopses & Reviews
This popular and successful text was originally written for a one-semester course in linear algebra at the sophomore undergraduate level. Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces. The goal of the first two editions was the principal axis theorem for real symmetric linear transformation. The principal axis theorem becomes the first of two goals for this new edition, which follows a straight path to its solution. A wide selection of examples of vector spaces and linear transformation is presented to serve as a testing ground for the theory. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, along with applications to differential systems. To achieve the principal axis theorem in one semester a straight path to these two goals is followed. As compensation, there is a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show students that linear algebra alone is not capable of solving these canonical forms problems. The book continues to offer a compact, but mathematically clean introduction to linear algebra with particular emphasis on topics that are used in abstract algebra, the theory of differential equations, and group representation theory.
Table of Contents
1: Vectors in the Plane and Space. 2: Vector Spaces. 3: Examples of Vector Spaces. 4: Subspaces. 5: Linear Independence and Dependence. 6: Finite Dimensional Vector Spaces and Bases. 7: The Elements of Vector Spaces: A Summing Up. 8: Linear Transformations. 9: Linear Transformations: Examples and Applications. 10: Linear Transformations and Matrices. 11: Representing Linear Transformations by Matrices. 12: More on Representing Linear Transformations by Matrices. 13: Systems of Linear Equations. 14: The Elements of Eigenvalue and Eigenvector Theory. 15: Inner Product Spaces. 16: The Spectral Theorem and Quadratic Forms. 17: Jordan Canonical Form. 18: Application to Differential Equations. 19: The Similarity Problem. Appendix A: Multilinear Algebra and Determinants. B: Complex Numbers.