Linear Algebra [Paperback]

This popular and successful text was originally written for a one-semester course in linear algebra at the sophomore undergraduate level. Consequently, 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. 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, after the principal axis theorem. To achieve these goals in one semester it is necessary to follow a straight path, but this is compensated by a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show that linear algebra alone is incapable of solving these canonical forms problems. This bookis a compact but mathematically clean introduction to linear algebra with particular emphasis on topics in abstract algebra, the theory of differential equations, and group representation theory.1. Vectors in the Plane and in Space.- 1.1 First Steps.- 1.2 Exercises.- 2. Vector Spaces.- 2.1 Axioms for Vector Spaces.- 2.2 Cartesian (or Euclidean) Spaces.- 2.3 Some Rules for Vector Algebra.- 2.4 Exercises.- 3. Examples of Vector Spaces.- 3.1 Three Basic Examples.- 3.2 Further Examples of Vector Spaces.- 3.3 Exercises.- 4. Subspaces.- 4.1 Basic Properties of Vector Subspaces.- 4.2 Examples of Subspaces.- 4.3 Exercises.- 5. Linear Independence and Dependence.- 5.1 Basic Definitions and Examples.- 5.2 Properties of Independent and Dependent Sets.- 5.3 Exercises.- 6. Finite-Dimensional Vector Spaces and Bases.- 6.1 Finite-Dimensional Vector Spaces.- 6.2 Properties of Bases.- 6.3 Using Bases.- 6.4 Exercises.- 7. The Elements of Vector SpacelÃ+