Page 9 - Advanced Linear Algebra

Preface to the First Edition

This book is a thorough introduction to linear algebra, for the graduate or
advanced undergraduate student. Prerequisites are limited to a knowledge of the
basic properties of matrices and determinants. However, since we cover the
basics of vector spaces and linear transformations rather rapidly, a prior course
(
in linear algebra even at the sophomore level), along with a certain measure of
“mathematical maturity,” is highly desirable.
Chapter 0 contains a summary of certain topics in modern algebra that are
required for the sequel. This chapter should be skimmed quickly and then used
primarily as a reference. Chapters 1–3 contain a discussion of the basic
properties of vector spaces and linear transformations.

Chapter 4 is devoted to a discussion of modules, emphasizing a comparison
between the properties of modules and those of vector spaces. Chapter 5
provides more on modules. The main goals of this chapter are to prove that any
two bases of a free module have the same cardinality and to introduce
Noetherian modules. However, the instructor may simply skim over this
chapter, omitting all proofs. Chapter 6 is devoted to the theory of modules over
a principal ideal domain, establishing the cyclic decomposition theorem for
finitely generated modules. This theorem is the key to the structure theorems for
finite-dimensional linear operators, discussed in Chapters 7 and 8.

Chapter 9 is devoted to real and complex inner product spaces. The emphasis
here is on the finite-dimensional case, in order to arrive as quickly as possible at
the finite-dimensional spectral theorem for normal operators, in Chapter 10.
However, we have endeavored to state as many results as is convenient for
vector spaces of arbitrary dimension.

The second part of the book consists of a collection of independent topics, with
the one exception that Chapter 13 requires Chapter 12. Chapter 11 is on metric
vector spaces, where we describe the structure of symplectic and orthogonal
geometries over various base fields. Chapter 12 contains enough material on
metric spaces to allow a unified treatment of topological issues for the basic