Advanced Linear AlgebraAuthor: Steven Roman Published by Springer New York ISBN: 978-1-4757-2180-5 DOI: 10.1007/978-1-4757-2178-2, 0 Preliminaries -- 1 Vector Spaces -- 2 Linear Transformations -- 3 The Isomorphism Theorems -- 4 Modules I -- 5 Modules II -- 6 Modules over Principal Ideal Domains -- 7 The Structure of a Linear Operator -- 8 Eigenvalues and Eigenvectors -- 9 Real and Complex Inner Product Spaces -- 10 The Spectral Theorem for Normal Operators -- 11 Metric Vector Spaces -- 12 Metric Spaces -- 13 Hilbert Spaces -- 14 Tensor Products -- 15 Affine Geometry -- 16 The Umbral Calculus -- References -- Index of Notation, 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