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Advanced Linear Algebra [electronic resource]

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Advanced Linear Algebra [electronic resource]
Original Title Advanced Linear Algebra [electronic resource]
Author Roman, Steven
Publication date

Topics Mathematics, Matrix theory
Publisher New York, NY : Springer New York
Collection folkscanomy_miscellaneous, folkscanomy, additional_collections
Language English
Book Type EBook
Material Type Book
File Type PDF
Downloadable Yes
Support Mobile, Desktop, Tablet
Scan Quality: Best No watermark
PDF Quality: Good
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Advanced Linear AlgebraAuthor: Steven Roman Published by Springer New York ISBN: 978-1-4757-2180-5 DOI: 10.1007/978-1-4757-2178-20 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 NotationThis 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
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