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Advanced Linear Algebra

  • Textbook
  • © 1992

Overview

Authors:
  1. Steven Roman

    1. Department of Mathematics, California State University at Fullerton, Fullerton, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 135)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xii
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  2. Preliminaries

    1. Preliminaries

      • Steven Roman
      Pages 1-24

  3. Basic Linear Algebra

    1. Front Matter

      Pages 25-25
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    2. Vector Spaces

      • Steven Roman
      Pages 27-43

    3. Linear Transformations

      • Steven Roman
      Pages 45-62

    4. The Isomorphism Theorems

      • Steven Roman
      Pages 63-81

    5. Modules I

      • Steven Roman
      Pages 83-95

    6. Modules II

      • Steven Roman
      Pages 97-106

    7. Modules over Principal Ideal Domains

      • Steven Roman
      Pages 107-119

    8. The Structure of a Linear Operator

      • Steven Roman
      Pages 121-133

    9. Eigenvalues and Eigenvectors

      • Steven Roman
      Pages 135-156

    10. Real and Complex Inner Product Spaces

      • Steven Roman
      Pages 157-174

    11. The Spectral Theorem for Normal Operators

      • Steven Roman
      Pages 175-202

  4. Topics

    1. Front Matter

      Pages 203-203
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    2. Metric Vector Spaces

      • Steven Roman
      Pages 205-237

    3. Metric Spaces

      • Steven Roman
      Pages 239-261

    4. Hilbert Spaces

      • Steven Roman
      Pages 263-290

    5. Tensor Products

      • Steven Roman
      Pages 291-314

    6. Affine Geometry

      • Steven Roman
      Pages 315-328

    7. The Umbral Calculus

      • Steven Roman
      Pages 329-352
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, California State University at Fullerton, Fullerton, USA

    Steven Roman

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