• Steven Roman
Textbook

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

1. Front Matter
Pages i-xii
2. ### Preliminaries

1. Steven Roman
Pages 1-24
3. ### Basic Linear Algebra

1. Front Matter
Pages 25-25
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Pages 27-43
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Pages 45-62
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Pages 63-81
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Pages 83-95
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Pages 97-106
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Pages 107-119
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Pages 121-133
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Pages 135-156
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Pages 157-174
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Pages 175-202
4. ### Topics

1. Front Matter
Pages 203-203
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Pages 205-237
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Pages 239-261
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Pages 291-314
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Pages 315-328
7. Steven Roman
Pages 329-352
5. Back Matter
Pages 353-366

### Introduction

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.

### Keywords

Eigenvalue Eigenvector algebra linear algebra matrices transformation

#### Authors and affiliations

• Steven Roman
• 1
1. 1.Department of MathematicsCalifornia State University at FullertonFullertonUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-2178-2
• Copyright Information Springer-Verlag New York 1992
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4757-2180-5
• Online ISBN 978-1-4757-2178-2
• Series Print ISSN 0072-5285
• Buy this book on publisher's site
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