Foundations of Linear Algebra

Jonathan S. Golan (auth.)0792336143, 9780792336143

This book is an extensively revised version of my textbook “¥esodot HaAlgebra HaLiniarit” (The Foundations of Linear Algebra) used at many universities in Israel. It is designed for a comprehensive one-year course in linear algebra (112 lecture hours) for mathematics majors. Therefore, I assume that the student already has a certain amount of mathematical background – including set theory, mathematical induction, basic analytic geometry, and elementary calculus – as wellas a modicum of mathematical sophistication. My intention is to provide not only a solid basis in the abstract theory of linear algebra, but also to provide examples of the application of this theory to other branches ofmathematics and computer science. Thus, for example, the introduction of finite fields is dictated by the needs of students studying algebraic coding theory as an immediate followup to their linear algebra studies. Many of the students studying linear algebra either are familiar with the care and feeding of computers before they begin their studies or are simultaneously en­ rolled in an introductory computer science course. Therefore, consideration of the more computational aspects of linear algebra – such as the solution of systems of linear equations and the calculation of eigenvalues – is delayed until all students are assumed able to write computer programs for this purpose. Beginning with Chap­ ter VII, there is an implicit assumption that the student has access to a personal computer and knows how to use it.

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Table of contents :
Front Matter….Pages i-viii
Basic Notation and Terminology….Pages 1-2
Fields and Integral Domains….Pages 3-11
Vector Spaces….Pages 12-23
Linear Independence and Dimension….Pages 24-40
Linear Transformations….Pages 41-54
Endomorphism Rings of Vector Spaces….Pages 55-64
Representation of Linear Transformations by Matrices….Pages 65-75
Rings of Square Matrices….Pages 76-92
Systems of Linear Equations….Pages 93-110
Determinants….Pages 111-126
Eigenvectors and Eigenvalues….Pages 127-143
The Jordan Canonical Form….Pages 144-153
The Dual Space….Pages 154-160
Inner Product Spaces….Pages 161-182
Endomorphisms of Inner Product Spaces….Pages 183-197
The Moore-Penrose Pseudoinverse….Pages 198-203
Bilinear Transformations and Forms….Pages 204-218
Algebras Over A Field….Pages 219-227
Back Matter….Pages 228-237