# Linear Algebra: A Pure Mathematical Approach

Springer Science & Business Media, 2002 - 250 pages
Linear algebra is one of the most important branches of mathematics - important because of its many applications to other areas of mathematics, and important because it contains a wealth of ideas and results which are basic to pure mathematics. This book gives an introduction to linear algebra, and develops and proves its fundamental properties and theorems taking a pure mathematical approach - linear algebra contains some fine pure mathematics. Main topics: - vector spaces and algebras, dimension, linear maps, direct sums, and (briefly) exact sequences - matrices and their connections with linear maps, determinants (properties proved using some elementary group theory), and linear equations - Cayley-Hamilton and Jordan theorems leading to the spectrum of a linear map - this provides a geometric-type description of these maps - Hermitian and inner product spaces introducing some metric properties (distance, perpendicularity etc.) into the theory, also unitary and orthogonal maps and matrices - applications to finite fields, mathematical coding theory, finite matrix groups, the geometry of quadratic forms, quaternions and Cayley numbers, and some basic group representation theory A large number of examples, exercises and problems are provided. Answers and/or sketch solutions to all of the problems are given in an appendix. Some of these are theoretical and some numerical, both types are important. No particular computer algebra package is discussed but a number of the exercises are intended to be solved using one of these packages chosen by the reader.The approach is pure-mathematical, and the intended readership is undergraduate mathematicians, also anyone who requires a more than basic understanding of the subject. This book will be most useful for a "second course" in linear algebra, that is for students that have seen some elementary matrix algebra. But as all terms are defined from scratch, the book can be used for a "first course" for more advanced students.

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### Contents

 Algebraic Preamble 1 Permutation Groups 6 Problems 1 9 Vector Spaces and Linear Maps 13 Bases and Dimension 17 Linear Maps 22 Direct Sums 28 Addendum Modules 32
 Problems 5 123 Hermitian and Inner Product Spaces 125 Hermitian and Inner Products and Norms 126 Unitary and Selfadjoint Maps 139 Orthogonal and Symmetric Maps 149 Problems 6 153 Selected Topics 159 Normed Algebras Quaternions and Cayley Numbers 169

 Problems 2 33 Matrices Determinants and Linear Equations 39 Determinants 51 Systems of Linear Equations 62 Problems 3 67 CayleyHamilton Theorem and Jordan Form 75 The CayleyHamilton and Spectral Theorems 81 Jordan Form 92 Problems 4 103 Interlude on Finite Fields 109 Applications 116
 Introduction to the Representation of Finite Groups 178 Problems 7 190 Set Theory 195 Problems A 202 Answers and Solutions to the Problems 205 Bibliography 239 Notation index 241 Definition Index 243 Theorem Index 244 Subject Index 245 Copyright