Linear Algebra: A Pure Mathematical Approach

Portada
Springer Science & Business Media, 2002 - 250 páginas
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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Contenido

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
148
Problems 6
151
Selected Topics
157
Normed Algebras Quaternions and Cayley Numbers
167

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
176
Problems 7
188
Set Theory
193
Problems A
200
Answers and Solutions to the Problems
203
Bibliography
237
Notation index
237
Definition Index
239
Theorem Index
240
Subject Index
241
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