Linear Algebra: A Pure Mathematical Approach

Front Cover
Springer Science & Business Media, 2002 - 250 pages
0 Reviews
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

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

Other editions - View all

Common terms and phrases

About the author (2002)

Rose's Linear Algebra is a highly sophisticated undergraduate work...This book would be excellent for mathematics majors or for non-majors with access to a second course in which applications were presented. Summing Up: Recommended for lower- and upper-division undergraduates."

--Choice

Bibliographic information