# Groups, Matrices, and Vector Spaces

## A Group Theoretic Approach to Linear Algebra

Textbook

1. Front Matter
Pages i-xvii
2. James B. Carrell
Pages 1-9
3. James B. Carrell
Pages 11-55
4. James B. Carrell
Pages 57-83
5. James B. Carrell
Pages 85-111
6. James B. Carrell
Pages 113-134
7. James B. Carrell
Pages 135-196
8. James B. Carrell
Pages 197-237
9. James B. Carrell
Pages 239-295
10. James B. Carrell
Pages 297-317
11. James B. Carrell
Pages 319-335
12. James B. Carrell
Pages 337-382
13. James B. Carrell
Pages 383-401
14. Back Matter
Pages 403-410

### Introduction

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group.

Applications involving symm

etry groups, determinants, linear coding theory and cryptography are interwoven throughout. Each section ends with ample practice problems assisting the reader to better understand the material.  Some of the applications are illustrated in the chapter appendices. The author's unique melding of topics evolved from a two semester course that he taught at the University of British Columbia consisting of an undergraduate honors course on abstract linear algebra and a similar course on the theory of groups. The combined content from both makes this rare text ideal for a year-long course, covering more material than most linear algebra texts. It is also optimal for independent study and as a supplementary text for various professional applications. Advanced undergraduate or graduate students in mathematics, physics, computer science and engineering will find this book both useful and enjoyable.

### Keywords

Determinants Eigentheory Linear Coding Theory Matrix Groups Prime Fields Vector Spaces

#### Authors and affiliations

1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

James B. Carrell is Professor Emeritus of mathematics at  the University of British Columbia. His research areas include algebraic transformation groups, algebraic geometry, and Lie theory.

### Bibliographic information

• Book Title Groups, Matrices, and Vector Spaces
• Book Subtitle A Group Theoretic Approach to Linear Algebra
• Authors James B. Carrell
• DOI https://doi.org/10.1007/978-0-387-79428-0
• Publisher Name Springer, New York, NY
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-0-387-79427-3
• Softcover ISBN 978-1-4939-7910-3
• eBook ISBN 978-0-387-79428-0
• Edition Number 1
• Number of Pages XVII, 410
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“This is an introductory text on linear algebra and group theory from a geometric viewpoint. The topics, largely standard, are presented in brief, well-organized one- and two-page subsections written in clear, if rather pedestrian, language, with detailed examples.” (R. J. Bumcrot, Mathematical Reviews, February, 2018)

“It is particularly applicable for anyone who is familiar with vector spaces and wants to learn about groups – and also for anyone who is familiar with groups and wants to learn about vector spaces. This book is well readable and therefore suitable for self-studying. Each chapter begins with a concise and informative summary of its content, guiding the reader to choose the chapters with most interest to him/her.” (Jorma K. Merikoski, zbMATH 1380.15001, 2018)