# Matrix Groups

• Morton L. Curtis
Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-xiv
2. Morton L. Curtis
Pages 1-22
3. Morton L. Curtis
Pages 23-34
4. Morton L. Curtis
Pages 35-44
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Pages 45-59
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Pages 60-72
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Pages 73-91
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Pages 92-105
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Pages 122-130
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Pages 131-142
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Pages 143-160
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Pages 161-181
14. Morton L. Curtis
Pages 182-200
15. Back Matter
Pages 201-210

### Introduction

These notes were developed from a course taught at Rice Univ- sity in the spring of 1976 and again at the University of Hawaii in the spring of 1977. It is assumed that the students know some linear algebra and a little about differentiation of vector-valued functions. The idea is to introduce students to some of the concepts of Lie group theory-- all done at the concrete level of matrix groups. As much as we could, we motivated developments as a means of deciding when two matrix groups (with different definitions) are isomorphic. In Chapter I "group" is defined and examples are given; ho- morphism and isomorphism are defined. For a field k denotes the algebra of n x n matrices over k We recall that A E Mn(k) has an inverse if and only if det A ~ 0 , and define the general linear group GL(n,k) We construct the skew-field lli of to operate linearly on llin quaternions and note that for A E Mn(lli) we must operate on the right (since we mUltiply a vector by a scalar n on the left). So we use row vectors for R , en, llin and write xA for the row vector obtained by matrix multiplication. We get a ~omplex-valued determinant function on Mn (11) such that det A ~ 0 guarantees that A has an inverse.

### Keywords

Abelian group Algebra Group theory Groups Matrizengruppe Vector space homomorphism

#### Authors and affiliations

• Morton L. Curtis
• 1
1. 1.Department of MathematicsRice University, Weiss School of Natural SciencesHoustonUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-5286-3
• Copyright Information Springer-Verlag New York 1984
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-96074-6
• Online ISBN 978-1-4612-5286-3
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675
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