# Matrix Groups

• Morton L. Curtis
Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-xii
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 133-144
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Pages 163-183
14. Back Matter
Pages 184-191

### 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 isomorphie. 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 E of quaternions and note that for A E Mn(E) to operate linearlyon Rn we must operate on the right (since we multiply a vector by a scalar n n on the left). So we use row vectors for Rn, c E and write xA , for the row vector obtained by matrix multiplication. We get a complex-valued determinant function on Mn (E) such that det A # 0 guarantees that A has an inverse.

### Keywords

Groups Matrizengruppe algebra clifford algebra field group theory homomorphism lie algebra lie group linear algebra manifold matrices matrix

#### Authors and affiliations

• Morton L. Curtis
• 1
1. 1.Department of MathematicsRice UniversityHoustonUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-0093-9
• Copyright Information Springer-Verlag New York 1979
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-90462-7
• Online ISBN 978-1-4684-0093-9
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675
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