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E-raamat: Matrix Groups for Undergraduates

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Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups.

Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots.

This second edition includes two new chapters that allow for an easier transition to the general theory of Lie groups.

Arvustused

This book offers a very nice introduction to the theory of matrix groups and their Lie algebras. The background is kept to a minimum, only basics of calculus, linear algebra and group theory are assumed, while background on topology (of subsets of Euclidean space) is developed in the text. While the text gives complete and exact proofs, it is easy to read, appeals to intuition, and contains many pictures and helpful exercises." - A. Cap, Monatshefte für Mathematik"[ T]he second edition is an expanded and improved version of the original. It can be strongly recommended for an undergraduate course in Lie groups, or as complementary reading for a course in group theory. Prerequisites are basic: knowledge of algebra, geometry, and analysis at an undergraduate level. Hence the book is suitable for a wide audience of readers who are meeting applications of group theory in other areas of mathematics and physics, or even further afield." - Alla S. Detinko, Mathematical Reviews"The author gives an inspiring presentation of the topics presented in this book." - Erich W. Ellers, Zentralblatt Math

Why study matrix groups? 1(4)
Chapter 1 Matrices 5(18)
1 Rigid motions of the sphere: a motivating example
5(2)
2 Fields and skew-fields
7(1)
3 The quaternions
8(3)
4 Matrix operations
11(4)
5 Matrices as linear transformations
15(2)
6 The general linear groups
17(1)
7 Change of basis via conjugation
18(2)
8 Exercises
20(3)
Chapter 2 All matrix groups are real matrix groups 23(10)
1 Complex matrices as real matrices
24(4)
2 Quaternionic matrices as complex matrices
28(2)
3 Restricting to the general linear groups
30(1)
4 Exercises
31(2)
Chapter 3 The orthogonal groups 33(20)
1 The standard inner product on 1K
33(3)
2 Several characterizations of the orthogonal groups
36(3)
3 The special orthogonal groups
39(1)
4 Low dimensional orthogonal groups
40(1)
5 Orthogonal matrices and isometries
41(2)
6 The isometry group of Euclidean space
43(2)
7 Symmetry groups
45(3)
8 Exercises
48(5)
Chapter 4 The topology of matrix groups 53(16)
1 Open and closed sets and limit points
54(5)
2 Continuity
59(2)
3 Path-connected sets
61(1)
4 Compact sets
62(2)
5 Definition and examples of matrix groups
64(2)
6 Exercises
66(3)
Chapter 5 Lie algebras 69(12)
1 The Lie algebra is a subspace
70(2)
2 Some examples of Lie algebras
72(3)
3 Lie algebra vectors as vector fields
75(2)
4 The Lie algebras of the orthogonal groups
77(2)
5 Exercises
79(2)
Chapter 6 Matrix exponentiation 81(14)
1 Series in K
81(3)
2 Series in MM (K)
84(2)
3 The best path in a matrix group
86(2)
4 Properties of the exponential map
88(4)
5 Exercises
92(3)
Chapter 7 Matrix groups are manifolds 95(22)
1 Analysis background
96(4)
2 Proof of part (1) of Theorem 7.1
100(2)
3 Proof of part (2) of Theorem 7.1
102(3)
4 Manifolds
105(3)
5 More about manifolds
108(4)
6 Exercises
112(5)
Chapter 8 The Lie bracket 117(22)
1 The Lie bracket
117(4)
2 The adjoint representation
121(3)
3 Example: the adjoint representation for SO(3)
124(1)
4 The adjoint representation for compact matrix groups
125(3)
5 Global conclusions
128(2)
6 The double cover Sp(1) SO(3)
130(3)
7 Other double covers
133(2)
8 Exercises
135(4)
Chapter 9 Maximal tori 139(24)
1 Several characterizations of a torus
140(4)
2 The standard maximal torus and center of SO(n), SU (n), U(n) and Sp(n)
144(5)
3 Conjugates of a maximal torus
149(7)
4 The Lie algebra of a maximal torus
156(1)
5 The shape of SO(3)
157(2)
6 The rank of a compact matrix group
159(2)
7 Exercises
161(2)
Chapter 10 Homogeneous manifolds 163(34)
1 Generalized manifolds
163(6)
2 The projective spaces
169(3)
3 Coset spaces are manifolds
172(3)
4 Group actions
175(2)
5 Homogeneous manifolds
177(5)
6 Riemannian manifolds
182(5)
7 Lie groups
187(5)
8 Exercises
192(5)
Chapter 11 Roots 197(38)
1 The structure of su(3)
198(3)
2 The structure of g = su(n)
201(3)
3 An invariant decomposition of g
204(2)
4 The definition of roots and dual roots
206(4)
5 The bracket of two root spaces
210(2)
6 The structure of so(2n)
212(2)
7 The structure of so(2n 1)
214(1)
8 The structure of sp(n)
215(1)
9 The Weyl group
216(5)
10 Towards the classification theorem
221(4)
11 Complexified Lie algebras
225(5)
12 Exercises
230(5)
Bibliography 235(2)
Index 237
Kristopher Tapp, Saint Joseph's University, Philadelphia, PA, USA.
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