Matrix Groups for Undergraduates : Second Edition

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, ma. Read more... Abstract: 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, ma 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. From reviews of the First Edition: This book could be used as an excellent textbook for a one semester course at university and it will prepare students for a graduate course on Lie groups, Lie algebras, etc. ... The book combines an intuitive style of writing with rigorous definitions and proofs, giving examples from fields of mathematics, physics, and other sciences where matrices are successfully applied. The book will surely be interesting and helpful for students in algebra and their teachers. —European Mathematical Society Newsletters This is an excellent, well-written textbook which is strongly recommended to a wide audience of readers interested in mathematics and its applications. The book is suitable for a one semester undergraduate lecture course in matrix groups, and would also be useful supplementary reading for more general group theory courses. —MathSciNet (or Mathematical Reviews) Title page 4 Copyright 5 Contents 6 Why study matrix groups? 10 1. Matrices 14 2. All matrix groups are real matrix groups 32 3. The orthogonal groups 42 4. The topology of matrix groups 62 5. Lie algebras 78 6. Matrix exponentiation 90 7. Matrix groups are manifolds 105 8. The Lie bracket 126 9. Maximal tori 148 10. Homogeneous manifolds 172 11. Roots 206 Bibliography 244 Index 246 1. Matrices -- 2. All Matrix Groups Are Real Matrix Groups -- 3. The Orthogonal Groups -- 4. The Topology Of Matrix Groups -- 5. Lie Algebras -- 6. Matrix Exponentiation -- 7. Matrix Groups Are Manifolds -- 8. The Lie Bracket -- 9. Maximal Tori -- 10. Homogenous Manifolds -- 11. Roots. Kristopher Tapp. Includes Bibliographical References And Index.