Matrix Groups for Undergraduates

Description

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 Content: Cover Title page Why study matrix groups? Chapter 1. Matrices 1. Rigid motions of the sphere: a motivating example 2. Fields and skew-fields 3. The quaternions 4. Matrix operations 5. Matrices as linear transformations 6. The general linear groups 7. Change of basis via conjugation 8. Exercises Chapter 2. All matrix groups are real matrix groups 1. Complex matrices as real matrices 2. Quaternionic matrices as complex matrices 3. Restricting to the general linear groups 4. Exercises Chapter 3. The orthogonal groups 1. The standard inner product on \Kⁿ 2. Several characterizations of the orthogonal groups3. The special orthogonal groups 4. Low dimensional orthogonal groups 5. Orthogonal matrices and isometries 6. The isometry group of Euclidean space 7. Symmetry groups 8. Exercises Chapter 4. The topology of matrix groups 1. Open and closed sets and limit points 2. Continuity 3. Path-connected sets 4. Compact sets 5. Definition and examples of matrix groups 6. Exercises Chapter 5. Lie algebras 1. The Lie algebra is a subspace 2. Some examples of Lie algebras 3. Lie algebra vectors as vector fields 4. The Lie algebras of the orthogonal groups5. Exercises Chapter 6. Matrix exponentiation 1. Series in \K 2. Series in _{ }(\K) 3. The best path in a matrix group 4. Properties of the exponential map 5. Exercises Chapter 7. Matrix groups are manifolds 1. Analysis background 2. Proof of part (1) of Theorem 7.1 3. Proof of part (2) of Theorem 7.1 4. Manifolds 5. More about manifolds 6. Exercises Chapter 8. The Lie bracket 1. The Lie bracket 2. The adjoint representation 3. Example: the adjoint representation for (3) 4. The adjoint representation for compact matrix groups5. Global conclusions 6. The double cover (1)\ra (3) 7. Other double covers 8. Exercises Chapter 9. Maximal tori 1. Several characterizations of a torus 2. The standard maximal torus and center of ( ), ( ), ( ) and ( ) 3. Conjugates of a maximal torus 4. The Lie algebra of a maximal torus 5. The shape of (3) 6. The rank of a compact matrix group 7. Exercises Chapter 10. Homogeneous manifolds 1. Generalized manifolds 2. The projective spaces 3. Coset spaces are manifolds 4. Group actions 5. Homogeneous manifolds6. Riemannian manifolds 7. Lie groups 8. Exercises Chapter 11. Roots 1. The structure of (3) 2. The structure of \mg= ( ) 3. An invariant decomposition of \mg 4. The definition of roots and dual roots 5. The bracket of two root spaces 6. The structure of (2 ) 7. The structure of (2 +1) 8. The structure of ( ) 9. The Weyl group 10. Towards the classification theorem 11. Complexified Lie algebras 12. Exercises Bibliography Index Back Cover