Linear Algebra (Pure and Applied Undergraduate Texts)

This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material. The text starts with vector spaces, over the sets of real and complex numbers, and linear transformations between such vector spaces. Later on, this setting is extended to general fields. The reader will be in a position to appreciate the early material on this more general level with minimal effort. Notable features of the text include a treatment of determinants, which is cleaner than one often sees, and a high degree of contact with geometry and analysis, particularly in the chapter on linear algebra on inner product spaces. In addition to studying linear algebra over general fields, the text has a chapter on linear algebra over rings. There is also a chapter on special structures, such as quaternions, Clifford algebras, and octonions. Table of contents : Cover Title Page Preface Some basic notation Chapter 1. Vector spaces, linear transformations, and matrices 1.1. Vector spaces 1.2. Linear transformations and matrices 1.3. Basis and dimension 1.4. Matrix representation of a linear transformation 1.5. Determinants and invertibility 1.6. Applications of row reduction and column reduction Chapter 2. Eigenvalues, eigenvectors, and generalized eigenvectors 2.1. Eigenvalues and eigenvectors 2.2. Generalized eigenvectors and the minimal polynomial 2.3. Triangular matrices and upper triangularization 2.4. The Jordan canonical form Chapter 3. Linear algebra on inner product spaces 3.1. Inner products and norms 3.2. Norm, trace, and adjoint of a linear transformation 3.3. Self-adjoint and skew-adjoint transformations 3.4. Unitary and orthogonal transformations 3.5. Schur’s upper triangular representation 3.6. Polar decomposition and singular value decomposition 3.7. The matrix exponential 3.8. The discrete Fourier transform Chapter 4. Further basic concepts: duality, convexity, quotients, positivity 4.1. Dual spaces 4.2. Convex sets 4.3. Quotient spaces 4.4. Positive matrices and stochastic matrices Chapter 5. Multilinear algebra 5.1. Multilinear mappings 5.2. Tensor products 5.3. Exterior algebra 5.4. Isomorphism ????kern .5??(?)≈Λ2? and the Pfaffian Chapter 6. Linear algebra over more general fields 6.1. Vector spaces over more general fields 6.2. Rational matrices and algebraic numbers Chapter 7. Rings and modules 7.1. Rings and modules 7.2. Modules over principal ideal domains 7.3. The Jordan canonical form revisited 7.4. Integer matrices and algebraic integers 7.5. Noetherian rings and Noetherian modules 7.6. Polynomial rings over UFDs Chapter 8. Special structures in linear algebra 8.1. Quaternions and matrices of quaternions 8.2. Algebras 8.3. Clifford algebras 8.4. Octonions Appendix A. Complementary results A.1. The fundamental theorem of algebra A.2. Averaging rotations A.3. Groups A.4. Finite fields and other algebraic field extensions Bibliography Index Selected Published Titles in This Series Back Cover Explores linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. "The pdf contains a draft title page, draft copyright page and a draft manuscript"-- Provided by publisher