Advanced Linear Algebra

Advanced Linear Algebra features a student-friendly approach to the theory of linear algebra. The author’s emphasis on vector spaces over general fields, with corresponding current applications, sets the book apart. He focuses on finite fields and complex numbers, and discusses matrix algebra over these fields. The text then proceeds to cover vector spaces in depth. Also discussed are standard topics in linear algebra including linear transformations, Jordan canonical form, inner product spaces, spectral theory, and, as supplementary topics, dual spaces, quotient spaces, and tensor products. Written in clear and concise language, the text sticks to the development of linear algebra without excessively addressing applications. A unique chapter on "How to Use Linear Algebra" is offered after the theory is presented. In addition, students are given pointers on how to start a research project. The proofs are clear and complete and the exercises are well designed. In addition, full solutions are included for almost all exercises.

Fields and Matrix Algebra The Field Z3 The Field AxiomsField ExamplesMatrix Algebra over Different FieldsExercisesVector SpacesDefinition of a Vector SpaceVector Spaces of FunctionsSubspaces and More Examples of Vector Spaces Linear Independence, Span, and Basis Coordinate Systems ExercisesLinear TransformationsDefinition of a Linear Transformation Range and Kernel of Linear Transformations Matrix Representations of Linear Maps ExercisesThe Jordan Canonical Form The Cayley-Hamilton Theorem Jordan Canonical Form for Nilpotent Matrices An Intermezzo about Polynomials The Jordan Canonical Form The Minimal Polynomial Commuting Matrices Systems of Linear Differential EquationsFunctions of Matrices The Resolvent ExercisesInner Product and Normed Vector Spaces Inner Products and Norms Orthogonal and Orthonormal Sets and Bases The Adjoint of a Linear Map Unitary Matrices, QR, and Schur Triangularization Normal and Hermitian Matrices Singular Value Decomposition Exercises Constructing New Vector Spaces from Given Ones The Cartesian Product The Quotient Space The Dual Space Multilinear Maps and Functionals The Tensor Product Anti-Symmetric and Symmetric Tensors Exercises How to Use Linear Algebra Matrices You Can't Write Down, but Would Still Like to Use Algorithms Based on Matrix Vector Products Why Use Matrices When Computing Roots of Polynomials? How to Find Functions with Linear Algebra? How to Deal with Incomplete Matrices Solving Millennium Prize Problems with Linear Algebra How Secure Is RSA Encryption? Quantum Computation and Positive Maps ExercisesHow to Start Your Own Research ProjectAnswers to Exercises

Hugo J. Woerdeman, PhD, professor, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, USA