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 Axioms
Field Examples
Matrix Algebra over Different Fields
Exercises
Vector Spaces
Definition of a Vector Space
Vector Spaces of Functions
Subspaces and More Examples of Vector Spaces
Linear Independence, Span, and Basis
Coordinate Systems
Exercises
Linear Transformations
Definition of a Linear Transformation
Range and Kernel of Linear Transformations
Matrix Representations of Linear Maps
Exercises
The 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 Equations
Functions of Matrices
The Resolvent
Exercises
Inner 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
Exercises
How to Start Your Own Research Project
Answers to Exercises
Biography
Hugo J. Woerdeman, PhD, professor, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, USA
Woerdeman’s work requires background knowledge of linear algebra. Students should be familiar with matrix computations, solving systems, eigenvalues, eigenvectors, finding a basis for the null space, row and column spaces, determinants, and inverses. This text provides a more general approach to vector spaces, developing these over complex numbers and finite fields. Woerdeman (mathematics, Drexel Univ.) provides a review of complex numbers and some basic results for finite fields. This book will help build on previous knowledge obtained from an earlier course and introduce students to numerous advanced topics. A few of these topics are Jordan canonical form, the Cayley-Hamilton Theorem, nilpotent matrices, functions of matrices, Hermitian matrices, the tensor product, quotient space, and dual space. The last chapter, which discusses how to use linear algebra, illustrates some applications, such as finding roots of polynomials, algorithms based on matrix vector products, RSA public key inscription, and theoretical topics, such as the Riemann hypothesis and the “P versus NP problem.” Copious exercises are provided, and most give complete solutions. The text will provide a solid foundation for any further work in linear algebra.
--R. L. Pour, Emory and Henry College
