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
