Advanced Linear Algebra

Advanced Linear Algebra focuses on vector spaces and the maps between them that preserve their structure (linear transformations). It starts with familiar concepts and then slowly builds to deeper results. Along with including many exercises and examples, each section reviews what students need to know before studying the material.

The book first introduces vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension. After covering linear transformations, it discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. The author then develops the whole structure theory of a linear operator on a finite dimensional vector space from a collection of some simple results. He also explores the entire range of topics associated with inner product spaces, from the Gram-Schmidt process to the spectral theorems for normal and self-adjoint operators on an inner product space. The text goes on to rigorously describe the trace and determinant of linear operators and square matrices. The final two chapters focus on bilinear forms and tensor products and related material.

Designed for advanced undergraduate and beginning graduate students, this textbook shows students the beauty of linear algebra. It also prepares them for further study in mathematics.

Bruce Cooperstein is a professor of mathematics at the University of California, Santa Cruz, USA. He was a visiting scholar at the Carnegie Foundation for the Advancement of Teaching (spring 2007) and a recipient of the Kellogg National Fellowship (1982-1985) and the Pew National Fellowship for Carnegie Scholars (1999-2000). Dr. Cooperstein has authored over fifty papers in referred mathematics journals.

Vector Spaces
Fields
The Space Fn
Vector Spaces over an Arbitrary Field
Subspaces of Vector Spaces
Span and Independence
Bases and Finite Dimensional Vector Spaces
Bases and Infinite Dimensional Vector Spaces
Coordinate Vectors


Linear Transformations
Introduction to Linear Transformations
The Range and Kernel of a Linear Transformation
The Correspondence and Isomorphism Theorems
Matrix of a Linear Transformation
The Algebra of L(V, W) and Mmn(F)
Invertible Transformations and Matrices


Polynomials
The Algebra of Polynomials
Roots of Polynomials


Theory of a Single Linear Operator
Invariant Subspaces of an Operator
Cyclic Operators
Maximal Vectors
Indecomposable Linear Operators
Invariant Factors and Elementary Divisors
Canonical Forms
Operators on Real and Complex Vector Spaces


Inner Product Spaces
Inner Products
Geometry in Inner Product Spaces
Orthonormal Sets and the Gram-Schmidt Process
Orthogonal Complements and Projections
Dual Spaces
Adjoints


Linear Operators on Inner Product Spaces
Self-Adjoint and Normal Operators
Spectral Theorems
Normal Operators on Real Inner Product Spaces
Unitary and Orthogonal Operators
Polar Decomposition and Singular Value Decomposition


Trace and Determinant of a Linear Operator
Trace of a Linear Operator
Determinant of a Linear Operator and Matrix
Uniqueness of the Determinant of a Linear Operator


Bilinear Maps and Forms
Basic Properties of Bilinear Maps
Symplectic Spaces
Quadratic Forms and Orthogonal Space
Real Quadratic Forms


Tensor Products
Introduction to Tensor Products
Properties of Tensor Products
The Tensor Algebra
The Symmetric and Exterior Algebras

Appendix A: Answers to Selected Exercises
Appendix B: Hints to Selected Problems



Index