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# Advanced Linear Algebra

## Preview

## Book Description

**Advanced Linear Algebra, Second Edition** takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress.

The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate students, the book discusses the structure theory of an operator, various topics on inner product spaces, and the trace and determinant functions of a linear operator. It addresses bilinear forms with a full treatment of symplectic spaces and orthogonal spaces, as well as explains the construction of tensor, symmetric, and exterior algebras.

Featuring updates and revisions throughout, **Advanced Linear Algebra, Second Edition**:

- Contains new chapters covering sesquilinear forms, linear groups and groups of isometries, matrices, and three important applications of linear algebra
- Adds sections on normed vector spaces, orthogonal spaces over perfect fields of characteristic two, and Clifford algebras
- Includes several new exercises and examples, with a solutions manual available upon qualifying course adoption

The book shows students the beauty of linear algebra while preparing them for further study in mathematics.

## Table of Contents

Preface to the Second Edition

Preface to the First Edition

Acknowledgments

List of Figures

Symbol Description

Vector Spaces

Fields

The Space ₣^{n}

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 *₤*(V,W) and M_{mn}(₣)

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

Normed and Inner Product Spaces

Inner Products

Geometry in Inner Product Spaces

Orthonormal Sets and the Gram-Schmidt Process

Orthogonal Complements and Projections

Dual Spaces

Adjoints

Normed Vector Spaces

Linear Operators on Inner Product Spaces

Self-Adjoint and Normal Operators

Spectral Theorems

Normal Operators on Real Inner Product Spaces

Unitary and Orthogonal Operators

The 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 Forms

Basic Properties of Bilinear Maps

Symplectic Spaces

Quadratic Forms and Orthogonal Space

Orthogonal Space, Characteristic Two

Real Quadratic Forms

Sesquilinear Forms and Unitary Geometry

Basic Properties of Sesquilinear Forms

Unitary Space

Tensor Products

Introduction to Tensor Products

Properties of Tensor Products

The Tensor Algebra

The Symmetric Algebra

The Exterior Algebra

Clifford Algebras, char ₣ ≠ 2

Linear Groups and Groups of Isometries

Linear Groups

Symplectic Groups

Orthogonal Groups, char ₣ ≠ 2

Unitary Groups

Additional Topics in Linear Algebra

Matrix Norms

The Moore–Penrose Inverse of a Matrix

Nonnegative Matrices

The Location of Eigenvalues

Functions of Matrices

Applications of Linear Algebra

Least Squares

Error Correcting Codes

Ranking Webpages for Search Engines

Appendices

Concepts from Topology and Analysis

Concepts from Group Theory

Answers to Selected Exercises

Hints to Selected Problems

Bibliography

Index

## Author(s)

### Biography

**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 numerous papers in refereed mathematics journals.

## Reviews

"This is the substantially extended second edition of a book comprising an advanced course in linear algebra …"

—Zentralblatt MATH1319

Praise for the First Edition:"The book is well written, and the examples are appropriate. … Each section contains relevant problems at the end. The ‘What You Need to Know’ feature at the beginning of each section outlining the knowledge required to grasp the material is useful. Summing Up: Recommended."

—CHOICE, January 2011"Pedagogically, a structural and general approach is taken, and topically, the material has been chosen in order to cover the material a beginning graduate student would be expected to know when taking a first course in group or field theory or functional analysis."

—SciTech Book News, February 2011