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.
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 Mmn(₣)
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
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.
"This is the substantially extended second edition of a book comprising an advanced course in linear algebra …"
—Zentralblatt MATH 1319Praise 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
