Advanced Linear Algebra

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

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.