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.