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| Preface to the Instructor Preface to the Student Acknowledgments CHAPTER 1 Vector Spaces Complex Numbers Definition of Vector Space Properties of Vector Spaces Subspaces Sums and Direct Sums Exercises CHAPTER 2 Finite-Dimenslonal Vector Spaces Span and Linear Independence Bases Dimension Exercises CHAPTER 3 Linear Maps Definitions and Examples Null Spaces and Ranges The Matrix of a Linear Map Invertibility Exercises CHAPTER 4 Potynomiags Degree Complex Coefficients Real Coefflcients Exercises CHAPTER 5 Eigenvalues and Eigenvectors lnvariant Subspaces Polynomials Applied to Operators Upper-Triangular Matrices Diagonal Matrices Invariant Subspaces on Real Vector Spaces Exercises CHAPTER 6 Inner-Product spaces Inner Products Norms Orthonormal Bases Orthogonal Projections and Minimization Problems Linear Functionals and Adjoints Exercises CHAPTER 7 Operators on Inner-Product Spaces Self-Adjoint and Normal Operators The Spectral Theorem Normal Operators on Real Inner-Product Spaces Positive Operators Isometries Polar and Singular-Value Decompositions Exercises CHAPTER 8 Operators on Complex Vector Spaces Generalized Eigenvectors The Characteristic Polynomial Decomposition of an Operator Square Roots The Minimal Polynomial Jordan Form Exercises CHAPTER 9 Operators on Real Vector Spaces Eigenvalues of Square Matrices Block Upper-Triangular Matrices The Characteristic Polynomial Exercises CHAPTER 10 Trace and Determinant Change of Basis Trace Determinant of an Operator Determinant of a Matrix Volume Exercises Symbol Index Index |
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