Table of Contents * Sections denoted by an asterisk are optional. Vector Spaces 1.1 Introduction1.2 Vector Spaces1.3 Subspaces1.4 Linear Combinations and Systems of Linear Equations1.5 Linear Dependence and Linear Independence1.6 Bases and Dimension1.7* Maximal Linearly Independent Subsets Index of DefinitionsLinear Transformations and Matrices 2.1 Linear Transformations, Null Spaces, and Ranges2.2 The Matrix Representation of a Linear Transformation2.3 Composition of Linear Transformations and Matrix Multiplication2.4 Invertibility and Isomorphisms2.5 The Change of Coordinate Matrix2.6* Dual Spaces2.7* Homogeneous Linear Differential Equations with Constant Coefficients Index of DefinitionsElementary Matrix Operations and Systems of Linear Equations 3.1 Elementary Matrix Operations and Elementary Matrices3.2 The Rank of a Matrix and Matrix Inverses3.3 Systems of Linear Equations – Theoretical Aspects3.4 Systems of Linear Equations – Computational Aspects Index of DefinitionsDeterminants 4.1 Determinants of Order 24.2 Determinants of Order n4.3 Properties of Determinants4.4 Summary|Important Facts about Determinants4.5* A Characterization of the Determinant Index of DefinitionsDiagonalization 5.1 Eigenvalues and Eigenvectors5.2 Diagonalizability5.3* Matrix Limits and Markov Chains5.4 Invariant Subspaces and the Cayley–Hamilton Theorem Index of DefinitionsInner Product Spaces 6.1 Inner Products and Norms6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements6.3 The Adjoint of a Linear Operator6.4 Normal and Self-Adjoint Operators6.5 Unitary and Orthogonal Operators and Their Matrices6.6 Orthogonal Projections and the Spectral Theorem6.7* The Singular Value Decomposition and the Pseudoinverse6.8* Bilinear and Quadratic Forms6.9* Einstein's Special Theory of Relativity6.10* Conditioning and the Rayleigh Quotient6.11* The Geometry of Orthogonal Operators Index of DefinitionsCanonical Forms 7.1 The Jordan Canonical Form I7.2 The Jordan Canonical Form II7.3 The Minimal Polynomial7.4* The Rational Canonical Form Index of Definitions Appendices SetsFunctionsFieldsComplex NumbersPolynomials Answers to Selected Exercises Index

New and updated features of this title A streamlined presentation, with clarified exposition informed by extensive reviews from instructors.Revised proofs of some theorems.Additional examples and new exercises throughout.Online solutions to selected theoretical exercises in each section of the book: These exercises each have their exercise number printed within a gray box, and the last sentence of each of these exercises gives a short URL for its online solution.4 new applications available online of the content in Sections 2.3, 5.3, 6.5, and 6.6. Short URLs at point of use provide easy access to this material.