Linear Algebra, 5th Edition, By Stephen Friedberg, Arnold Insel, Lawrence Spense ( )

Linear Algebra Contents Preface Suggested Course Outlines Overview of Contents Differences Between the Fourth and Fifth Editions To the Student 1 Vector Spaces 1.1 Introduction Exercises 1.2 Vector Spaces Exer ...

Linear Algebra Contents Preface Suggested Course Outlines Overview of Contents Differences Between the Fourth and Fifth Editions To the Student 1 Vector Spaces 1.1 Introduction Exercises 1.2 Vector Spaces Exercises 1.3 Subspaces Exercises Definition. Definition. 1.4 Linear Combinations and Systems of Linear Equations Exercises 1.5 Linear Dependence and Linear Independence Exercises 1.6 Bases and Dimension An Overview of Dimension and Its Consequences The Dimension of Subspaces Corollary. The Lagrange Interpolation Formula Exercises 1.7* Maximal Linearly Independent Subsets Exercises Index of Definitions for Chapter 1 2 Linear Transformations and Matrices 2.1 Linear Transformations, Null Spaces, and Ranges Exercises Definitions. Definitions. 2.2 The Matrix Representation of a Linear Transformation Exercises 2.3 Composition of Linear Transformations and Matrix Multiplication Applications* Exercises 2.4 Invertibility and Isomorphisms Exercises 2.5 The Change of Coordinate Matrix Exercises 2.6* Dual Spaces Exercises 2.7* Homogeneous Linear Differential Equations with Constant Coefficients Exercises Index of Definitions for Chapter 2 3 Elementary Matrix Operations and Systems of Linear Equations 3.1 Elementary Matrix Operations and Elementary Matrices Exercises 3.2 The Rank of a Matrix and Matrix Inverses The Inverse of a Matrix Definition. Exercises 3.3 Systems of Linear Equations—Theoretical Aspects An Application Exercises 3.4 Systems of Linear Equations—Computational Aspects Exercises Index of Definitions for Chapter 3 4 Determinants 4.1 Determinants of Order 2 Exercises 4.2 Determinants of Order n Exercises 4.3 Properties of Determinants Exercises Definition. 4.4 Summary—Important Facts about Determinants Properties of the Determinant Exercises 4.5* A Characterization of the Determinant Exercises Index of Definitions for Chapter 4 5 Diagonalization 5.1 Eigenvalues and Eigenvectors Exercises 5.2 Diagonalizability Test for Diagonalizability Systems of Differential Equations Direct Sums* Definition. Definition. Proof. Proof. Exercises Definitions. 5.3* Matrix Limits and Markov Chains Applications* Exercises Definition. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem The Cayley-Hamilton Theorem Proof. Corollary (Cayley-Hamilton Theorem for Matrices). Proof. Invariant Subspaces and Direct Sums3 Proof. Definition. Proof. Exercises Definition. Index of Definitions for Chapter 5 6 Inner Product Spaces 6.1 Inner Products and Norms Exercises Definition. 6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements Exercises 6.3 The Adjoint of a Linear Operator Proof. Proof. Proof. Corollary. Proof. Proof. Corollary. Proof. Least Squares Approximation Lemma 1. Proof. Lemma 2. Proof. Corollary. Minimal Solutions to Systems of Linear Equations Proof. Exercises Definition. 6.4 Normal and Self-Adjoint Operators Exercises Definitions. 6.5 Unitary and Orthogonal Operators and Their Matrices Rigid Motions* Definition. Proof. Orthogonal Operators on R2 Proof. Corollary. Conic Sections Exercises Definition. 6.6 Orthogonal Projections and the Spectral Theorem Exercises 6.7* The Singular Value Decomposition and the Pseudoinverse The Polar Decomposition of a Square Matrix Proof. The Pseudoinverse Definition. The Pseudoinverse of a Matrix The Pseudoinverse and Systems of Linear Equations Proof. Proof. Exercises 6.8* Bilinear and Quadratic Forms Bilinear Forms Definition. Definitions. Proof. Definition. Proof. Corollary 1. Proof. Corollary 2. Corollary 3. Definition. Proof. Corollary. Proof. Symmetric Bilinear Forms Definition. Proof. Definition. Corollary. Proof. Proof. Proof. Corollary. Proof. Diagonalization of Symmetric Matrices Quadratic Forms Definition. Quadratic Forms on a Real Inner Product Space Proof. Corollary. Proof. The Second Derivative Test for Functions of Several Variables Proof. Sylvester’s Law of Inertia Proof. Definitions. Corollary 1 (Sylvester’s Law of Inertia for Matrices). Definitions. Corollary 2. Proof. Corollary 3. Exercises 6.9* Einstein’s Special Theory of Relativity Time Contraction Exercises 6.10* Conditioning and the Rayleigh Quotient Exercises 6.11* The Geometry of Orthogonal Operators Exercises Index of Definitions for Chapter 6 7 Canonical Forms 7.1 The Jordan Canonical Form I Exercises 7.2 The Jordan Canonical Form II Exercises Definitions. Definition. 7.3 The Minimal Polynomial Exercises Definition. 7.4* The Rational Canonical Form Uniqueness of the Rational Canonical Form Lemma 1. Lemma 2. Corollary. Definitions. Direct Sums* Proof. Proof. Exercises Index of Definitions for Chapter 7 Appendices Appendix A Sets Appendix B Functions Appendix C Fields Definitions. Proof. Corollary. Proof. Proof. Corollary. Appendix D Complex Numbers Definitions. Proof. Definition. Proof. Definition. Proof. Proof. Corollary. Proof. Appendix E Polynomials Definition. Proof. Corollary 1. Proof. Corollary 2. Proof. Definition. Lemma. Proof. Proof. Definitions. Proof. Proof. Proof. Definitions. Proof. Proof. Proof. Corollary. Proof. Proof. Proof. Answers to Selected Exercises Chapter 1 Section 1.1 Section 1.2 Section 1.3 Section 1.4 Section 1.5 Section 1.6 Section 1.7 Chapter 2 Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5 Section 2.6 Section 2.7 Chapter 3 Section 3.1 Section 3.2 Section 3.3 Section 3.4 Chapter 4 Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Chapter 5 Section 5.1 Section 5.2 Section 5.3 Section 5.4 Chapter 6 Section 6.1 Section 6.2 Section 6.3 Section 6.4 Section 6.5 Section 6.6 Section 6.7 Section 6.8 Section 6.9 Section 6.10 Section 6.11 Chapter 7 Section 7.1 Section 7.2 Section 7.3 Section 7.4 Index List of Symbols