Elementary Linear Algebra

Preface1 Introduction to Linear Equations and Matrices 1.1 Introduction to Linear Systems and Matrices 1.2 Gaussian Elimination 1.3 The Algebra of Matrices 1.4 Inverses and Elementary Matrices 1.5 Gaussian Elimination as a Matrix Factorization 1.6 Transposes, Symmetry, and Band Matrices; An Application 1.7 Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems Review Exercises2 Vector Spaces 2.1 Vectors in 2- and 3- Spaces 2.2 Euclidean n-Space 2.3 General Vector Spaces 2.4 Subspaces, Span, Null Spaces 2.5 Linear Independence 2.6 Basis and Dimension 2.7 The Fundamental Subspaces of a Matrix; Rank 2.8 An Application: Error-Correcting Codes Review Exercises3 Linear Transformations, Orthogonal Projections, and Least Squares 3.1 Matrices as Linear Transformations 3.2 Relationships Involving Inner Products 3.3 Least Squares and Orthogonal Projections 3.4 Orthogonal Bases and the Gram-Schmidt Process 3.5 Orthogonal Matrices, QR Decompositions, and Least Squares (Revisited) 3.6 Encoding the QR Decomposition-A Geometric Approach Review Exercises4 Eigenvectors and Eigenvalues 4.1 A Brief Introduction to Determinants 4.2 Eigenvalues and Eigenvectors 4.3 Diagonalization 4.4 Symmetric Matrices 4.5 An Application-Difference Equations: Fibonacci Sequences and Markov Processes 4.6 An Application-Differential Equations 4.7 An Application-Quadratic Forms 4.8 Solving the Eigenvalue Problem Numerically Review Exercises5 Determinantst 5.1 The Determinant Function 5.2 Evaluating Determinants 5.3 Properties of Determinants 5.4 Cofactor Expansion; Cramer's Rule Review Exercises6 Further Directions 6.1 Function Spaces 6.2 Singular Value Decomposition and Generalized Inverses 6.3 General Vector Spaces and Linear Transformations Over an Arbitrary Field Review ExercisesAppendices Appendix A. More on LU-Decompositions Appendix B. Counting Operations and Gauss-Jordan Elimination Appendix C. Another Application Appendix D. Software and Codes for Linear AlgebraBibliography and Further ReadingsAnswers to Odd-Numbered ExercisesIndex