Elementary Linear Algebra

Preface


1 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 Exercises


2 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 Exercises


3 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 Exercises


4 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 Exercises


5 Determinantst


5.1 The Determinant Function


5.2 Evaluating Determinants


5.3 Properties of Determinants


5.4 Cofactor Expansion; Cramer's Rule

Review Exercises


6 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 Exercises

Appendices

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 Algebra

Bibliography and Further Readings

Answers to Odd-Numbered Exercises

Index