Table of Contents
for
The Keys to Linear Algebra
1 EUCLIDEAN VECTORS
1.1 Vectors in Euclidean Space and Their Applications
1.2 Arithmetic Operations on Vectors
1.3 Lines and Planes
1.4 Problem Solving with Vectors
Chapter Summary
2 USING MATRICES TO SOLVE (m x n) LINEAR EQUATIONS
2.1 Applications of Solving Linear Equations
2.2 The Problem of Solving Linear Equations
2.3 Solving Linear Equations by Row Operations
2.4 Matrices and Their Operations
2.5 Problem Solving with Linear Equations
Chapter Summary
3 USING MATRICES TO SOLVE (n x n) LINEAR EQUATIONS
3.1 Applications of (n x n) Linear Equations
3.2 Using the Inverse of a Matrix to Solve (n x n) Linear Equations
3.3 Using Determinants to Solve Linear Equations
3.4 Problem Solving with Linear Equations
Chapter Summary
4 VECTOR SPACES
4.1 Unifying n-Vectors and Matrices into a Vector Space
4.2 Basic Properties and Subspaces of Vector Spaces
4.3 Span, Linear Independence, and Basis
4.4 The Dimension of a Vector Space
4.5 Problem Solving with Vector Spaces
Chapter Summary
5 LINEAR TRANSFORMATIONS
5.1 A Review of Functions
5.2 Linear Transformations and Their Applications
5.3 The Matrix of a Linear Transformation
5.4 Linear Transformations from V to V
5.5 Solving Equations with Linear Transformations
5.6 Problem Solving with Linear Transformations
Chapter Summary
6 EIGENVALUES AND EIGENVECTORS
6.1 What Are Eigenvalues and Eigenvectors?
6.2 Applications to Dynamical Systems
6.3 Solving a Dynamical System
6.4 Diagonalization
6.5 Problem Solving with Eigenvalues and Eigenvectors
Chapter Summary
7 ORTHOGONALITY AND INNER PRODUCT SPACES
7.1 Orthogonality in R^n
7.2 Orthogonal Projections and the Gram-Schmidt Process
7.3 Applications to Regression Models
7.4 Inner Product Spaces
7.5 Problem Solving with Orthogonality
Chapter Summary
8 NUMERICAL METHODS IN LINEAR ALGEBRA
8.1 General Computational Concerns
8.2 Matrix Factorizations
8.3 Finding Eigenvalues and Eigenvectors
8.4 Iterative Methods
8.5 Using Numerical Methods \newline in Problem Solving
Chapter Summary
APPENDIX A. PROOF TECHNIQUES
APPENDIX B. MATHEMATICAL THINKING PROCESSES
SOLUTIONS TO SELECTED EXERCISES
INDEX