**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 Equations3.2 Using the Inverse of a Matrix to Solve (

n x n) Linear Equations3.3 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 Space4.2 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

VtoV5.5 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^n7.2 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