**Chapter 1 Linear Equations in Linear Algebra** ** **

INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering

1.1 Systems of Linear Equations

1.2 Row Reduction and Echelon Forms

1.3 Vector Equations

1.4 The Matrix Equation Ax = b

1.5 Solution Sets of Linear Systems

1.6 Applications of Linear Systems

1.7 Linear Independence

1.8 Introduction to Linear Transformations

1.9 The Matrix of a Linear Transformation

1.10 Linear Models in Business, Science, and Engineering

Supplementary Exercises

**Chapter 2 Matrix Algebra**

INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design

2.1 Matrix Operations

2.2 The Inverse of a Matrix

2.3 Characterizations of Invertible Matrices

2.4 Partitioned Matrices

2.5 Matrix Factorizations

2.6 The Leontief Input=Output Model

2.7 Applications to Computer Graphics

2.8 Subspaces of R^n

2.9 Dimension and Rank

Supplementary Exercises

**Chapter 3 Determinants**

INTRODUCTORY EXAMPLE: Determinants in Analytic Geometry

3.1 Introduction to Determinants

3.2 Properties of Determinants

3.3 Cramer's Rule, Volume, and Linear Transformations

Supplementary Exercises

**Chapter 4 Vector Spaces**

INTRODUCTORY EXAMPLE: Space Flight and Control Systems

4.1 Vector Spaces and Subspaces

4.2 Null Spaces, Column Spaces, and Linear Transformations

4.3 Linearly Independent Sets; Bases

4.4 Coordinate Systems

4.5 The Dimension of a Vector Space

4.6 Rank

4.7 Change of Basis

4.8 Applications to Difference Equations

4.9 Applications to Markov Chains

Supplementary Exercises

**Chapter 5 Eigenvalues and Eigenvectors** ** **

INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls

5.1 Eigenvectors and Eigenvalues

5.2 The Characteristic Equation

5.3 Diagonalization

5.4 Eigenvectors and Linear Transformations

5.5 Complex Eigenvalues

5.6 Discrete Dynamical Systems

5.7 Applications to Differential Equations

5.8 Iterative Estimates for Eigenvalues

Supplementary Exercises

**Chapter 6 Orthogonality and Least Squares**

INTRODUCTORY EXAMPLE: Readjusting the North American Datum

6.1 Inner Product, Length, and Orthogonality

6.2 Orthogonal Sets

6.3 Orthogonal Projections

6.4 The Gram-Schmidt Process

6.5 Least-Squares Problems

6.6 Applications to Linear Models

6.7 Inner Product Spaces

6.8 Applications of Inner Product Spaces

Supplementary Exercises

**Chapter 7 Symmetric Matrices and Quadratic Forms**

INTRODUCTORY EXAMPLE: Multichannel Image Processing

7.1 Diagonalization of Symmetric Matrices

7.2 Quadratic Forms

7.3 Constrained Optimization

7.4 The Singular Value Decomposition

7.5 Applications to Image Processing and Statistics

Supplementary Exercises

*ONLINE ONLY *Chapter 8 The Geometry of Vector Spaces ** **

INTRODUCTORY EXAMPLE: The Platonic Solids

8.1 Affine Combinations

8.2 Affine Independence

8.3 Convex Combinations

8.4 Hyperplanes

8.5 Polytopes

8.6 Curves and Surfaces

Supplementary Exercises

*ONLINE ONLY ***Chapter 9 Optimization**

** **

INTRODUCTORY EXAMPLE: The Berlin Airlift

9.1 Matrix Games

9.2 Linear Programming - Geometric Method

**9.3 **Linear Programming - Simplex Method

**9.4 **Duality

Supplementary Exercises

** **

**Appendices** ** **

A Uniqueness of the Reduced Echelon Form

B Complex Numbers

Glossary

Answers to Odd-Numbered Exercises

Index