## Table of Contents

**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 *A***x = 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

Projects

Supplementary Exercises

**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 Rn

2.9 Dimension and Rank

Projects

Supplementary Exercises

**3. Determinants**

Introductory Example: Random Paths and Distortion

3.1 Introduction to Determinants

3.2 Properties of Determinants

3.3 Cramer’s Rule, Volume, and Linear Transformations

Projects

Supplementary Exercises

**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 Change of Basis

4.7 Digital Signal Processing

4.8 Applications to Difference Equations

Projects

Supplementary Exercises

**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

5.9 Markov Chains

Projects

Supplementary Exercises

**6. Orthogonality and Least Squares**

Introductory Example: The North American Datum and GPS Navigation

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 Machine Learning and Linear Models

6.7 Inner Product Spaces

6.8 Applications of Inner Product Spaces

Projects

Supplementary Exercises

**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

Projects

Supplementary Exercises

**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

Projects

Supplementary Exercises

**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

Projects

Supplementary Exercises

**10. Finite-State Markov Chains (Online Only)**

Introductory Example: Googling Markov Chains

10.1 Introduction and Examples

10.2 The Steady-State Vector and Google's PageRank

10.3 Communication Classes

10.4 Classification of States and Periodicity

10.5 The Fundamental Matrix

10.6 Markov Chains and Baseball Statistics

**Appendices**

A. Uniqueness of the Reduced Echelon Form

B. Complex Numbers