**C H A P T E R 1** Systems of Linear Equations and Matrices

1.1 Introduction to Systems of Linear Equations

1.2 Gaussian Elimination

1.3 Matrices and Matrix Operations

1.4 Inverses; Algebraic Properties of Matrices

1.5 Elementary Matrices and a Method for Finding *A*−1

1.6 More on Linear Systems and Invertible Matrices

1.7 Diagonal, Triangular, and Symmetric Matrices

1.8 Matrix Transformations

1.9 **Applications of Linear Systems**

• Network Analysis (Traffic Flow)

• Electrical Circuits

• Balancing Chemical Equations

• Polynomial Interpolation

1.10 **Application:** Leontief Input-Output Models

**C H A P T E R 2** Determinants

2.1 Determinants by Cofactor Expansion

2.2 Evaluating Determinants by Row Reduction

2.3 Properties of Determinants; Cramer’s Rule

**C H A P T E R 3** Euclidean Vector Spaces

3.1 Vectors in 2-Space, 3-Space, and *n*-Space

3.2 Norm, Dot Product, and Distance in *R**n*

3.3 Orthogonality

3.4 The Geometry of Linear Systems

3.5 Cross Product

**C H A P T E R 4** General Vector Spaces

4.1 Real Vector Spaces

4.2 Subspaces

4.3 Linear Independence

4.4 Coordinates and Basis

4.5 Dimension

4.6 Change of Basis

4.7 Row Space, Column Space, and Null Space

4.8 Rank, Nullity, and the Fundamental Matrix Spaces

4.9 Basic Matrix Transformations in *R*2 and *R*3

4.10 Properties of Matrix Transformations

4.11 **Application:** Geometry of Matrix Operators on *R*2

**C H A P T E R 5** Eigenvalues and Eigenvectors

5.1 Eigenvalues and Eigenvectors

5.2 Diagonalization

5.3 Complex Vector Spaces

5.4 **Application:** Differential Equations

5.5 **Application:** Dynamical Systems and Markov Chains

**C H A P T E R 6** Inner Product Spaces

6.1 Inner Products

6.2 Angle and Orthogonality in Inner Product Spaces

6.3 Gram–Schmidt Process; *QR*-Decomposition

6.4 Best Approximation; Least Squares

6.5 **Application:** Mathematical Modeling Using Least Squares

6.6 **Application:** Function Approximation; Fourier Series

**C H A P T E R 7** Diagonalization and Quadratic Forms

7.1 Orthogonal Matrices

7.2 Orthogonal Diagonalization

7.3 Quadratic Forms

7.4 Optimization Using Quadratic Forms

7.5 Hermitian, Unitary, and Normal Matrices

**C H A P T E R 8** General Linear Transformations

8.1 General Linear Transformation

8.2 Compositions and Inverse Transformations

8.3 Isomorphism

8.4 Matrices for General Linear Transformations

8.5 Similarity

**C H A P T E R 9** Numerical Methods

9.1 *LU*-Decompositions

9.2 The Power Method

9.3 Comparison of Procedures for Solving Linear Systems

9.4 Singular Value Decomposition

9.5 **Application:** Data Compression Using Singular Value Decomposition

**A P P E N D I X A** Working with Proofs

**A P P E N D I X B** Complex Numbers

Answers to Exercises

Index