Preface.

Features of the Text.

**1. Systems of Linear Equations.**

1.1 The Vector Space of m x n Matrices.

The Space R^{n}.

Linear Combinations and Linear Dependence.

What Is a Vector Space?

Why Prove Anything?

True-False Questions.

Exercises.

1.1.1 Computer Projects.

Exercises.

1.1.2 Applications to Graph Theory I.

Self-Study Questions.

Exercises.

1.2 Systems.

Rank: The Maximum Number of Linearly Independent Equations.

True-False Questions.

Exercises.

1.2.1 Computer Projects.

Exercises.

1.2.2 Applications to Circuit Theory.

Self-Study Questions.

Exercises.

1.3 Gaussian Elimination.

Spanning in Polynomial Spaces.

Computational Issues: Pivoting.

True-False Questions.

Exercises.

Computational Issues: Flops.

1.3.1 Computer Projects.

Exercises.

1.3.2 Applications to Traffic Flow.

Self-Study Questions.

Exercises.

1.4 Column Space and Nullspace.

Subspaces.

Subspaces of Functions.

True-False Questions.

Exercises.

1.4.1 Computer Projects.

Exercises.

1.4.2 Applications to Predator-Prey Problems.

Self-Study Questions.

Exercises.

Chapter Summary.

**2. Linear Independence and Dimension.**

2.1 The Test for Linear Independence.

Bases for the Column Space.

Testing Functions for Independence.

True-False Questions.

Exercises.

2.1.1 Computer Projects.

2.2 Dimension.

True-False Questions.

Exercises.

2.2.1 Computer Projects.

Exercises.

2.2.2 Applications to Calculus.

Self-Study Questions.

Exercises.

2.2.3 Applications to Differential Equations.

Self-Study Questions.

Exercises.

2.2.4 Applications to the Harmonic Oscillator.

Self-Study Questions.

Exercises.

2.2.5 Computer Projects.

Exercises.

2.3 Row Space and the Rank-Nullity Theorem.

Bases for the Row Space.

Rank-Nullity Theorem.

Computational Issues: Computing Rank.

True-False Questions.

Exercises.

2.3.1 Computer Projects.

Chapter Summary.

**3. Linear Transformations.**

3.1 The Linearity Properties.

True-False Questions.

Exercises.

3.1.1 Computer Projects.

3.1.2 Applications to Control Theory.

Self-Study Questions.

Exercises.

3.2 Matrix Multiplication (Composition).

Partitioned Matrices.

Computational Issues: Parallel Computing.

True-False Questions.

Exercises.

3.2.1 Computer Projects.

3.2.2 Applications to Graph Theory II.

Self-Study Questions.

Exercises.

3.3 Inverses.

Computational Issues: Reduction vs. Inverses.

True-False Questions.

Exercises.

Ill Conditioned Systems.

3.3.1 Computer Projects.

Exercises.

3.3.2 Applications to Economics.

Self-Study Questions.

Exercises.

3.4 The LU Factorization.

Exercises.

3.4.1 Computer Projects.

Exercises.

3.5 The Matrix of a Linear Transformation.

Coordinates.

Application to Differential Equations.

Isomorphism.

Invertible Linear Transformations.

True-False Questions.

Exercises.

3.5.1 Computer Projects.

Chapter Summary.

**4. Determinants.**

4.1 Definition of the Determinant.

4.1.1 The Rest of the Proofs.

True-False Questions.

Exercises.

4.1.2 Computer Projects.

4.2 Reduction and Determinants.

Uniqueness of the Determinant.

True-False Questions.

Exercises.

4.2.1 Application to Volume.

Self-Study Questions.

Exercises.

4.3 A Formula for Inverses.

Cramer’s Rule.

True-False Questions.

Exercises 273.

Chapter Summary.

**5. Eigenvectors and Eigenvalues.**

5.1 Eigenvectors.

True-False Questions.

Exercises.

5.1.1 Computer Projects.

5.1.2 Application to Markov Processes.

Exercises.

5.2 Diagonalization.

Powers of Matrices.

True-False Questions.

Exercises.

5.2.1 Computer Projects.

5.2.2 Application to Systems of Differential Equations.

Self-Study Questions.

Exercises.

5.3 Complex Eigenvectors.

Complex Vector Spaces.

Exercises.

5.3.1 Computer Projects.

Exercises.

Chapter Summary.

**6. Orthogonality.**

6.1 The Scalar Product in R^{n}.

Orthogonal/Orthonormal Bases and Coordinates.

True-False Questions.

Exercises.

6.1.1 Application to Statistics.

Self-Study Questions.

Exercises.

6.2 Projections: The Gram-Schmidt Process.

The QR Decomposition 334.

Uniqueness of the *Q*R-factoriaition.

True-False Questions.

Exercises.

6.2.1 Computer Projects.

Exercises.

6.3 Fourier Series: Scalar Product Spaces.

Exercises.

6.3.1 Computer Projects.

Exercises.

6.4 Orthogonal Matrices.

Householder Matrices.

True-False Questions.

Exercises.

6.4.1 Computer Projects.

Exercises.

6.5 Least Squares.

Exercises.

6.5.1 Computer Projects.

Exercises.

6.6 Quadratic Forms: Orthogonal Diagonalization.

The Spectral Theorem.

The Principal Axis Theorem.

True-False Questions.

Exercises.

6.6.1 Computer Projects.

Exercises.

6.7 The Singular Value Decomposition (SVD).

Application of the SVD to Least-Squares Problems.

True-False Questions.

Exercises.

Computing the SVD Using Householder Matrices.

Diagonalizing Symmetric Matrices Using Householder Matrices.

6.8 Hermitian Symmetric and Unitary Matrices.

True-False Questions.

Exercises.

Chapter Summary.

**7. Generalized Eigenvectors.**

7.1 Generalized Eigenvectors.

Exercises.

7.2 Chain Bases.

Jordan Form.

True-False Questions.

Exercises.

The Cayley-Hamilton Theorem.

Chapter Summary.

**8. Numerical Techniques.**

8.1 Condition Number.

Norms.

Condition Number.

Least Squares.

Exercises.

8.2 Computing Eigenvalues.

Iteration.

The *Q*R Method.

Exercises.

Chapter Summary.

Answers and Hints.

Index.