Linear Algebra and Its Applications

1. MATRICES AND GAUSSIAN ELIMINATION. Introduction. The Geometry of Linear Equations. An Example of Gaussian Elimination. Matrix Notation and Matrix Multiplication. Triangular Factors and Row Exchanges. Inverses and Transposes. Special Matrices and Applications. Review Exercises. 2. VECTOR SPACES. Vector Spaces and Subspaces. The Solution of m Equations in n Unknowns. Linear Independence, Basis, and Dimension. The Four Fundamental Subspaces. Networks and Incidence Matrices. Linear Transformations. Review Exercises. 3. ORTHOGONALITY. Perpendicular Vectors and Orthogonal Subspaces. Inner Products and Projections onto Lines. Least Squares Approximations. Orthogonal Bases, Orthogonal Matrices, and Gram-Schmidt Orthogonalization. The Fast Fourier Transform. Review and Preview. Review Exercises. 4. DETERMINANTS. Introduction. Properties of the Determinant. Formulas for the Determinant. Applications of Determinants. Review Exercises. 5. EIGENVALUES AND EIGENVECTORS. Introduction. Diagonalization of a Matrix. Difference Equations and the Powers Ak. Differential Equations and the Exponential eAt. Complex Matrices: Symmetric vs. Hermitian. Similarity Transformations. Review Exercises. 6. POSITIVE DEFINITE MATRICES. Minima, Maxima, and Saddle Points. Tests for Positive Definiteness. The Singular Value Decomposition. Minimum Principles. The Finite Element Method. 7. COMPUTATIONS WITH MATRICES. Introduction. The Norm and Condition Number. The Computation of Eigenvalues. Iterative Methods for Ax = b. 8. LINEAR PROGRAMMING AND GAME THEORY. Linear Inequalities. The Simplex Method. Primal and Dual Programs. Network Models. Game Theory. Appendix A: Computer Graphics. Appendix B: The Jordan Form. References. Solutions to Selected Exercises. Index.

1. MATRICES AND GAUSSIAN ELIMINATION.

Introduction.

The Geometry of Linear Equations.

An Example of Gaussian Elimination.

Matrix Notation and Matrix Multiplication.

Triangular Factors and Row Exchanges.

Inverses and Transposes.

Special Matrices and Applications.

Review Exercises.

2. VECTOR SPACES.

Vector Spaces and Subspaces.

The Solution of m Equations in n Unknowns.

Linear Independence, Basis, and Dimension.

The Four Fundamental Subspaces.

Networks and Incidence Matrices.

Linear Transformations.

Review Exercises.

3. ORTHOGONALITY.

Perpendicular Vectors and Orthogonal Subspaces.

Inner Products and Projections onto Lines.

Least Squares Approximations.

Orthogonal Bases, Orthogonal Matrices, and Gram-Schmidt Orthogonalization.

The Fast Fourier Transform.

Review and Preview.

Review Exercises.

4. DETERMINANTS.

Introduction.

Properties of the Determinant.

Formulas for the Determinant.

Applications of Determinants.

Review Exercises.

5. EIGENVALUES AND EIGENVECTORS.

Introduction.

Diagonalization of a Matrix.

Difference Equations and the Powers Ak.

Differential Equations and the Exponential eAt.

Complex Matrices: Symmetric vs. Hermitian.

Similarity Transformations.

Review Exercises.

6. POSITIVE DEFINITE MATRICES.

Minima, Maxima, and Saddle Points.

Tests for Positive Definiteness.

The Singular Value Decomposition.

Minimum Principles.

The Finite Element Method.

7. COMPUTATIONS WITH MATRICES.

Introduction.

The Norm and Condition Number.

The Computation of Eigenvalues.

Iterative Methods for Ax = b.

8. LINEAR PROGRAMMING AND GAME THEORY.

Linear Inequalities.

The Simplex Method.

Primal and Dual Programs.

Network Models.

Game Theory.

Appendix A: Computer Graphics.

Appendix B: The Jordan Form.

References. Solutions to Selected Exercises.

Index.

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