《LINEAR ALGEBRA AND LTS APPLICATIONS》

1SYSTEMS OF LINEAR EQUATIONS1

Introductory Example：Linear Models in Economics and Engineering1

1.1 Introduction to Systems of Linear Equations2

1.2 Row Reduction and Echelon Forms13

1.3 Applications of Linear Systems25

Supplementary Exercises34

2VECTOR AND MATRIX EQUATIONS37

Introductory Example：Nutrition Problems37

2.1 Vectors in R”38

2.2 The Equation Ax＝b48

2.3 Solution Sets of Linear Systems56

2.4 Linear Independence63

2.5 Introduction to Linear Transformations71

2.6 The Matrix of a Linear Transformation79

2.7 Applications to Nutrition and Population Movement85

Supplementary Exercises92

3MATRIX ALGEBRA95

Introductory Example：Computer Graphics in Automotive Design95

3.1 Matrix Operations96

3.2 The Inverse of a Matrix107

3.3 Characterizations of Invertible Matrices116

3.4 Partitioned Matrices121

3.5 Matrix Factorizations128

3.6 Iterative Solutions of Linear Systems137

3.7 The Leontief Input-Output Model142

3.8 Applications to Computer Graphics148

Supplementary Exercises158

4DETERMINANTS161

Introductory Example：Determinants in Analytic Geometry161

4.1 Introduction to Determinants162

4.2 Properties of Determinants168

4.3 Cramer’s Rule，Volume，and Linear Transformations176

Supplementary Exercises186

5VECTOR SPACES189

Introductory Example：Space Flight and Control Systems189

5.1 Vector Spaces and Subspaces190

5.2 Null Spaces，Column Spaces，and Linear Transformations200

5.3 Linearly Independent Sets； Bases211

5.4 Coordinate Systems219

5.5 The Dimension of a Vector Space229

5.6 Rank235

5.7 Change of Basis243

5.8 Applications to Difference Equations248

5.9 Applications to Markov Chains259

Supplementary Exercises269

6EIGENVALUES AND EIGENVECTORS271

Introductory Example：Dynamical Systems and Spotted Owls271

6.1 Eigenvectors and Eigenvalues273

6.2 The Characteristic Equation280

6.3 Diagonalization288

6.4 Eigenvectors and Linear Transformations296

6.5 Complex Eigenvalues303

6.6 Applications to Dynamical Systems310

6.7 Iterative Estimates for Eigenvalues321

Supplementary Exercises329

7ORTHOGONALITY AND LEAST-SQUARES331

Introductory Example：Readjusting the North American Datum331

7.1 Inner Product，Length，and Orthogonality333

7.2 Orthogonal Sets342

7.3 Orthogonal Projections352

7.4 The Gram-Schmidt Process359

7.5 Least-Squares Problems366

7.6 Applications to Linear Models376

7.7 Inner Product Spaces384

7.8 Applications of Inner Product Spaces393

Supplementary Exercises401

8SYMMETRIC MATRICES AND QUADRATIC FORMS403

Introductory Example：Multichannel Image Processing403

8.1 Diagonalization of Symmetric Matrices405

8.2 Quadratic Forms411

8.3 Constrained Optimization419

8.4 The Singular Value Decomposition426

8.5 Applications to Image Processing and Statistics435

Supplementary Exercises444

APPENDICES447

A Uniqueness of the Reduced Echelon Form447

B Complex Numbers449

GLOSSARY455

ANSWERS TO ODD-NUMBERED EXERCISES467