CHAPTER 1  Linear Equations in Linear Algebra  1
Introductory Example: Linear Models in Economics and Engineering  1
1.1	Systems of Linear Equations 2
1.2	Row Reduction and Echelon Forms  14
1.3	Vector Equations  28
1.4	The Matrix Equation Ax = b  40
1.5	Solution Sets of Linear Systems  50
1.6	Applications of Linear Systems  57
1.7	Linear Independence  65
1.8	Introduction to Linear Transformations  73
1.9	The Matrix of a Linear Transformations  82
1.10	Linear Models in Business, Science, and Engineering  92
Supplementary Exercises  102

CHAPTER 2  Matrix Algebra  105
Introductory Example: Computer Models in Aircraft Design  105
2.1	Matrix Operations  107
2.2	The Inverse of a Matrix  118
2.3	Characterizations of Invertible Matrices  128
2.4	Partioned Matrices  134
2.5	Matrix Factorizations  142
2.6	The Leontief Input-Output Modes  152
2.7	Applications to Computer Graphics  158
2.8	Subspaces of Rn  167
2.9	Dimension and Rank  176
Supplementary Exercises  183

CHAPTER 3  Determinants  185
Introductory Example: Determinants in Analytic Geometry  185
3.1	Introduction to Determinants  186
3.2	Properties of Determinants  192
3.3	Cramer’s Rule, Volume, and Linear Transformations  201
Supplementary Exercises  211

CHAPTER 4  Vector Spaces  215
Introductory Example: Space Flight and Control Systems  215
4.1	Vector Spaces and Subspaces  216
4.2	Null Space, Column Spaces, and Linear Transformations  226
4.3	Linearly Independent Sets: Bases  237
4.4	Coordinate Systems  246
4.5	The Dimension of a Vector Space  256
4.6	Rank  262
4.7	Change of Basis  271
4.8	Applications to Difference Equations  277
4.9	Applications to Markov Chains  288
Supplementary Exercises  299

CHAPTER 5  Eigenvalues and Eigenvectors  301
Introductory Example: Dynamical Systems and Spotted Owls  301
5.1	Eigenvectors and Eignevalues  302
5.2	The Characteristic Equation  310
5.3	Diagonalization  319
5.4	Eigenvectors and Linear Transformations  327
5.5	Complex Eigenvalues  335
5.6	Discrete Dynamical Systems  342
5.7	Applications to Differential Equations  353
5.8	Iterative Estimates for Eigenvalues  363
Supplementary Exercises  370

CHAPTER 6  Orthogonality and Least Squares  373
Introductory Example: Readjusting the North American Datum  373
6.1	Inner Product, Length, and Orthogonality  375
6.2	Orthogonal Sets  384
6.3	Orthogonal Projections  394
6.4	The Gram-Schmidt Process  402
6.5	Least-Squares Problems  409
6.6	Applications to Linear Models  419
6.7	Inner Product Spaces  427
6.8	Applications of Inner Product Spaces  436
Supplementary Exercises  444

CHAPTER 7  Symmetric Matrices and Quadratic Forms  447
Introductory Example: Multichannel Image Processing  447
7.1	Diagonalization of Symmetric Matices  449
7.2	Quadratic Forms  455
7.3	Constrained Optimization  463
7.4	The Singular Value Decomposition  471
7.5	Applications to Image Processing and Statistics  482
Supplementary Exercises  444

Appendixes
A  Uniqueness of the Reduced Echelon Form  A1
B  Complex Numbers  A3

Glossary  A9
Answers to Odd-Numbered Exercises  A19
Index  I1