Preface 

xiii  
A Note To Students 

xix  

Linear Equations in Linear Algebra 


1  (96) 

Introductory Example: Linear Models in Economics and Engineering 


1  (96) 

Systems of Linear Equations 


2  (11) 

Row Reduction and Echelon Forms 


13  (14) 


27  (12) 

The Matrix Equation Ax = b 


39  (9) 

Solution Sets of Linear Systems 


48  (10) 


58  (8) 

Introduction to Linear Transformations 


66  (10) 

The Matrix of a Linear Transformation 


76  (9) 

Linear Models in Business, Science, and Engineering 


85  (10) 


95  (2) 


97  (82) 

Introductory Example: Computer Graphics in Automotive Design 


97  (82) 


98  (12) 


110  (10) 

Characterizations of Invertible Matrices 


120  (5) 


125  (8) 


133  (10) 

Iterative Solutions of Linear Systems 


143  (5) 

The Leontief InputOutput Model 


148  (7) 

Applications to Computer Graphics 


155  (10) 


165  (12) 


177  (2) 


179  (30) 

Introductory Example: Determinants in Analytic Geometry 


179  (30) 

Introduction to Determinants 


180  (7) 

Properties of Determinants 


187  (8) 

Cramer's Rule, Volume, and Linear Transformations 


195  (11) 


206  (3) 


209  (86) 

Introductory Example: Space Flight and Control Systems 


209  (86) 

Vector Spaces and Subspaces 


210  (10) 

Null Spaces, Column Spaces, and Linear Transformations 


220  (11) 

Linearly Independent Sets; Bases 


231  (9) 


240  (10) 

The Dimension of a vector Space 


250  (7) 


257  (8) 


265  (6) 

Applications to Difference Equations 


271  (11) 

Applications to Markov Chains 


282  (10) 


292  (3) 

Eigenvalues and Eigenvectors 


295  (72) 

Introductory Example: Dynamical Systems and Spotted Owls 


295  (72) 

Eigenvectors and Eigenvalues 


296  (9) 

The Characteristic Equation 


305  (8) 


313  (8) 

Eigenvectors and Linear Transformations 


321  (8) 


329  (7) 

Discrete Dynamical Systems 


336  (11) 

Applications to Differential Equations 


347  (10) 

Iterative Estimates for Eigenvalues 


357  (8) 


365  (2) 

Orthogonality and LeastSquares 


367  (74) 

Introductory Example: Readjusting the North American Datum 


367  (74) 

Inner Product, Length, and Orthogonality 


369  (9) 


378  (11) 


389  (8) 


397  (7) 


404  (10) 

Applications to Linear Models 


414  (8) 


422  (9) 

Applications of Inner Product Spaces 


431  (8) 


439  (2) 

Symmetric Matrices and Quadratic Forms 


441  

Introductory Example: Multichannel Image Processing 


441  

Diagonalization of Symmetric Matrices 


443  


450  


458  

The Singular Value Decomposition 


466  

Applications to Image Processing and Statistics 


477  


485  
Appendixes 


A Uniqueness of the Reduced Echelon Form 


A1  


A3  
Glossary 

A9  
Answers to OddNumbered Exercises 

A21  
Index 

I1  