Preface 

ix  
A Note to Students 

xv  

Linear Equations in Linear Algebra 


1  (104) 

Introductory Example: Linear Models in Economics and Engineering 


1  (104) 

Systems of Linear Equations 


2  (12) 

Row Reduction and Echelon Forms 


14  (14) 


28  (12) 

The Matrix Equation Ax = b 


40  (10) 

Solution Sets of Linear Systems 


50  (7) 

Applications of Linear Systems 


57  (8) 


65  (8) 

Introduction to Linear Transformations 


73  (9) 

The Matrix of a Linear Transformation 


82  (10) 

Linear Models in Business, Science, and Engineering 


92  (10) 


102  (3) 


105  (80) 

Introductory Example: Computer Models in Aircraft Design 


105  (80) 


107  (11) 


118  (10) 

Characterizations of Invertible Matrices 


128  (6) 


134  (8) 


142  (10) 

The Leontief InputOutput Model 


152  (6) 

Applications to Computer Graphics 


158  (9) 


167  (9) 


176  (7) 


183  (2) 


185  (30) 

Introductory Example: Determinants in Analytic Geometry 


185  (30) 

Introduction to Determinants 


186  (6) 

Properties of Determinants 


192  (9) 

Cramer's Rule, Volume, and Linear Transformations 


201  (10) 


211  (4) 


215  (86) 

Introductory Example: Space Flight and Control Systems 


215  (86) 

Vector Spaces and Subspaces 


216  (10) 

Null Spaces, Column Spaces, and Linear Transformations 


226  (11) 

Linearly Independent Sets: Bases 


237  (9) 


246  (10) 

The Dimension of a Vector Space 


256  (6) 


262  (9) 


271  (6) 

Applications to Difference Equations 


277  (11) 

Applications to Markov Chains 


288  (10) 


298  (3) 

Eigenvalues and Eigenvectors 


301  (72) 

Introductory Example: Dynamical Systems and Spotted Owls 


301  (72) 

Eigenvectors and Eigenvalues 


302  (8) 

The Characteristic Equation 


310  (9) 


319  (8) 

Eigenvectors and Linear Transformations 


327  (8) 


335  (7) 

Discrete Dynamical Systems 


342  (11) 

Applications to Differential Equations 


353  (10) 

Iterative Estimates for Eigenvalues 


363  (7) 


370  (3) 

Orthogonality and Least Squares 


373  (74) 

Introductory Example: Readjusting the North American Datum 


373  (74) 

Inner Product, Length, and Orthogonality 


375  (9) 


384  (10) 


394  (8) 

The GramSchmidt Process 


402  (7) 


409  (10) 

Applications to Linear Models 


419  (8) 


427  (9) 

Applications of Inner Product Spaces 


436  (8) 


444  (3) 

Symmetric Matrices and Quadratic Forms 


447  

Introductory Example: Multichannel Image Processing 


447  

Diagonalization of Symmetric Matrices 


449  (6) 


455  (8) 


463  (8) 

The Singular Value Decomposition 


471  (11) 

Applications to Image Processing and Statistics 


482  (9) 


491  

The Geometry of Vector Spaces 



Introductory Example: The Platonic Solids 

















Introductory Example: The Berlin Airlift 





Linear ProgrammingGeometric Method 



Linear ProgrammingSimplex Method 







A Uniqueness of the Reduced Echelon Form 


1  (2) 


3  (6) 
Glossary 

9  (10) 
Answers to OddNumbered Exercises 

19  
Index 

1  