What is Linear Algebra? 

xiii  

Systems of Linear Equations 


1  (41) 

Biographical Sketch of Carl Friedrich Gauss 


1  (1) 

Introduction to Systems of Linear Equations 


1  (12) 

Gaussian Elimination and GaussJordan Elimination 


13  (14) 

Applications of Systems of Linear Equations 


27  (15) 


37  (1) 


38  (2) 


40  (2) 


42  (69) 

Biographical Sketch of Arthur Cayley 


42  (1) 


42  (13) 

Properties of Matrix Operations 


55  (11) 


66  (13) 


79  (11) 

Applications of Matrix Operations 


90  (21) 


105  (1) 


106  (3) 


109  (2) 


111  (49) 

Biographical Sketch of AugustinLouis Cauchy 


111  (1) 

The Determinant of a Matrix 


111  (9) 

Evaluation of a Determinant Using Elementary Operations 


120  (9) 

Properties of Determinants 


129  (8) 

Applications of Determinants 


137  (23) 


152  (1) 


153  (1) 


153  (5) 

Cumulative Test for Chapters 13 


158  (2) 


160  (90) 

Biographical Sketch of William Rowan Hamilton 


160  (1) 


160  (10) 


170  (8) 

Subspaces of Vector Spaces 


178  (8) 

Spanning Sets and Linear Independence 


186  (12) 


198  (11) 

Rank of a Matrix and Systems of Linear Equations 


209  (15) 

Coordinates and Change of Basis 


224  (11) 

Applications of Vector Spaces 


235  (15) 


244  (1) 


245  (3) 


248  (2) 


250  (75) 

Biographical Sketch of JeanBaptiste Joseph Fourier 


250  (10) 

Length and Dot Product in Rn 


260  (4) 


264  (12) 

Orthonormal Bases: GramSchmidt Process 


276  (12) 

Mathematical Models and Least Squares Analysis 


288  (14) 

Applications of Inner Product Spaces 


302  (23) 


316  (1) 


317  (2) 


319  (3) 

Cumulative Test for Chapters 4 and 5 


322  (3) 


325  (54) 

Biographical Sketch of Emmy Noether 


325  (1) 

Introduction to Linear Transformations 


325  (12) 

The Kernel and Range of a Linear Transformation 


337  (12) 

Matrices for Linear Transformations 


349  (11) 

Transition Matrices and Similarity 


360  (6) 

Applications of Linear Transformations 


366  (13) 


374  (1) 


375  (2) 


377  (2) 

Eigenvalues and Eigenvectors 


379  (1) 

Biographical Sketch of James Joseph Sylvester 


379  (1) 

Eigenvalues and Eigenvectors 


379  (13) 


392  (10) 

Symmetric Matrices and Orthogonal Diagonalization 


402  (12) 

Applications of Eigenvalues and Eigenvectors 


414  (15) 


429  (1) 


430  (2) 


432  (2) 

Cumulative Test for Chapters 6 and 7 


434  
Appendices 

A1  

A Mathematical Induction and Other Forms of Proofs 


A1  

B Computer Algebra Systems and Graphing Calculators 


A9  
Answer Key 

A20  
Index 

A58  