Test Bank for Linear Algebra and Its Applications, 6th edition by David C. Lay.pdf

,Linear Algebra and Its Applications
B B B B




Fourth Edition B




Gilbert Strang B




x = y +z
B B B B




y
Ax = b B B




Ay = b B B




0
Az = 0 B B

0

,Contents



Preface iv

1 Matrices and Gaussian Elimination
B B B 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B


1.2 The Geometry of Linear Equations . . . . . . . . . . . . . . . . . . . . 4
B B B B B B B B B B B B B B B B B B B B B B B B


1.3 An Example of Gaussian Elimination .......................................................... 13
B B B B


1.4 Matrix Notation and Matrix Multiplication ................................................... 21
B B B B


1.5 Triangular Factors and Row Exchanges ......................................................... 36
B B B B


1.6 Inverses and Transposes ................................................................................. 50
B B


1.7 Special Matrices and Applications ................................................................ 66
B B B


Review Exercises .......................................................................................... 72
B




2 Vector Spaces
B 77
2.1 Vector Spaces and Subspaces ........................................................................ 77
B B B


2.2 Solving Ax = 0 and Ax = b ............................................................................ 86
B B B B B B B


2.3 Linear Independence, Basis, and Dimension............................................... 103
B B B B


2.4 The Four Fundamental Subspaces ................................................................ 115
B B B


2.5 Graphs and Networks .................................................................................. 129
B B


2.6 Linear Transformations ............................................................................. 140
B


Review Exercises ........................................................................................ 154
B




3 Orthogonality 159
3.1 Orthogonal Vectors and Subspaces .............................................................. 159
B B B


3.2 Cosines and Projections onto Lines ............................................................. 171
B B B B


3.3 Projections and Least Squares ...................................................................... 180
B B B


3.4 Orthogonal Bases and Gram-Schmidt.......................................................... 195
B B B


3.5 The Fast Fourier Transform ........................................................................ 211
B B B


Review Exercises ........................................................................................ 221
B



i

, ii CONTENTS

4 Determinants 225
4.1 Introduction . . . . . . . . . . B B B B B B B B B B . B . B .
B . B .
B . B .
B . B . B . B . B . B . B . B . B . B . B . B . B . B . B . 225
4.2 Properties of the Determinant . B B B B . B . B .
B . B .
B . B .
B . B . B . B . B . B . B . B . B . B . B . B . B . B . B . 227
4.3 Formulas for the Determinant . B B B B . B . B .
B . B .
B . B .
B . B . B . B . B . B . B . B . B . B . B . B . B . B . B . 236
4.4 Applications of Determinants . B B B . B . B .
B . B .
B . B .
B . B . B . B . B . B . B . B . B . B . B . B . B . B . B . 247
Review Exercises . . . . . . . B B B B B B B B . B . B .
B . B .
B . B .
B . B . B . B . B . B . B . B . B . B . B . B . B . B . B . 258

5 Eigenvalues and Eigenvectors B B 260
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B . .B 260
5.2 Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . . . B B B B B B B B B B B B B B B B B B B B B B B B B B . .B 273
5.3 Difference Equations and Powers Ak . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B . .
B 283
5.4 Differential Equations and eAt . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B . .
B 296
5.5 Complex Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B . .B 312
5.6 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B . .
B 325
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B . .B 341

6 Positive Definite Matrices
B B 345
6.1 Minima, Maxima, and Saddle Points . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B .B . 345
6.2 Tests for Positive Definiteness . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B .B . 352
6.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B .B . 367
6.4 Minimum Principles . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B .B . 376
6.5 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B .B . 384

7 Computations with Matrices B B 390
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B . B . 390
7.2 Matrix Norm and Condition Number . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B .B . 391
7.3 Computation of Eigenvalues . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B .B . 399
7.4 Iterative Methods for Ax = b . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B .B . 407

8 Linear Programming and Game Theory
B B B B 417
8.1 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B .B .B . 417
8.2 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . .
B B BB B B B B B B B B B B B B B B B B B B B B B B B B . B . B . 422
8.3 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B . B . B . 434
8.4 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . .
B B B B B B B B B B B B B B B B B B B B B B B B B B B .B .B . 444
8.5 Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B BB B B B B B B B B B B B B B B B B B B B B B B B B B B B B . B . B . 451

A Intersection, Sum, and Product of Spaces
B B B 459 B B


A.1 The Intersection of Two Vector Spaces ........................................................ 459
B B B B B


A.2 The Sum of Two Vector Spaces ................................................................... 460
B B B B B


A.3 The Cartesian Product of Two Vector Spaces .............................................. 461
B B B B B B


A.4 The Tensor Product of Two Vector Spaces................................................... 461
B B B B B B


A.5 The Kronecker Product A ⊗ B of Two Matrices ........................................... 462
B B B B B B B B