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Contents
Preface ix
1 Matrices and Systems of Equations 1
1.1Systems of Linear Equations 1
1.2 RowEchelonForm11
1.3Matrix Arithmetic 27
1.4Matrix Algebra 46
1.5Elementary Matrices 60
1.6Partitioned Matrices 70
MATLABExercises80
ChapterTestA¡ªTrueorFalse84
Chapter Test B 85
2 Determinants87
2.1The Determinant of a Matrix 87
2.2Properties of Determinants 94
2.3Additional Topics and Applications 101
MATLABExercises109
ChapterTestA¡ªTrueorFalse111
Chapter Test B 111
3 Vector Spaces112
3.1 Definition and Examples 112
3.2 Subspaces 119
3.3 Linear Independence 130
3.4 Basis and Dimension 141
3.5 Change of Basis 147
3.6 Row Space and Column Space 157
MATLABExercises165
ChapterTestA¡ªTrueorFalse166
Chapter Test B 167
4 Linear Transformations169
4.1De.nition and Examples 169
4.2Matrix Representations of Linear Transformations 178
4.3Similarity 192
MATLABExercises198
ChapterTestA¡ªTrueorFalse199
Chapter Test B 200
5 Orthogonality201
5.1The Scalar Product in Rn 202
5.2Orthogonal Subspaces 217
5.3Least Squares Problems 225
5.4Inner Product Spaces 238
5.5Orthonormal Sets 247
5.6The Gram¨CSchmidt Orthogonalization Process 266
5.7Orthogonal Polynomials 275
MATLABExercises283
ChapterTestA¡ªTrueorFalse285
Chapter Test B 285
6 Eigenvalues287
6.1Eigenvaluesand Eigenvectors288
6.2Systems of Linear Differential Equations 301
6.3Diagonalization 312
6.4Hermitian Matrices 330
6.5The Singular Value Decomposition 342
6.6Quadratic Forms 356
6.7Positive De.nite Matrices 370
6.8Nonnegative Matrices 377
MATLABExercises387
ChapterTestA¡ªTrueorFalse393
Chapter Test B 393
7 Numerical Linear Algebra395
7.1Floating-Point Numbers 396
7.2Gaussian Elimination 404
7.3PivotingStrategies409
7.4Matrix Norms and Condition Numbers 415
7.5Orthogonal Transformations 429
7.6The Eigenvalue Problem 440
7.7Least Squares Problems 451
MATLABExercises463
ChapterTestA¡ªTrueorFalse468
Chapter Test B 468
8 Iterative MethodsOnline.
8.1Basic Iterative Methods
9 Canonical FormsOnline.
9.1Nilpotent Operators
9.2The Jordan Canonical Form
Appendix:MATLAB471
Bibliography 483
Answers to Selected Exercises 486
Index 499
. Online: The supplemental Chapters 8 and 9 can be downloaded from the Internet. See the section of the Preface on supplementary materials.

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