A Modern Introduction to Linear Algebra

665 Pages·2009·5.983 MB·English

by  Henry Ricardo

A Modern Introduction to
Linear Algebra
© 2010 by Taylor & Francis Group, LLC
© 2010 by Taylor & Francis Group, LLC
A Modern Introduction to
Linear Algebra
HENRY RICARDO

© 2010 by Taylor & Francis Group, LLC
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© 2010 by Taylor & Francis Group, LLC
For my wife, Catherine—‘‘Age cannot wither her,
h nor custom stale her infinite variety.’’— i
and for
Henry and Marta Ricardo Cathy and Mike Corcoran
Tom(cid:1)as and Nicholas Ricardo Christopher Corcoran
Christine and Greg Gritmon

© 2010 by Taylor & Francis Group, LLC
© 2010 by Taylor & Francis Group, LLC
Contents
Anasterisk * marks optionalsections, those discussions notneededfor later work.
Author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Vectors inRn 1
1.1.1 Euclidean n-Space 3
1.1.2 Vector Addition=Subtraction 4
1.1.3 Scalar Multiplication 6
1.1.4 GeometricVectors in R2 and R3 7
1.1.5 AlgebraicProperties 8
1.2 The Inner Productand Norm 12
1.2.1 The Angle between Vectors 16
1.3 SpanningSets 27
1.4 Linear Independence 36
1.5 Bases 45
1.5.1 Coordinates=Change of Basis 53
1.6 Subspaces 58
1.7 Summary 64
2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67
2.1 The Geometry ofSystems of Equationsin R2 and R3 68
2.2 Matrices and EchelonForm 77
2.2.1 The Matrix of Coefficients 77

2.2.2 ElementaryRow Operations 79
2.3 Gaussian Elimination 91
*2.4 Computational Considerations—Pivoting 97
2.5 Gauss–Jordan Elimination and Reduced Row EchelonForm 104
*2.6 Ill-Conditioned Systems of Linear Equations 116
2.7 Rank and Nullity ofa Matrix 123
2.7.1 Row Spaces and Column Spaces 123
2.7.2 The Null Space 132
2.8 Systems ofm Linear Equations in n Unknowns 137
2.8.1 The Solution ofLinear Systems 138
2.9 Summary 154
vii
© 2010 by Taylor & Francis Group, LLC
viii Contents
3 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
3.1 Addition and Subtraction of Matrices 157
3.1.1 Scalar Multiplication 160
3.1.2 Transpose of aMatrix 161
3.2 Matrix–Vector Multiplication 165
3.2.1 Matrix–Vector Multiplication as a Transformation 172
3.3 The Productof Two Matrices 182
3.3.1 The Column View of Multiplication 185
3.3.2 The Row View of Multiplication 188
3.3.3 Positive Powers ofa SquareMatrix 193
3.4 Partitioned Matrices 203
3.5 Inverses ofMatrices 210
3.5.1 Negative Powers of aSquare Matrix 217
3.6 Elementary Matrices 220
3.7 The LUFactorization 234
3.8 Summary 249
4 Eigenvalues, Eigenvectors, and Diagonalization. . . . . . . . . . . . . .253
4.1 Determinants 254
4.1.1 Cramer’s Rule 268
*4.2 Determinants and Geometry 277
4.3 The Manual Calculation of Determinants 284
4.4 Eigenvalues and Eigenvectors 293
4.5 Similar Matrices and Diagonalization 311
4.6 Algebraicand GeometricMultiplicities of Eigenvalues 321
*4.7 The Diagonalization ofRealSymmetric Matrices 326
4.8 The Cayley–Hamilton Theorem (aFirst Look)=the Minimal Polynomial 333
4.9 Summary 344
5 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .347
5.1 Vector Spaces 347
5.2 Subspaces 353

5.2.1 The Sum and Intersection ofSubspaces 357
5.3 Linear Independence and theSpan 365
5.4 Bases and Dimension 373
5.5 Summary 386
6 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389
6.1 Linear Transformations 389
6.2 The Rangeand NullSpace ofa Linear Transformation 399
6.3 The Algebra of Linear Transformations 408
6.4 Matrix Representation of aLinearTransformation 414
6.5 InvertibleLinear Transformations 425
6.6 Isomorphisms 434
6.7 Similarity 441
6.8 SimilarityInvariants ofOperators 453
6.9 Summary 457
© 2010 by Taylor & Francis Group, LLC
Contents ix
7 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461
7.1 Complex Vector Spaces 461
7.2 Inner Products 468
7.3 Orthogonality and OrthonormalBases 477
7.4 The Gram–SchmidtProcess 483
7.5 Unitary Matrices and OrthogonalMatrices 493
7.6 Schur Factorizationand theCayley–HamiltonTheorem 505
7.7 The QRFactorization and Applications 512
7.8 Orthogonal Complements 520
7.9 Projections 530
7.10 Summary 535
8 Hermitian Matrices and Quadratic Forms . . . . . . . . . . . . . . . . . .539
8.1 Linear Functionalsand theAdjoint of an Operator 539
8.2 HermitianMatrices 547
8.3 Normal Matrices 554
*8.4 Quadratic Forms 560
*8.5 Singular Value Decomposition 569
8.6 The Polar Decomposition 579
8.7 Summary 585
Appendix A: Basics of Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .587
Appendix B: Summation and Product Notation . . . . . . . . . . . . . . . . . . .593
Appendix C: Mathematical Induction. . . . . . . . . . . . . . . . . . . . . . . . . .597
Appendix D: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .601
Answers=Hints to Odd-Numbered Problems. . . . . . . . . . . . . . . . . . . . . . .605
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .645

© 2010 by Taylor & Francis Group, LLC
© 2010 by Taylor & Francis Group, LLC