Linear algebra and its applications

Gilbert Strang : Linear Algebra and It _Applications 4ed ... 1
Contents ... 3
Preface ... 6
Chapter 1 Matrices and Gaussian Elimination ... 11
	1.1 Introduction ... 11
	1.2 The Geometry of Linear Equations ... 14
		Column Vectors and Linear Combinations ... 16
		The Singular Case ... 18
	1.3 An Example of Gaussian Elimination ... 23
		The Breakdown of Elimination ... 24
		The Cost of Elimination ... 25
	1.4 Matrix Notation and Matrix Multiplication ... 31
		Multiplication of a Matrix and a Vector ... 32
		The Matrix Form of One Elimination Step ... 34
		Matrix Multiplication ... 35
	1.5 Triangular Factors and Row Exchanges ... 46
		One Linear System = Two Triangular Systems ... 50
		Row Exchanges and Permutation Matrices ... 51
		Elimination in a Nutshell: PA = LU ... 53
	1.6 Inverses and Transposes ... 60
		The Calculation of A^?1: The Gauss-Jordan Method ... 62
		Invertible = Nonsingular (n pivots) ... 64
		The Transpose Matrix ... 65
		Symmetric Matrices ... 66
		Symmetric Products R^TR, RR^T, and LDL^T ... 67
	1.7 Special Matrices and Applications ... 76
		Roundoff Error ... 79
	Review Exercises ... 82
Chapter 2 Vector Spaces ... 87
	2.1 Vector Spaces and Subspaces ... 87
		The Column Space of A ... 89
		The Nullspace of A ... 91
	2.2 Solving Ax = 0 and Ax = b ... 96
		Echelon Form U and Row Reduced Form R ... 97
		Pivot Variables and Free Variables ... 99
		Solving Ax = b, Ux = c, and Rx = d ... 101
		Another Worked Example ... 104
	2.3 Linear Independence, Basis, and Dimension ... 113
		Spanning a Subspace ... 116
		Basis for a Vector Space ... 117
		Dimension of a Vector Space ... 118
	2.4 The Four Fundamental Subspaces ... 125
		Existence of Inverses ... 131
		Matrices of Rank 1 ... 133
	2.5 Graphs and Networks ... 139
		Spanning Trees and Independent Rows ... 142
		The Ranking of Football Teams ... 143
		Networks and Discrete Applied Mathematics ... 145
	2.6 Linear Transformations ... 150
		Transformations Represented by Matrices ... 153
		Rotations Q, Projections P, and Re?ections H ... 156
	Review Exercises ... 164
Chapter 3 Orthogonality ... 169
	3.1 Orthogonal Vectors and Subspaces ... 169
		Orthogonal Vectors ... 170
		Orthogonal Subspaces ... 172
		The Matrix and the Subspaces ... 175
	3.2 Cosines and Projections onto Lines ... 181
		inner products and cosines ... 182
		Projection onto a Line ... 183
		Projection Matrix of Rank 1 ... 185
		Transposes from Inner Products ... 186
	3.3 Projections and Least Squares ... 190
		Least Squares Problems with Several Variables ... 191
		The Cross-Product Matrix A^TA ... 193
		Projection Matrices ... 194
		Least-Squares Fitting of Data ... 195
		Weighted Least Squares ... 198
	3.4 Orthogonal Bases and Gram-Schmidt ... 205
		Orthogonal Matrices ... 206
		Rectangular Matrices with Orthogonal Columns ... 208
		The Gram-Schmidt Process ... 211
		The Factorization A = QR ... 213
		Function Spaces and Fourier Series ... 214
	3.5 The Fast Fourier Transform ... 221
		Complex Roots of Unity ... 222
		The Fourier Matrix and Its Inverse ... 224
		The Fast Fourier Transform ... 226
		The Complete FFT and the Butter?y ... 228
	Review Exercises ... 231
Chapter 4 Determinants ... 235
	4.1 Introduction ... 235
	4.2 Properties of the Determinant ... 237
	4.3 Formulas for the Determinant ... 246
		Expansion of detA in Cofactors ... 249
	4.4 Applications of Determinants ... 257
	Review Exercises ... 268
Chapter 5 Eigenvalues and Eigenvectors ... 270
	5.1 Introduction ... 270
		The Solution of Ax =?x ... 272
		Summary and Examples ... 274
		Eigshow ... 277
	5.2 Diagonalization of a Matrix ... 283
		Examples of Diagonalization ... 285
		Powers and Products: A^k and AB ... 286
	5.3 Difference Equations and Powers A^k ... 293
		Fibonacci Numbers ... 293
		Markov Matrices ... 296
		Stability of uk+1 = Auk ... 298
		Positive Matrices and Applications in Economics ... 299
	5.4 Differential Equations and e^At ... 306
		stability of differential equations ... 310
		Second-Order Equations ... 314
	5.5 Complex Matrices ... 322
		Complex Numbers and Their Conjugates ... 322
		Lengths and Transposes in the Complex Case ... 324
		Hermitian Matrices ... 325
		Unitary Matrices ... 328
	5.6 Similarity Transformations ... 335
		Change of Basis = Similarity Transformation ... 337
		Triangular Forms with a Unitary M ... 339
		Diagonalizing Symmetric and Hermitian Matrices ... 340
		The Jordan Form ... 342
	Review Exercises ... 351
Chapter 6 Positive De?nite Matrices ... 355
	6.1 Minima, Maxima, and Saddle Points ... 355
		De?nite versus Inde?nite: Bowl versus Saddle ... 357
		Higher Dimensions: Linear Algebra ... 358
	6.2 Tests for Positive De?niteness ... 362
		Positive De?nite Matrices and Least Squares ... 365
		Semide?nite Matrices ... 365
		Ellipsoids in n Dimensions ... 367
		The Law of Inertia ... 369
		The Generalized Eigenvalue Problem ... 370
	6.3 Singular Value Decomposition ... 377
		Application of the SVD ... 378
	6.4 Minimum Principles ... 386
		Minimizing with Constraints ... 387
		Least Squares Again ... 389
		The Rayleigh quotient ... 389
		Intertwining of the Eigenvalues ... 390
	6.5 The Finite Element Method ... 394
		Trial Functions ... 395
		Linear Finite Elements ... 396
		Eigenvalue Problems ... 397
Chapter 7 Computations with Matrices ... 400
	7.1 Introduction ... 400
	7.2 Matrix Norm and Condition Number ... 401
		Unsymmetric Matrices ... 403
		A Formula for the Norm ... 405
	7.3 Computation of Eigenvalues ... 409
		Tridiagonal and Hessenberg Forms ... 411
		The QR Algorithm for Computing Eigenvalues ... 414
	7.4 Iterative Methods for Ax = b ... 417
Chapter 8 Linear Programming and Game Theory ... 427
	8.1 Linear Inequalities ... 427
		The Feasible Set and the Cost Function ... 428
		Slack Variables ... 430
		The Diet Problem and Its Dual ... 430
		Typical Applications ... 431
	8.2 The Simplex Method ... 432
		The Geometry: Movement Along Edges ... 433
		The Simplex Algorithm ... 435
		The Tableau ... 437
		The Organization of a Simplex Step ... 439
		Karmarkar’s Method ... 441
	8.3 The Dual Problem ... 444
		The Proof of Duality ... 447
		Shadow Prices ... 448
		Interior Point Methods ... 449
		The Theory of Inequalities ... 450
	8.4 Network Models ... 454
		The Marriage Problem ... 456
		Spanning Trees and the Greedy Algorithm ... 458
		Further Network Models ... 459
	8.5 Game Theory ... 461
		Matrix Games ... 463
		The Minimax Theorem ... 464
		Real Games ... 465
Appendix A Intersection, Sum, and Product of Spaces ... 469
	A.1 The Intersection of Two Vector Spaces ... 469
	A.2 The Sum of Two Vector Spaces ... 470
	A.3 The Cartesian Product of Two Vector Spaces ... 471
	A.4 The Tensor Product of Two Vector Spaces ... 471
	A.5 The Kronecker Product A?B of Two Matrices ... 472
	Problem Set A ... 474
Appendix B The Jordan Form ... 476
Appendix C Matrix Factorizations ... 483
Appendix D Glossary: A Dictionary for Linear Algebra ... 485
Appendix E MATLAB Teaching Codes ... 494
Solutions to Selected Exercises ... 497
	Problem Set 1.2, page 9 ... 497
	Problem  Set 1.4,  page  26 ... 498
	Problem Set  1.5  page  39 ... 500
	Problem Set  1.6,  page  52 ... 502
	Problem Set 1.7,  page  63 ... 505
	Problem Set 2.1, page 73 ... 505
	Problem Set 2.2, page 85 ... 506
	Problem Set 2.3, page 98 ... 509
	Problem Set 2.4, page 110 ... 511
	Problem Set 2.5, page 122 ... 512
	Problem Set 2.6, page 133 ... 513
	Problem Set 3.1, page 148 ... 515
	Problem Set 3.2, page 157 ... 516
	Problem Set 3.3, page 170 ... 517
	Problem Set 3.4, page 185 ... 518
	Problem Set 3.5, page 196 ... 519
	Problem Set 4.3, page 206 ... 520
	Problem Set 4.3, page 215 ... 521
	Problem Set 4.4, page 225 ... 523
	Problem Set 5.1, page 240 ... 524
	Problem Set 5.2, page 250 ... 525
	Problem Set 5.3, page 262 ... 527
	Problem Set 5.4, page 275 ... 528
	Problem Set 5.5, page 288 ... 529
	Problem Set 5.6, page 302 ... 531
	Problem Set 6.1, page 316 ... 532
	Problem Set 6.2, page 326 ... 535
	Problem Set 6.3, page 327 ... 537
	Problem Set 6.4, page 344 ... 538
	Problem Set 6.5, page 350 ... 538
	Problem Set 7.2, page 357 ... 539
	Problem Set 7.3, page 365 ... 540
	Problem Set 7.4, page 372 ... 540
	Problem Set 8.1, page 381 ... 541
	Problem Set 8.2, page 391 ... 542
	Problem Set 8.3, page 399 ... 542
	Problem Set 8.4, page 406 ... 543
	Problem Set 8.5, page 413 ... 543
	Problem Set A, page 420 ... 544
	Problem Set B, 427 ... 544