Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than deduces. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.
Author(s): Gilbert Strang
Edition: 4th
Publisher: Brooks Cole
Year: 2006
Language: English
Pages: 487
Tags: Linear;Algebra;Pure Mathematics;Mathematics;Science & Math;Calculus;Pure Mathematics;Mathematics;Science & Math;Algebra & Trigonometry;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique;Calculus;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique
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
