Linear Algebra and Its Applications Fourth Edition by Gilbert S...

Linear Algebra and Its Applications Fourth Edition Gilbert Strang Preface 5 1 Matrices and Gaussian Elimination 1 1.1 Introduction 1 1.2 The Geometry of Linear Equations 4 1.3 An Example of Gaussian Elimination 13 1.4 Matrix Notation and Matrix Multiplication 21 1.5 Triangular Factors and Row Exchanges 36 1.6 Inverses and Transposes 50 1.7 Special Matrices and Applications 66 1.8 Review Exercises 72 2 Vector Spaces 77 2.1 Vector Spaces and Subspaces 77 2.2 Solving Ax = 0 and Ax = b 86 2.3 Linear Independence, Basis, and Dimension 103 2.4 The Four Fundamental Subspaces 115 2.5 Graphs and Networks 129 2.6 Linear Transformations 140 2.7 Review Exercises 154 3 Orthogonality 159 3.1 Orthogonal Vectors and Subspaces 159 3.2 Cosines and Projections onto Lines 171 3.3 Projections and Least Squares 180 3.4 Orthogonal Bases and Gram-Schmidt 195 3.5 The Fast Fourier Transform 211 3.6 Review Exercises 221 4 Determinants 225 4.1 Introduction 225 4.2 Properties of the Determinant 227 4.3 Formulas for the Determinant 236 4.4 Applications of Determinants 247 4.5 Review Exercises 258 5 Eigenvalues and Eigenvectors 260 5.1 Introduction 260 5.2 Diagonalization of a Matrix 273 5.3 Difference Equations and Powers Ak 283 5.4 Differential Equations and eAt 296 5.5 Complex Matrices 312 5.6 Similarity Transformations 325 5.7 Review Exercises 341 6 Positive Definite Matrices 345 6.1 Minima, Maxima, and Saddle Points 345 6.2 Tests for Positive Definiteness 352 6.3 Singular Value Decomposition 367 6.4 Minimum Principles 376 6.5 The Finite Element Method 384 7 Computations with Matrices 390 7.1 Introduction 390 7.2 Matrix Norm and Condition Number 391 7.3 Computation of Eigenvalues 399 7.4 Iterative Methods for Ax = b 407 8 Linear Programming and Game Theory 417 8.1 Linear Inequalities 417 8.2 The Simplex Method 422 8.3 The Dual Problem 434 8.4 Network Models 444 8.5 Game Theory 451 A Intersection, Sum, and Product of Spaces 459 A.1 The Intersection of Two Vector Spaces 459 A.2 The Sum of Two Vector Spaces 460 A.3 The Cartesian Product of Two Vector Spaces 461 A.4 The Tensor Product of Two Vector Spaces 461 A.5 The Kronecker Product A­B of Two Matrices 462 B The Jordan Form 466 C Matrix Factorizations 473 D Glossary: A Dictionary for Linear Algebra 475 E MATLAB Teaching Codes 484 F Linear Algebra in a Nutshell 486