Praxis Institute - BUSINESS ANALYTICS C121Introduction to linear Algebra- Gilbert Strang

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Table of Contents 1 Introduction to Vectors 1 1.1 Vectors and Linear Combinations. 2 1.2 Lengths and Dot Products. 11 1.3 Matrices . ......... 22 2 Solving Linear Equations 31 2.1 Vectors a... nd Linear Equations . 31 2.2 The Idea of Elimination . . . 45 2.3 Elimination Using Matrices . 57 2.4 Rules for Matrix Operations 68 2.5 Inverse Matrices. . . . . . . 82 2.6 Elimination = Factorization: A = L U 96 2.7 Transposes and Permutations . . .... 108 3 Vector Spaces and Subspaces 121 3.1 Spaces of Vectors . . . . . . . ....3.2 The Nullspace of A: Solving Ax = 0 133 3.3 The Rank and the Row Reduced Form 145 3.4 The Complete Solution to Ax = b . 156 \ 3.5 Independence, Basis and Dimension 169 3.6 Dimensions of the Four Subspaces 185 4 Orthogonality 196 4.1 Orthogonality of the Four Subspaces . 196 4.2 Projections . . . . . . . . . . . . . . 207 4.3 Least Squares Approximations . . . . 219 4.4 Orthogonal Bases and Gram-Schmidt 231 5 Determinants 245 5.1 The Properties of Determinants . 245 5.2 Permutations and Cofactors . . . 256 5.3 Cramer's Rule, Inverses, and Volumes 270 [Show More]

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