Linear Algebra

Description

Title page Preface CHAPTER 1. INTRODUCTION 1-1 Linear models 1-2 Linear algebra 1-3 Leontief's interindustry model of an economy 1-4 Linear programming 1-5 Graphical solution of a Iinear programming problem in two variab1es 1-6 Regression analysis 1-7 Linear circuit theory 1-8 Other linear models 1-9 The road ahead CHAPTER 2. VECTORS 2-1 Physical motivation for the vector concept 2-2 Operations with vectors 2-3 The scalar product 2-4 Generalization to higher dimensions 2-5 Generalized vector operations 2-6 Euclidean space and the scalar product 2-7 Linear depedence . ' 2-8 The concept of a basis 2-9 Changing a single vector in a basis 2-10 Number of vectors in a basis for En 2-11 Orthogonal bases 2-12 Generalized coordinate systems 2-13 Vector spaces and subspaces CHAPTER 3. MATRICES AND DETERMINANTS 3-1 Matrices 3-2 Matrix operations 3-3 Matrix multiplication-introduction 3-4 Matrix multiplication-further development 3-5 Vectors and matrices 3-6 Identity, scalar, diagonal, and null matrices 3-7 The transpose 3-8 Symmetric and skew-symmetric matrices 3-9 Partitioning of matriees 3-10 Basic notion of a determinant 3-11 General definition of a determinant 3-12 Some properties of determinants 3-13 Expansion by cofactors 3-14 Additional properties of determinants 3-15 Laplace expansion 3-16 Multiplication of determinants 3-17 Determinant of the product of rectangular matrices 3-18 The matrix inverse 3-19 Properties of the inverse 3-20 Computation of the inverse by partitioning 3-21 Product form of the inverse 3-22 Matrix series and the Leontief inverse CHAPTER 4. LINEAR TRANSFORMATIONS, RANK, AND ELEMENTARY TRANSFORMATIONS 4-1 Definition of linear transformations 4-2 Properties of linear transformations 4-3 Rank 4-4 Rank and determinants 4-5 Elementary transformations 4-6 Echelon matrices and rank CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS 5-1 Introduction 5-2 Gaussian elimination 5-3 Cramer' s rule 5-4 Rules of rank 5-5 Further properties 5-6 Homogeneous linear equations 5-7 Geometric interpretation 5-8 Basic solutions CHAPTER 6. CONVEX SETS AND n-DIMENSIONAL GEOMETRY 6-1 Sets 6-2 Point sets 6-3 Lines and hyperplanes 6-4 Convex sets 6-5 The convex hull 6-6 Theorems on separating hyperplanes 6-7 A basic result in linear programming 6-8 Convex hull of extreme points 6-9 Introduction to convex cones 6-10 Convex polyhedral cones 6-11 Linear transformations of regions CHAPTER 7. CHARACTERISTIC VALUE PROBLEMS AND QUADRATIC FORMS 7-1 Characteristic value problems 7-2 Similarity 7-3 Characteristic value problems for symmetric matrices 7-4 Additional properties of the eigenvectors of a symmetric matrix 7-5 Diagonalization of symmetric matrices 7-6 Characteristic value problems for nonsymmetric matrices 7-7 Quadratic forms 7-8 Change of variables 7-9 Definite quadratic forms 7-10 Diagonalization of quadratic-forms 7-11 Diagonalization by completion of the square 7-12 Another set of necessary and suffieient conditions for positive and negative definite forms 7-13 Simultaneous diagonalization of two quadratic fors 7-14 Geometric interpretation; coordinates and bases 7-15 Equivalence and similarity 7-16 Rotation of coordinates; orthogonal transformations Index