Linear Algebra

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