- 1 Introduction
- 1.1 Mathematics and OR
- 1.2 Mathematics as a language
- 1.3 The art of making proofs
- 1.3.1 Forward-Backward method
- 1.3.2 Induction Method
- 1.3.3 Contradiction Method
- 1.3.4 Theorem of alternatives
- 1.3.1 Forward-Backward method
- Problems
- Web material
- 2 Preliminary Linear Algebra
- 2.1 Vector Spaces
- 2.1.1 Fields and linear spaces
- 2.1.2 Subspaces
- 2.1.3 Bases
- 2.2 Linear transformations, matrices and change of basis
- 2.2.1 Matrix multiplication
- 2.2.2 Linear transformation
- 2.3 Systems of Linear Equations
- 2.3.1 Gaussian elimination
- 2.3.2 Gauss-Jordan method for inverses
- 2.3.3 The most general case
- 2.4 The four fundamental subspaces
- 2.4.1 The row space of A
- 2.4.2 The column space of A
- 2.4.3 The null space (kernel) of A
- 2.4.4 The left null space of A
- 2.4.5 The Fundamental Theorem of Linear Algebra
- Problems
- Web material
- 2.1 Vector Spaces
- 3 Orthogonality X Contents
- 3.1 Inner Products
- 3.1.1 Norms
- 3.1.2 Orthogonal Spaces
- 3.1.3 Angle between two vectors
- 3.1.4 Projection
- 3.1.5 Symmetric Matrices
- 3.2 Projections and Least Squares Approximations
- 3.2.1 Orthogonal bases
- 3.2.2 Gram-Schmidt Orthogonalization
- 3.2.3 Pseudo (Moore-Penrose) Inverse
- 3.2.4 Singular Value Decomposition
- 3.3 Summary for Ax = b
- Problems
- Web material
- 4 Eigen Values and Vectors
- 4.1 Determinants
- 4.1.1 Preliminaries
- 4.1.2 Properties
- 4.2 Eigen Values and Eigen Vectors
- 4.3 Diagonal Form of a Matrix
- 4.3.1 All Distinct Eigen Values
- 4.3.2 Repeated Eigen Values with Full Kernels
- 4.3.3 Block Diagonal Form
- 4.4 Powers of A
- 4.4.1 Difference equations
- 4.4.2 Differential Equations
- 4.5 The Complex case
- Problems
- Web material
- 4.1 Determinants
- 5 Positive Definiteness
- 5.1 Minima, Maxima, Saddle points
- 5.1.1 Scalar Functions
- 5.1.2 Quadratic forms
- 5.2 Detecting Positive-Definiteness
- 5.3 Semidefinite Matrices
- 5.4 Positive Definite Quadratic Forms
- Problems
- Web material
- 5.1 Minima, Maxima, Saddle points
- 3.1 Inner Products
- 6 Computational Aspects Contents XI
- 6.1 Solution of Ax = b
- 6.1.1 Symmetric and positive definite
- 6.1.2 Symmetric and not positive definite
- 6.1.3 Asymmetric
- 6.2 Computation of eigen values
- Problems
- Web material
- 7 Convex Sets
- 7.1 Preliminaries
- 7.2 Hyperplanes and Polytopes
- 7.3 Separating and Supporting Hyperplanes
- 7.4 Extreme Points
- Problems
- Web material
- 8 Linear Programming
- 8.1 The Simplex Method
- 8.2 Simplex Tableau
- 8.3 Revised Simplex Method
- 8.5 Farkas' Lemma 8.4 Duality Theory Ill
- Problems
- Web material
- 9 Number Systems
- 9.1 Ordered Sets
- 9.2 Fields
- 9.3 The Real Field
- 9.4 ' The Complex Field
- 9.5 Euclidean Space
- 9.6 Countable and Uncountable Sets
- Problems
- Web material
- 6.1 Solution of Ax = b
- 10 Basic Topology
- 10.1 Metric Spaces
- 10.2 Compact Sets
- 10.3 The Cantor Set
- 10.4 Connected Sets
- Problems
- Web material
- 11 Continuity XII Contents
- 11.1 Introduction I
- 11.2 Continuity and Compactness
- 11.3 Uniform Continuity
- 11.4 Continuity and Connectedness
- 11.5 Monotonic Functions
- Problems
- Web material
- 12 Differentiation
- 12.1 Derivatives
- 12.2 Mean Value Theorems
- 12.3 Higher Order Derivatives
- Problems
- Web material
- 13 Power Series and Special Functions
- 13.1 Series
- 13.1.1 Notion of Series
- 13.1.2 Operations on Series
- 13.1.3 Tests for positive series
- 13.2 Sequence of Functions
- 13.3 Power Series
- 13.4 Exponential and Logarithmic Functions
- 13.5 Trigonometric Functions
- 13.6 Fourier Series
- 13.7 Gamma Function
- Problems
- Web material
- 13.1 Series
- 14 Special Transformations
- 14.1 Differential Equations
- 14.2 Laplace Transforms
- 14.3 Difference Equations
- 14.4 Z Transforms
- Problems
- Web material
- Solutions
- Index
rick simeone
(Rick Simeone)
#1