Computational Mathematics

Contents

1. System of Linear Equation
1. Introduction
1.1 Dot Product and Matrix Multiplication
2. Determinant of a Square Matrix
3. Properties of the Determinant Function
4. Cramer’s Rule
5. System of Linear equations and solutions
5.1 Matrix Representation of System of Linear Equations
6. Tree Diagram of Linear Equations
6.1 Definitions: Equivalent System
6.2 Elementary Row Operations
6.3 Algorithm to reduce a matrix into row echelon form
6.4 Gaussian Elimination
6.5 Gauss Jordan Elimination

2. Vector Spaces
1. Introduction
2. Real Vector Spaces
3. Euclidean n-space
4. Subspaces of a Vector Space
5. Linear Combination and Linear Span
6. Linear Independence
7. Basis and Dimension
8. Row Space, Column Space, Null Space
9. Rank and Nullity

3. Eigen Values and Eigen Vectors
1. Introduction
2. Eigen Values and Eigen Vectors
2.1 Procedure to find Eigen Values, Eigen Vectors and Eigen Space
2.2 Method for Finding Roots of the Characteristic Polynomial
2.3 Properties of Eigen Values
2.4 Properties of Eigen Vectors
3. Diagonalization
3.1 Some Important Theorems
3.2 Geometric and Algebraic Multiplicity

4. Boolean Function
1. Introduction
1.1 Types of Relations
2. Digraphs of Relations
2.1 Matrix Representation and Composition of Relations
2.2 Composition of Relations
2.3 Transitive Closure and Warshall’s Algorithm
2.4 Warshall’s Algorithm
3. Poset (Partially Ordered Set)
3.1 Partially Ordered Set (or Poset)
3.2 Comparibility
3.3 Totally Ordered Set
3.4 Hasse Diagram
4. Boolean Functions
4.1 Operations in Boolean Algebra
4.2 Boolean Expressions and Boolean Functions
4.3 Boolean Expression
4.4 Identities of Boolean Algebra
5. Duality
5.1 Definition of Boolean Algebra
6. Representing Boolean Function
6.1 Disjunctive Normal Form (DNF): [or Sum of Products Expansion]
6.2 Conjunctive Normal Form (CNF): [or Product-of-Sums Expansion]

Appendix A: Matrices and Linear Algebra
1. Introduction
2. Matrices
2.1 Algebra of Matrices
2.2 Operations on Matrices
2.3 Columnspace, Nullspace, Rank and Nullity
Appendix B: Linear Equations
1. Introduction
2. Vector Spaces
3. Eigen Values and Eigen Vectors
4. Diagonalization