1 Linear Algebra Notations and Fundamental Concepts Matrix Multiplication Operational Properties Linear Space Quadratic Form and Positive Definite Matrix Matrix Calculus 2 Probability Theory Probability Space Random Variable and Its Distribution Numerical Characteristics of Random Variables Conditional Expectation Jenson’s Inequality
The need for a richer language In P, it is not possible to express assertions about elements of a structure. First Order Logic is a considerably richer logic than propositional logic, but yet enjoys many nice mathematical properties
Some Properties 1. ∆n is consistent. 2. Γ ⊆ ∆n ⊆ ∆n+1 ⊆ ∆Γ 3. ∆Γ is complete. 4. If ∆Γ ` A then there exists n ∈ N such that ∆n ` A. 5. A ∈ ∆Γ iff ∆Γ ` A 6. ∆Γ is consistent
