E600 Graduate Mathematics
Chapter 0
Introduction • The Fundamentals of Mathematics
Slides
Notes
- Mathematical notation and the basics of formal logic
- Fundamental definitions of set theory
- Terminology and fundamentals of functions
- Basic definitions of convergence and continuity
- Archetypes of proofs
Chapter 1
Linear Algebra I • Vector Spaces
- The basic concept of the general, formal vector space concept
- The details of the widely used Euclidean Vector Space
- Mathematical distance functions and their properties
- Limits and continuity beyond univariate real-valued functions
- Key properties of general sets (open/closed, bounded, convex)
Chapter 2
Linear Algebra II • Matrix Algebra
- The formal matrix concept and key definitions/types of matrices
- The matrix-based linear independence test
- Matrix inversion and its usefulness for solving equation systems
- Elementary operations and the Gauss-Jordan algorithm
- Key concepts related to matrices: rank, determinant, eigenvalues, definiteness
Chapter 3
Analysis I • Multivariate Calculus
- A formal introduction to multi-dimensional functions
- Key function properties: invertability, convexity (and concavity)
- Multivariate differentiation: Formal definition and derivation, Application
- Multivariate integration: concept and key theorems
Chapter 4
Analysis II • Optimization
- The formal basics of mathematical optimization
- Unconstrained optimization and its justification
- Optimization with one equality constraint
- Generalization to more complex problems (multiple constraints, inequalities)
- Solution techniques: Simplification, Lagrange, Karush-Kuhn-Tucker
Chapter 5
Statistics • Introduction to Probability Theory & Econometrics
- Basics from probability theory: outcomes, event spaces, probability spaces
- Random variables and their properties
- Matrix inversion and its usefulness for solving equation systems
- Stochastic and propabilistic convergence
- Weak Law of Large Numbers, Central Limit Theorem