E600 Graduate Mathematics
An Introduction to Graduate-Level Mathematics
This course provides you with an introduction to the mathematical tools you need for your Master’s studies at the University of Mannheim.
It consists of five chapters (plus an additional introductory chapter) that cover all the basics from Real Analysis and Linear Algebra to Statistics. It is recommended to read at least the slides before each lecture (further detail, particularly for later reference, can be found in the Companion Script).
Each lecture is accompanied by a Problem Set that helps you practice each chapter’s contents. Problem sets are voluntary, can be worked on individually or in groups and will be discussed in class, after which solutions will be uploaded here. There is no need to hand in your solutions.
Chapter 0
Introduction • The Fundamentals of Mathematics
- 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