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Mathematik für Informatiker I

8 ECTS
Semester 1

Linear Algebra

Overview

Foundation of vector spaces, matrices, and linear transformations

Learning Objectives

  • Understand vector spaces and subspaces
  • Master matrix operations and their applications
  • Apply linear transformations in computer graphics
  • Solve systems of linear equations
  • Understand basis and dimension
  • Work with eigenvalues and eigenvectors

Practical Applications

Computer Graphics

3D transformations and animations

Example: Game engines use matrices for object positioning

Machine Learning

Data transformations and neural networks

Example: Neural network layers use matrix operations

Practice Problems

  • Solve systems of linear equations using Gaussian elimination
  • Find eigenvalues and eigenvectors of transformation matrices
  • Apply matrix operations to transform 3D objects

Complex Numbers

Overview

Understanding and working with complex numbers and their applications

Learning Objectives

  • Master complex number operations
  • Understand geometric interpretation
  • Apply complex numbers in signal processing
  • Work with polar form and exponentials

Practical Applications

Signal Processing

Fourier transforms and signal analysis

Example: Audio processing in digital systems

Control Systems

System stability analysis

Example: Robotics control systems

Practice Problems

  • Perform complex number arithmetic
  • Convert between polar and rectangular forms
  • Solve equations with complex numbers

Mathematical Logic

Overview

Foundations of mathematical reasoning and proof techniques

Learning Objectives

  • Understand propositional and predicate logic
  • Master different proof techniques
  • Apply logical reasoning to problem-solving
  • Work with quantifiers and logical operators

Practical Applications

Program Verification

Proving program correctness

Example: Formal verification of critical systems

Database Theory

Query optimization and validation

Example: SQL query logical analysis

Practice Problems

  • Construct truth tables
  • Write formal proofs
  • Analyze logical statements

Number Theory

Overview

Properties of integers, prime numbers, and modular arithmetic

Learning Objectives

  • Understand divisibility and prime numbers
  • Master modular arithmetic
  • Apply number theory to cryptography
  • Work with congruences

Practical Applications

Cryptography

Public key encryption systems

Example: RSA algorithm implementation

Hash Functions

Data integrity verification

Example: Blockchain technology

Practice Problems

  • Find GCD using Euclidean algorithm
  • Solve linear congruences
  • Work with prime factorizations

Discrete Mathematics

Overview

Logic, proofs, sets, relations, and combinatorics

Learning Objectives

  • Master mathematical proof techniques
  • Understand set theory and operations
  • Apply combinatorial principles
  • Work with relations and functions

Practical Applications

Algorithm Analysis

Proving algorithm correctness

Example: Using induction to verify recursive algorithms

Cryptography

Number theory for encryption

Example: RSA encryption using modular arithmetic

Practice Problems

  • Prove statements using mathematical induction
  • Solve counting problems using combinatorics
  • Apply set operations to solve problems

Calculus Foundations

Overview

Limits, derivatives, and integrals with CS applications

Learning Objectives

  • Understand limits and continuity
  • Master differentiation rules
  • Apply integration techniques
  • Solve optimization problems

Practical Applications

Optimization

Finding optimal solutions

Example: Gradient descent in machine learning

Computer Graphics

Smooth animations and curves

Example: Bezier curves in graphics software

Practice Problems

  • Find limits of complex functions
  • Solve optimization problems using derivatives
  • Calculate areas using integration