## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Course MAT-02-11-K-1

## Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U) 8.0 CP 156 h
4 V Lecture 56 h
2 U Exercise class (in small groups) 28 h
(4V+2U) 8.0 CP 84 h 156 h

## Basedata

SWS 4V+2U 8.0 CP = 240 h 1 Sem. in WiSe/SuSe  Bachelor (General) [DE] German Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT) Böhm, Janko, Dr. (WMA | DEPT: MAT) Kunte, Michael, Dr. (WMA | DEPT: MAT) [MAT-Service] Mathematics for other Departments [NORM] Active

## Possible Study achievement

• Verification of study performance: proof of successful participation in the exercise classes (ungraded)
• Details of the examination (type, duration, criteria) will be announced at the beginning of the course.

## Contents

• propositions, sets, methods of proof, mappings, semi-orderings and equivalence relations,
• integers, division with remainder, greatest common divisor and Euclidean algorithm, fundamental theorem of arithmetic, Chinese remainder theorem over ℤ,
• groups, orbit equation, symmetry groups, normal subgroup and quotient group, applications (e.g. counting isomorphism classes of graphs),
• rings, polynomial rings, unity group of ℤ/nℤ, applications (e.g. public key cryptography, Pollard factorization, Diffie-Hellman key exchange), ideals and quotient rings, integral domains and fields, finite fields, Euclidean rings, Chinese remainder theorem, applications (e.g. modular computing, interpolation),
• vector spaces, Gaussian algorithm, bases and dimension, vector space homomorphisms, solving linear systems of equations, representing matrix of a homomorphism, algorithms for kernel and image,
• isomorphisms, base change, application (e.g. wavelet transformation), classification of homomorphisms, homomorphism theorem, applications (e.g. linear codes), determinants, eigenvectors, applications (e.g. page-rank algorithm).

## Literature

• G. Fischer: Lineare Algebra,
• S. Bosch: Lineare Algebra,
• K. Jänich: Linear Algebra,
• J. Böhm: Grundlagen der Algebra und Zahlentheorie,
• B. Kreußler, G. Pfister: Mathematik für Informatiker: Algebra, Analysis, Diskrete Strukturen,
• V. Shoup: A Computational Introduction to Number Theory and Algebra.

## Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

none

None

## References to Course [MAT-02-11-K-1]

Module Name Context
[INF-82-51-M-2] Formal Foundations of Computer Science P: Obligatory 4V+2U, 8.0 LP
[MAT-02-11-M-1] Mathematics for Computer Science Students: Algebraic Structures P: Obligatory 4V+2U, 8.0 LP