Algebra 2
Overview
- Credit value: 30 credits at Level 5
- Convenor and tutor: Steven Noble
- Assessment: short, problem-based assignments (20%) and a three-hour examination (80%)
Module description
This module will give you a thorough grounding in the concepts and techniques of linear algebra, in a more general context than in first-year undergraduate modules. You’ll also gain an understanding of the key definitions and results of group theory. It will also prepare you for the Level 6 option Algebra 3.
Indicative module syllabus
Linear Algebra
- Vector spaces and subspaces
- Linear independence, spanning sets, basis and dimension
- Linear transformations and matrices
- Image, kernel and the rank-nullity formula
Groups
- Revision of binary operations
- Definition of a group with examples from geometry, permutations, matrices and number sets
- Homomorphisms and isomorphisms
- Cyclic groups and abelian groups
- Subgroups and Lagrange’s Theorem
Graphs
- Definitions of graphs and classes of graphs
- Trees, Cayley’s theorem and finding minimum weight spanning trees
- Eulerian and Hamiltonian graphs
- The Travelling Salesman Problem
- Connectedness and Menger’s Theorem
- Flows in networks
- Matchings in graphs and Hall’s Theorem
- Stable matchings, optimal assignments and the Hungarian algorithm
Learning objectives
By the end of this module, you will have:
- knowledge and understanding of, and the ability to use, mathematical techniques
- knowledge and understanding of a range of results in mathematics
- an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments
- an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation
- an appreciation of the power of generalisation and abstraction in the development of mathematical theories
- a deeper knowledge of some particular areas of mathematics, in particular, linear algebra and group theory.