Algebra 2

• Credit value: 30 credits at Level 5
• Convenor: Professor Steven Noble
• Assessment: two problem sets (10% each), a short test (10%) and a three-hour examination (70%)

In this module you will gain a thorough grounding in the concepts and techniques of linear algebra, including the key definitions and results of group theory.

Indicative 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

By the end of this module, you will be able to:

• use mathematical techniques
• understand a range of results in mathematics
• appreciate the need for proof in mathematics, and follow and construct mathematical arguments
• understand the importance of assumptions and of where they are used and the possible consequences of their violation
• appreciate the power of generalisation and abstraction in the development of mathematical theories
• show a deeper knowledge of particular areas of mathematics, in particular, linear algebra, group theory and graph theory.