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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%)

Module description

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


  • 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


  • 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 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.