Algebra 2

• Credit value: 30 credits at Level 5
• Convenor and tutor: Steven Noble
• Assessment: short, problem-based assignments (20%) and a three-hour examination (80%)

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

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.