Algebra 2
Overview
- Credit value: 30 credits at Level 5
- Convenor and tutor: Professor Sarah Hart
- 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
- Fields, vector spaces, subspaces
- Linear independence, spanning sets, basis, the Steinitz exchange lemma and dimension
- Linear transformations and matrices
- Image, kernel and the rank-nullity formula
- Determinants of square matrices, inverses and Cramer’s rule
- Systems of linear equations and Gaussian elimination
Groups
- Binary operations, the properties of commutativity and associativity
- Identity elements and inverses
- Cayley tables
- Definition of a group
- Examples from geometry, permutations, matrices and number sets
- Homomorphisms and isomorphisms
- Cyclic groups and abelian groups
- Orders of elements and groups
- Subgroups and Lagrange’s Theorem
- Actions, G-sets, the orbit-stabilizer theorem, the orbit-counting lemma and applications such as colouring problems
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.