MAT-30045 - Module Specification
School of Computing and Mathematics
Faculty of Natural Sciences

Programme/Approved Electives for 2020/21

Available as a Free Standing Elective

Co-requisites

None

Prerequisites

MAT-20025 Abstract Algebra

Barred Combinations

None

Description for 2020/21

The module builds on the knowledge of vector spaces and rings gained from second year Abstract Algebra and Exploring Algebra and Analysis. Concepts such as linear independence, span basis are generalised from Euclidean space to other vector spaces, such as function spaces, in such a way that seemingly disparate results from different branches of mathematics are sometimes just different specialisations of the same general concept. The module also studies some of the properties of an algebraic object called a ring, again concentrating on generalisations and structure as in the case of vector spaces.

This module will introduce students to more advanced ideas in vector spaces and rings, building on the introduction to these mathematical structures at level 5. The module aims to broaden the scope of these ideas and to prepare students for more advanced study of these topics at Level 7.

Intended Learning Outcomes

recall the definition of a linear transformation and be able to prove and apply associated results, including the use of linear transformations to change between bases in a vector space: 1,2
recall the definitions of eigenvalue and eigenvector of a linear transformation and apply these concepts to, inter alia, the diagonalisation of square matrices: 1,2
define inner products on vector spaces, be able to prove and apply associated results and/or use these to construct orthonormal bases and orthogonal polynomials: 1,2
apply the Gram-Schmidt process to an inner product space: 1,2
define different types of ring, and state and prove associated results: 1,2
recognise and define ideals and Euclidean domains, prove associated results and/or solve associated problems: 2
define polynomial rings and solve associated problems: 2

Study hours

Learning/teaching comprises 30 hours video lectures, 5 hours flipped examples classes, and 2 hours final exam.
Independent study comprises 30 hours examples class preparation, 10 hours for completion of assignment, 20 hours preparation for examination, and 53 hours consolidation of lecture material.

School Rules

None

Description of Module Assessment