Module Catalogue 2025/26

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To provide students with an introduction to modern abstract linear algebra. Building on their existing knowledge of matrix methods, students will experience the benefits of the abstract and rigorous mathematical theory of vector spaces for the deeper understanding of a mathematical subject.

Module Summary

Linear algebra is a fundamental subject that pervades many areas of modern mathematics. On the one hand it is often convenient to replace a complicated problem by a linear approximation which is easier to solve. On the other hand, linear algebra has beautiful applications in coding theory, projective geometry, and many other areas of mathematics and statistics. Initially linear algebra aims to solve systems of linear equations. In the first-year courses this led naturally to matrix algebra. In this module abstraction and generalisation are pushed one level further with the formal introduction of vector spaces and linear maps as a replacement for real n-dimensional space and matrices, respectively. This allows us to consider analogous problems in different settings simultaneously and eventually makes explanations easier and faster. We will need to introduce notions of dimension and basis in this general setting. A guiding question is how to transform matrices (or linear maps) to a simple form in which essential properties can be immediately read off.

Outline Of Syllabus

Vector spaces, span and bases, linear maps and their properties, inner product spaces and change of basis.

Intended Knowledge Outcomes

Students will be able to demonstrate a reasonable understanding of abstract linear algebra. They will be able to reproduce definitions of elementary notions such as vector space, linear map, basis, dimension and inner product space.

Intended Skill Outcomes

Students should have a reasonable grasp of the knowledge outcomes and perform mathematical arguments with these notions. Students should be able to perform calculations related to vector spaces, bases and dimension and use them in problem-solving. The calculational skills include the ability to
i) Calculate the dimension and find bases for various vector spaces.
ii) Write down matrices representing linear maps.
iii) Identify kernel and image of linear maps.
iv) Perform Gram-Schmidt orthonormalisation in explicit examples.



Students will develop skills across the cognitive domain (Bloom’s taxonomy 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Approximately three assignments of roughly equal weight allow the students to develop their problem solving techniques, to practise the methods learnt in the module and to receive feedback; this is thus formative as well as summative assessment. The coursework assignments may be written assignments, computer based assessments or a combination of the two.

General Notes

N/A