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.

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. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.