Module Content

Matrices: Basic definitions, matrix addition, scalar multiplication, matrix multiplication, transposes, symmetric and antisymmetric matrices, traces, determinants, inverses using cofactors, eigenvalues and eigenvectors, orthogonal diagonalisation of symmetric matrices.

Coordinate geometry: Equations of straight lines in 2D, gradients, parallel and perpendicular lines, distances, midpoints, equations of circles, equations of lines and planes in 3D.

Transformations: Properties and types of transformations. Matrix representations of transformations in 2D and 3D. Isometries, symmetries and rigid motions in 2D and 3D.

Conics and Quadrics: Algebraic definitions of conic sections and quadric surfaces, basic properties, matrix representations, transformations, classification. 

Permutations: Permutations as mappings, properties of sets of permutations, representations in two-row and cycle notation, orders, inverses, transpositions, conjugation, representation of transformations as permutation groups. 

Colouring Problems: Expressing colouring problems of 2D and 3D shapes algebraically. Fixed sets. Using Burnside’s (Colouring) Theorem to solve colouring problems in 2D and 3D. 

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials. 

Assessment Methods

The module is assessed through a Portfolio of exercises and an examination.