Module Content

Algebra 

Set Theory: Definition of a set, equality, subsets, power sets (including order). Venn diagrams. Union, intersection, difference, complement, Cartesian product (and properties). 

Mappings: Domain, codomain and range. Surjective, injective and bijective mappings. Sums, products and compositions of mappings. Inverses. 

Equivalence Relations and Classes: Definition and properties of Congruence relations and classes. 

The Integers: Multiples & Divisors. Divisibility. Greatest Common Divisor (and properties of). Euclid’s algorithm. Linear combinations of integers. Prime numbers. Fundamental Theorem of Arithmetic. 

Polynomials: Definition, sum, difference, product. Factor and division. Euclid’s algorithm. Statement of unique factorisation in real and complex numbers. Remainder theorem and its consequences. Fundamental Theorem of Algebra, irreducible polynomials in real and complex numbers, Nth roots of unity and integers. 

Linear Algebra 

Matrices: Definition, order, equality. Special types. Addition, scalar multiplication and transpose. Matrix multiplication. Determinants and basic properties. Inverse of a matrix using cofactors and determinants. 

Systems of Linear Equations: Elementary row operations and reduced echelon form, method for finding inverses. Solution of linear equations by Gaussian Elimination and matrix inversion. 

Eigenvalues and Eigenvectors: Definition. Characteristic equation and polynomial. Independence of eigenvectors. Matrix diagonalization. 

Vector spaces: Definition, subspaces, linear combinations and spanning, linear dependence and independence, basis and dimension. 

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.