Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2025/26
| MATH1000 | Core Mathematics |
MMATH2230, MATH2231 and MATH2080 as well as parts of MATH2022
This module is not approved as a discovery module
This module develops the more abstract ideas of vector spaces and linear transformations, together with introducing the area of discrete mathematics.
This module will provide an introduction to key concepts in Discrete Mathematics and a rigorous foundation to Linear Algebra. It will emphasise connections between these two main themes.
On successful completion of the module students will have demonstrated the following learning outcomes:
1. State basic results about vector spaces over arbitrary fields, subspaces, bases and dimension, reproduce short proofs, and find similar proofs;
2. Test whether certain maps between vector spaces are linear transformations, calculate ranks and nullity, and present the Rank and Nullity Theorem as an example of the First Isomorphism Theorem for vector spaces;
3. State and prove criteria for diagonalising matrices, and find diagonal forms where feasible;
4. Solve counting problems involving binomials, permutations, and the inclusion-exclusion principle;
5. Formulate counting problems as linear difference equations and know some applications;
6. Solve linear difference equations;
7. Apply various elementary graph-theoretic algorithms.
Skills Learning Outcomes
On successful completion of the module students will have demonstrated the following skills learning outcomes:
a. Appreciate their own strengths and weaknesses.
b. Use appropriate sources to aid learning.
c. Communicate using written and spoken media.
d. Manage workload and deadlines through prioritisation and productivity skills.
e. Maintain and uphold academic integrity and professional ethics.
1. Vector spaces over arbitrary fields, subspaces, linear independence, spanning and bases; projections and dimensionality reduction.
2. Linear transformations and their representation by a matrix, rank and nullity theorem.
3. Eigenvalues, characteristic and minimal polynomials, diagonalizability, SVD factorisation, principal eigenvector, and applications such as Markov chains, network centrality and PageRank.
4. Inner product spaces, Gram-Schmidt, real symmetric matrices.
5. Combinatorics: counting problems and their relevance for calculating probability; number of functions between finite sets; the binomial theorem and applications to the number of surjections and derangements;
6. Difference equations: combinatorial problems solvable by difference equations; linear difference equations; some linearizable difference equations; applications.
7. Graph Theory: graphs and networks; adjacency matrices; handshaking lemma; subgraphs; isomorphism of graphs; connected graphs and communities; trees; search algorithms.
Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar:
8. (Linear algebra) Quotient spaces; Cayley-Hamilton Theorem; Jordan form; quadratic forms; determinants of matrices (Leibniz formula, with links to permutations); rotations in real Euclidean space; Perron-Frobenius Theorem; distance types.
9. (Discrete mathematics) Planar graphs, Euler's formula, and planarity tests; Cayley's formula and matrix-tree theorem for spanning trees.
Methods of assessment
The assessment details for this module will be provided at the start of the academic year
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Seminar | 10 | 1 | 10 |
| Private study hours | 146 | ||
| Total Contact hours | 54 | ||
| Total hours (100hr per 10 credits) | 200 | ||
The reading list is available from the Library website
Last updated: 30/04/2025
Errors, omissions, failed links etc should be notified to the Catalogue Team