Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching cycle |
---|---|
A | Autumn Term 2018-19 to Spring Term 2018-19 |
Linear Algebra underpins a very significant part of mathematical modelling, and to be successfully applied in the sciences it is necessary to understand both the theory and the practice of using linear algebra. This module will cover fundamental material and address applications and techniques which students will subsequently be able to draw on in various contexts.
Subject content
Definition of abstract vector spaces over R and C: axioms and simple consequences thereof. Connection with R^n and C^n; examples of vector spaces of functions.
Subspaces
Linear combinations: the span, linear dependence and linear independence.
Bases and dimension.
Co-ordinates: change of co-ordinate matrix when basis is changed.
Linear transformations: Kernel, Image, characterisation of 1-1 and onto; rank-nullity theorem; isomorphisms; every finite-dimensional vector space is isomorphic to some F^n.
Co-ordinates again: matrix representation of linear maps; isomorphisms represented by invertible matrices.
Eigenvalues and eigenvectors: diagonalisation of a linear map on a finite-dimensional space; criterion for diagonalisability in terms of eigenvectors;
Revision of determinants; characteristic polynomials and eigenvalues.
Real inner product spaces: length, Cauchy-Schwarz and triangle inequality; orthogonal vectors, orthogonal bases; isometries correspond to orthogonal matrices.
Hermitian inner product spaces: unitary matrices and results of previous section adapted to this setting; every Hermitian symmetric matrix is diagonalisable by a unitary transformation; simultaneous diagonalisation of commuting Hermitian symmetric matrices.
Orthogonal subspaces: projections, orthogonal projections and decompositions.
Use of numerical algorithms (such Gauss-Jordan elimination, QR decomposition) in Linear Algebra, using MATLAB or equivalent software.
Academic and graduate skills
It is hard to overstate the importance of linear algebra in a mathematician’s toolkit. Techniques and results from linear algebra are used across the full spectrum of mathematics and its applications, both in an academic setting and in the wider world. To take an example, as well as having concrete applications in all three of our second year streams, the theory of eigenvalues and eigenvectors is essential in Google’s PageRank algorithm.
The theoretical material is developed in the lectures in Autumn term. In Spring term there are 5 practical classes which focus on understanding numerical algorithms and the practical application of linear algebra to mathematical modelling.
Task | Length | % of module mark |
---|---|---|
Essay/coursework Computing Assignment |
N/A | 20 |
University - closed examination Linear Algebra for the Natural Sciences |
1.5 hours | 80 |
None
Task | Length | % of module mark |
---|---|---|
Essay/coursework Computing Assignment |
N/A | 20 |
University - closed examination Linear Algebra for the Natural Sciences |
1.5 hours | 80 |
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Coronavirus (COVID-19): changes to courses
The 2020/21 academic year will start in September. We aim to deliver as much face-to-face teaching as we can, supported by high quality online alternatives where we must.
Find details of the measures we're planning to protect our community.