Module Directory

This module gives an introduction to abstract linear algebra with a focus on the development of basic theory and its relation to matrix theory, including abstract vector spaces, bases, linear maps and diagonalization. There is a strong emphasis in examples that go beyond affine space.

Vectors are ubiquitous in mathematics and, for this reason, one sees vectors in the plane and space in school and our mathematical intuition builds around them. Can we interpret various mathematical structures, such as sets of functions, as spaces of vectors? This is the paradigm of Linear Algebra, which finds applications to many fields including computer vision and economics.

The aims of this module are:

• To understand the basic notions in abstract linear algebra and how they generalise notions they know from real vector spaces (e.g. bases and coordinates).


• To develop students’ critical understanding of some of the main results in linear algebra and how to apply them to specific problems.


• To carry out computations for abstract vector spaces and linear mappings to solve problems in a varied number of settings.

By the end of the module, students will be expected to:

1. Have an understanding of key definitions in linear algebra and awareness of how they interact and support each other.


2. Select and apply relevant theorems to examples and problems, with a special emphasis on abstract vector spaces (such as solutions to ODEs, vector spaces of polynomials, other subspaces of functions and other abstract examples that they have not encountered before in this context).


3. Be able to prove when a given set is a vector space and a given map is linear.


4. Be able to check whether a linear map is onto, one-to-one and an isomorphism, making use of both the definition and the structure of linear maps.


5. Check whether a set of vectors is linearly independent, spanning and/or a basis.


6. Determine the coordinates of vectors for a given basis and find the matrix of a change of basis, as well as using it to recalculate coordinate vectors.


7. Carry out the diagonalisation of an abstract linear map as well as understand the relation between algebraic and geometric multiplicities.


8. Be comfortable working with abstract constructions of vector spaces such as sums, direct sums, intersections, and the Hom functor.

Linear algebra is a fundamental mathematical subject, and its techniques can be applied throughout mathematics as a black box. The general aim of this module is to allow students to recognise when linear algebra techniques apply, through an understanding of the abstract concepts, as well as the practical ability to apply these techniques and solve problems.

Indicative syllabus:

Abstract definition and examples of vector spaces over arbitrary fields.
Subspaces, spans and related results.
Linearly dependent and linearly independent sets and related results.
Bases, dimension and related results.
Rank of a matrix and its equivalent definitions. Rouché-Capelli theorem.
Linear maps, rank of a linear map, image and kernel and related results, rank-nullity theorem, injections, surjections and isomorphisms.
Coordinates of vectors, matrices of linear maps, change of basis.
Diagonalisation of linear maps.
Examples of abstract vector spaces and maps, including spaces of polynomials, functions and solutions of ODEs. Sum, intersection and direct sum of vector spaces. Hom space.

Teaching in the School will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.