Linear Algebra

The details
Mathematical Sciences
Colchester Campus
Undergraduate: Level 5
Thursday 07 October 2021
Friday 17 December 2021
19 August 2021


Requisites for this module


MA204, MA225, MA301, MA316, MA323

Key module for

BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
MMATG198 Mathematics,
BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year),
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC G1F5 Mathematics with Physics (Including Foundation Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad),
MSCIG199 Mathematics and Data Science

Module description

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.

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.

Module aims

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. More specifically, the aims of the 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.

Module learning outcomes

On completion of the module students should be able to:

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

- 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).

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

- 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.

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

- 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.

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

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

Module information


- 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.

Learning and teaching methods

This module consists of 2 lectures & 1 class each week throughout the autumn term. There are 3 revision sessions given in the summer term.


The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework   Assignment 1   11/11/2021   
Coursework   Assignment 2   16/12/2021   
Exam  120 minutes during Summer (Main Period) (Main) 

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
20% 80%
Module supervisor and teaching staff
Prof Gerald Williams, email:
Professor Gerald Williams



External examiner

Prof Stephen Langdon
Brunel University London
Available via Moodle
Of 1336 hours, 30 (2.2%) hours available to students:
1306 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

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