## MA201-5-AU-CO:

Linear Algebra

## Key module for

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 GCF3 Mathematics with Physics (Including Year Abroad),

MSCIG199 Mathematics and Data Science

## Module description

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

- 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 studentsâ€™ 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

- 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. RouchĂ©-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

## Bibliography*

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 | ||

Coursework | Assignment 2 | ||

Exam | 120 minutes during Summer (Main Period) (Main) |

### Overall assessment

Coursework | Exam |
---|---|

20% | 80% |

### Reassessment

Coursework | Exam |
---|---|

20% | 80% |

## External examiner

**1254**hours,

**0 (0%)**hours available to students:

**1254**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s).

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

**Disclaimer:** The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can
be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements,
industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist
of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules.
The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.

The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.