Linear Algebra - 2025 entry

Abstract vector spaces are important objects in linear algebra, which has its origins in solving linear equations over a field such as the rational, real or complex numbers. The elements of a vector space can be somewhat abstract: for example, they can be functions. However, it is precisely this abstraction that makes the theory of vector spaces such a powerful tool. They arise in almost every area of (pure and applied) mathematics and statistics. For example, PDEs (partial differential equations) of some types are just ODEs (ordinary differential equations) in vector spaces of functions, and numerical and data analysis methods consider vector spaces of increasing dimension to approximate function spaces. 

Prerequisite modules: MTH1001, MTH1002 (or equivalent).

This module aims to develop the theories and techniques of modern algebra, particularly in relation to vector spaces and inner product spaces.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1. perform routine computations in linear algebra accurately;

2. state and apply key definitions and results in the theory of linear algebra;

3. prove core theorems in linear algebra;

4. translate problems that are linearisable into an appropriate format and interpret the solutions in the context of the original problem;

Discipline Specific Skills and Knowledge:

5. discuss and use material from this module in the context of the wider mathematics curriculum;

Personal and Key Transferable / Employment Skills and Knowledge:

6. communicate effectively your understanding of this topic;

7. work independently, monitor your own progress, and manage your time effectively to develop your knowledge and skills in this subject.

- vector spaces and subspaces;

- linear independence, spanning sets;

- bases, dimension of vector spaces;

- linear maps, matrices of linear maps, change of basis;

- kernel and image of linear maps;

- rank-nullity theorem;

- generalization of concepts and key results over arbitrary fields;

- characteristic and minimal polynomials; Cayley-Hamilton theorem; Jordan Canonical Form;

- normed and inner product spaces: bilinear forms and inner products; norms; Cauchy-Schwartz inequality; Gram-Schmidt;

- unitary matrices; self-adjoint operators, including the spectral theorem; diagonalisability; dual spaces and examples; adjoint maps.

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Deferrals: Reassessment will be by practical and/or written exams in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by practical exams only worth 40% of the module. As it is a referral, the mark will be capped at 40%.

Web based and Electronic Resources:

• ELE: http://vle.exeter.ac.uk

Reading list for this module: