This module aims to introduce the abstract concepts of vector spaces and linear maps, and show that familiar properties of matrices are special cases of more general results. Fundamental concepts such as basis, dimension, subspace and rank will be introduced and made rigorous.


- An abstract definition and examples of vector spaces
- Subspaces, spans and related results
- Linearly dependent and linearly independent sets and related results
- Bases, dimension and related results
- Linear mappings, the image and the kernel, and related results
- Coordinates of vectors, matrices of linear mappings, change of basis
- The concept of the rank of a matrix and of a linear mapping

On completion of the course students should be able to:

- Read and understand advanced abstract mathematical definitions in textbooks and other sources
- Prove simple properties of linear spaces from axioms
- Check whether a set of vectors is a basis
- Check whether a mapping is a linear mapping
- Check whether a linear mapping is onto and whether it is one-to-one
- Find a matrix of a linear mapping
- Change a basis and recalculate the coordinates of vectors and the matrices of mappings