Linear Algebra module (MA22006)

Linear algebra is one of the most useful languages in modern mathematics. It is the mathematics behind 3D graphics, machine learning, physical simulations, and the movement of a robot arm. It is also a beautiful subject in its own right, revealing how spaces are built, how transformations behave, and how the right perspective can make a difficult problem simple.

In this module, you will study vector spaces and learn how linear independence, basis, and dimension help identify the essential information in a space. You will also see how a good choice of basis can make a difficult problem easier to solve.

Matrices give a practical way to work with transformations. You will use them to describe rotations, reflections, projections, and changes of coordinates, building a geometric view of how linear algebra works in higher dimensions.

You will also study eigenvalues and eigenvectors, which reveal how a transformation behaves in its simplest directions. These ideas lead to diagonalisation and the Cayley-Hamilton theorem, helping you simplify matrix problems and understand repeated processes.

The module develops both the algebra and the geometry of the subject. Through inner products, orthogonality, and the Gram-Schmidt process, you will learn how to measure length and angle in vector spaces and construct orthogonal bases.

By the end of the module, you will understand why vectors, matrices, transformations, and high-dimensional spaces are central to modern mathematics and to applications across science and technology.

What you will learn

In this module, you will:

• study vector spaces, subspaces, basis, and dimension
• explore linear independence and how vectors can describe higher-dimensional spaces
• use matrices to represent linear transformations such as rotations, projections, and changes of coordinates
• work with inner products, orthogonality, and the Gram-Schmidt process
• study eigenvalues, eigenvectors, diagonalisation, and the Cayley-Hamilton theorem
• explore cases where Jordan blocks are needed to understand matrices that cannot be diagonalised in the usual way
• develop proof and problem-solving skills in linear algebra.

By the end of this module, you will be able to:

• work confidently with vector spaces, bases, and dimension
• determine whether vectors are linearly independent or span a space
• use matrices to represent and analyse linear transformations
• apply inner products and orthogonality to geometric problems
• find and use eigenvalues and eigenvectors
• use diagonalisation to simplify matrix problems
• explain how linear algebra supports applications in graphics, modelling, data analysis, and machine learning

Assignments / assessments

• Coursework (40%)
• Written exam (60%)

Teaching methods / timetable

• lectures, where the main ideas in linear algebra are introduced through definitions, examples, geometric interpretation, and proof
• worked examples, helping you see how vectors, matrices, transformations, eigenvalues, and orthogonality are used in practice
• tutorials, where you will practise solving problems individually and in groups
• guided support, helping you build confidence with abstract ideas, calculations, and mathematical reasoning