## Linear Algebra

### Module aims

In this module you will study linear algebra which underpins many applications of Computing that involve analysis of aggregated data in vector or matrix form. A core central topic is the formalisation and solution of linear systems of equations which are ubiquitous. Linear algebra features in many later modules, including graphics, robotics, performance engineering, operations research and statstical machine learning. The module also provides the background for follow-on mathematics modules that lay the foundations for advanced electives in the above topics and other mathematically-focused areas such as optimisation and finance.

### Learning outcomes

Upon successful completion of this module you will be able to:
- Solve systems of linear equations using Gaussian elimination
- Determine matrices representing linear mappings
- Compute matrix rank
- Compute Eigenvalues and Eigenvectors for simple matrices and explain their application
- Define projections and rotations in matrix form
- Find the spectral decomposition of real symmetric matrices

### Module syllabus

• Vectors and matrices
• Solution of linear systems of equations
• Gaussian elimination
• Vector spaces
• Linear transformations and matrix representation
• Change of basis
• Orthonormal Bases and Gram-Schmidt
• Rank and Nullity Theorem
• Scalar products
• Orthogonal subspaces
• Fundamental theorem
• Introduction to linear regression
• Eigenvalue and eigenvector problem
• Determinants and their properties
• Diagonalisability of matrices
• Caley-Hamilton theorem
• Projections
• Rotation matrices
• Symmetric matrices
• Spectral decomposition

### Teaching methods

The material will be taught through lectures that mix problem solving exercises and taught content.

There are regular unassessed, i.e. formative, tutorial exercises that are submitted for marking and feedback as part of separate Maths Methods Tutorials (MMTs), which run throughout the  Autumn and Spring terms. These tutorials encourage group discussions and group problem solving designed to reinforce your understanding of key topics in  calculus and linear algebra.

An online service will be used as a discussion forum for the module.

### Assessments

There is one assessed coursework which counts 15% of the mark for the module. There is also a written exam which counts for the remaining 85%. Formative assessment is via weekly mathematics methods tutorials (MMTs) which address topics in this module and Calculus. Approximately half of the MMT exercises will be dedicated to Linear Algebra.

A combination of written and verbal feedback will be provided for the MMT tutorial exercises and assessed coursework.