# Linear Algebra and Applications

## Course summary

Course code: MATH1105
Level: 5
Credits: 30
School: Architecture, Computing and Hums
Department: Mathematical Sciences
Course Coordinator(s): Alan Soper

## Specification

None.

### Aims

Linear algebra is a fundamental branch of mathematics which has found applications in a wide range of practical areas, including equation solving, finance, cryptography, computing, search engines, facial recognition and more. With its basis in the theory of matrices and vectors, it is also a highly relevant and widely-used area of mathematics in its own right.

The aim of the course is to develop the mathematical theory of linear algebra, demonstrating the concepts via various applications to modern-day practical situations.

### Learning outcomes

On successful completion of this course a student will be able to:

1. Understand and apply various techniques in linear algebra to mathematical problems in areas such as equation solving and vector spaces, and be able to use and compare different methods to solve a problem, including recognising their strengths and weaknesses.
2. Appreciate the importance of linear algebra to modern practical situations, and be able to creatively apply linear algebra techniques to practical problems.

### Indicative content

Matrices: Introduction to matrix algebra, determinants and inverses.

Equation solving: Using various methods (e.g. Cramer’s rule, Gaussian Elimination, LU factorisation) to solve systems of linear equations. Consistent and independent systems, rank and homogeneous equations. Numerical methods such as the Gauss-Seidel method and numerical errors.

Vector spaces: Vectors, vector spaces, subspaces, linear transformations, rank-nullity theorem. Bases and orthonormality. Methods to create orthonormal bases such as the Gram-Schmidt method.

Eigenvalues and eigenvectors: Determining eigenvalues and eigenvectors. The Cayley-Hamilton Theorem. Eigenspaces and principles of decomposition e.g. diagonalisation. Gershgorin’s Circle Theorem.

Practical applications: Practical applications of linear algebra, e.g. page ranking, facial recognition, least squares.

Abstract linear algebra: Abstract uses of the concepts such as inner product spaces and Fourier series, group representations and complex linear algebra.

The relevance to practical applications and employability will be built into the course material throughout the course.

### Teaching and learning activity

The material will be delivered through a flexible mix of lectures, tutorials and independent study.

### Assessment

Summative assessment:
Coursework 1 - 25%
Coursework covering both learning outcomes, which may include selected tutorial exercises. LO - 1, 2.

Coursework 2 - 25%
Coursework covering both learning outcomes, which may include selected tutorial exercises. LO - 1, 2.

Examination - 50%
Examination of 3 hours in duration covering both learning outcomes. LO - 1, 2.

Formative assessment: Weekly tutorial exercises