# Physics / Theoretical Physics BSc/MSci

## Mathematics 2B: Linear Algebra MATHS2004

**Academic Session:**2023-24**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 2 (SCQF level 8)**Typically Offered:**Semester 1**Available to Visiting Students:**Yes

### Short Description

This course covers the fundamentals of linear algebra that are applicable throughout science and engineering, and in particular in the physical, chemical and biological sciences, statistics and other parts of mathematics. It is an essential course for intending honours students. The emphasis is on methods and applications.

### Timetable

2 x lectures per week. There may be extra lectures in teaching week 1. Fortnightly tutorials.

### Assessment

One degree examination (80%) (1 hour 30 mins); coursework (20%).

**Main Assessment In:** December

**Are reassessment opportunities available for all summative assessments?** No

Reassessment will not be available for continuous assessment. Immediate feedback will be given to students at the assignment deadline, and it is not practical to set multiple versions of each of these exercises.

In total 80% of the assessment can be repeated for students not reaching the threshold grade.

Note that students are required to submit 75% of the assessments in order to receive a grade for the course. Accordingly, students who have a substantial period of illness and have all assignments and tutorial group work MV'ed, will not be in danger of failing to receive credits.

### Course Aims

This course covers the fundamentals of linear algebra that are applicable throughout science and engineering. The aim of the first part of the course is to introduce the idea of a finite dimensional vector space, including the concepts of linear independence, basis, dimension and linear map. The relation between linear maps and matrices will be explained, and this will motivate further study of matrices in the second part of the course, in which the determinant, eigenvalues and eigenvectors of a matrix will be studied. Symmetric matrices will be introduced and their relation to quadratic forms explained. Finally, we explore orthonormal basis of a vector space and describe the Gram-Schmidt algorithm used to construct such bases. Throughout, all new ideas will be illustrated by examples drawn from applications in low dimensions.

### Intended Learning Outcomes of Course

By the end of the course, students will be able to:

■ State and define basic properties of and operations on matrices and their entries and fluently apply these to solve suitable problems, including their abstract symbolic manipulation.

■ Determine when a given set is a vector (sub-)space over the real or complex numbers.

■ Given a set of vectors, determine when it is linearly independent and describe its span; determine a basis for a given subspace of a vector space.

■ Determine when a map between spaces is linear; given a linear map state its associated matrix with respect to fixed bases.

■ Calculate the determinant of a matrix using cofactor expansion along a row or column; state and prove properties of determinants of matrices related by row (or column) operations.

■ Given a square matrix, state its characteristic polynomial, determine its eigenvalues and eigenvectors, and apply these to solve suitable problems.

■ Given a quadratic form, determine its associated symmetric matrix, its rank, signature and diagonalisation.

■ Construct inner products on a real vector space. Apply the Gram-Schmidt algorithm to a given basis to construct an orthonormal basis of a Euclidean vector space.

■ State all definitions and results (lemmata, theorems, corollaries, propositions) covered in lectures, and apply these to solve suitable problems.

■ Prove the results covered in lectures, apply and adapt these proofs in novel situations.

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.