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### Course Overview

Linear algebra is the study of vector spaces and linear maps between them.

It provides foundations for almost all branches of mathematics and sciences in general. For example, special relativity and quantum mechanics are formulated within the framework of linear algebra.

### Course Details

Study Type Level Undergraduate 2 Second Sub Session 15 credits (7.5 ECTS credits) None. No Dr Ellen Henke

### Qualification Prerequisites

• Programme Level 2

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### Course Description

• Fields. Solving a linear system over a field; Definition and examples of fields (Q, R, C, Fp ); Elementary row operations, row echelon form, Gaussian algorithm for solving a linear system over a field.
• Vector spaces Definition of a vector space over a field; Examples; subspaces of a vector space, intersection and sum of subspaces; Span, spanning sets; Linear independence; Basis, dimension; Elementary results about bases and dimension; Change of basis matrix.
• Linear maps Definition of a linear map between two K-vector spaces; Kernel, image, injective, surjective linear maps; Matrix of a linear map; Rank of a matrix; Invertible matrices; Determinants; Change of basis and the matrix of a linear map.
• Eigenvalues, eigenvectors and diagonalisation Linear transformations, eigenvalues and eigenvectors of linear transformations; elementary properties; Minimal polynomial, characteristic polynomial, Cayley-Hamilton theorem; Triangularisation and diagonalisation.
• Inner product spaces, Euclidian/Hermitian spaces Basic definitions, examples; Constructing an orthogonal basis using Gram-Schmidt; Symmetric matrices, orthogonal transformations and matrices, unitary transformations and matrices.

### Degree Programmes for which this Course is Prescribed

• BSc Applied Mathematics
• BSc Mathematics
• BSc Mathematics & Engineering Mathematics
• BSc Mathematics with Gaelic
• MA Applied Mathematics
• MA Mathematics
• MA Mathematics with Computing
• Mathematics Joint
• Mathematics Major
• Mathematics Minor

### Contact Teaching Time

Sorry, we don't have that information available.

### Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).

### Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

### Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

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