## MATH2080 Further Linear Algebra

### 10 creditsClass Size: 98

Module manager: Chris Harlow
Email: c.j.halcrow@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisite qualifications

(MATH1012 or MATH1060), or MATH1331, or equivalent.

### This module is mutually exclusive with

 MATH2020 Algebraic Structures 1 MATH2022 Groups and Vector Spaces

This module is approved as a discovery module

### Module summary

This module develops the more abstract ideas of vector spaces and linear transformations. These ideas are then applied to questions about changing bases, so that the matrices become as simple as possible.

### Objectives

To introduce the idea of linear transformation and some of its applications, and to develop sufficient theory, eg diagonalisation, for applications in Pure and Applied Mathematics and Statistics.

On completion of this module, students should be able to reproduce the appropriate definitions accurately, reproduce short proofs that they have seen in the module and do examples on the material which are more challenging than those at level 1.

### Syllabus

1. Vector spaces over the real numbers, the complex numbers and the field of two elements. Revision of subspaces, linear independence, spanning sets and bases. Direct sums.
2. Revision of linear transformations, image and kernel, rank-nullity formula, the matrix of a linear transformation with respect to bases of the source and target vector spaces. Canonical form.
3. For vector spaces over R or C, revision of eigenvalues, eigenvectors, characteristic equation. Jordan canonical form, Cayley Hamilton Theorem, minimum polynomial.
4. Further topics may include inner products (Cauchy-Schwarz inequality, Gram Schmidt process, orthogonal decomposition, diagonalizability of real symmetric matrices) and positive matrices (Perron-Frobenius Theorem, application to Markov chains, PageRank algorithm).

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 22.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated