## MATH2080 Further Linear Algebra

### 10 creditsClass Size: 200

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH1060, or equivalent. Not with MATH2200

This module is approved as an Elective

### Objectives

To introduce the idea of linear transformation and some of its applications, and to develop sufficient theory, e.g. diagonalisation, for applications in Applied Mathematics and Statistics. On completion of this module, students should be able to: (a) determine whether a subset of a vector space is a subspace, or a direct summand; (b) recognise whether a mapping between vector spaces is a linear transformation; (c) associate a matrix with a linear transformation between vector spaces with bases; (d) calculate how the matrix changes when the bases change; (e) find the characteristic and minimum polynomials of a square matrix, and know the Cayley-Hamilton theorem; (f) find the Jordan canonical form of a square matrix, in straightforward cases; (g) establish basic properties of inner-product spaces and perform the Gram-Schmidt orthogonalisation process

### Syllabus

This course carries on from Linear Algebra, MATH 1060, and develops the more abstract ideas of vector spaces and linear transformations. These ideas are then applied to questions about 'changing variables', so that matrices look as simple as possible. 1. Revision of vector spaces and subspaces, including axioms for vector spaces over R. 2. Revision of linear dependence and independence. Spanning sets and bases, definition of a linear transformation, linear transformations and matrices I (from Rn x Rn to Rm a linear transformation is exactly multiplication by a matrix). Image and kernel of a linear transformation. 3. Linear transformations and matrices II. By taking bases of V and W, a linear transformation from V to W corresponds to a matrix. Similar matrices and AP = PB, where A and B represent a linear transformation with respect to different bases. Diagonalisation of a matrix. 4. Triangular matrices. A square matrix is similar to a triangular matrix. The Cayley-Hamilton Theorem. Minimum polynomial of a square matrix. 5. Jordan canonical form. Discussion of the Jordan canonical form. 6. Inner products and Euclidean spaces. Orthogonal vectors and the Gram-Schmidt process. Isometry and orthogonal matrices.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours) and examples classes (11 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (85%), coursework (15%).