# Undergraduate Catalogue of Courses 2012/2013

## MATHEMATICS

MA 2506

LINEAR ALGEBRA

CREDIT POINTS 15

*Course Co-ordinator:* Dr J Elmer

*Pre-requisite(s):*
MA 1006 and MA 2004 or, with permission of the Head of Mathematical Sciences, MA 1507 and MA 2004.

- 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.

12 week course - 3 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

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%).