Pre-requisite: MAST4001 (Algebraic Methods) ; MAST4010 (Real Analysis 1) ; MAST4006 (Mathematical Methods 1)
OverviewThis module is a sequel to Algebraic Methods. It considers the abstract theory of linear spaces together with applications to matrix algebra and other areas of Mathematics (and its applications). Since linear spaces are of fundamental importance in almost every area of mathematics, the ideas and techniques discussed in this module lie at the heart of mathematics.
Topics covered will include:
1 Vector Spaces: definition, examples, linearly independent and spanning sets, bases, dimension, subspaces.
2 Linear transformations: definition, examples, matrix of a linear transformation, change of basis, similar matrices.
3 Determinant of a linear transformation.
4 Eigenvalues/eigenvectors and diagonalisation: characteristic polynomial, invariant subspaces and upper triangular forms. Cayley-Hamilton Theorem.
5 Bilinear forms: inner products, norms, Cauchy-Schwarz inequality.
6 Orthonormal systems, the Gram-Schmidt process.
7 Symmetric Matrices. Every real symmetric matrix is diagonalisable.
8 Quadratic forms: Sylvester's Law of Inertia; signature of a quadratic form; application to conics (and quadrics if time permits).
This module appears in:
Method of assessment
80% examination and 20% coursework.
There is no essential reading or core text. Background reading includes:
A.G. Hamilton: Linear algebra: an introduction with concurrent examples. C.U.P, Cambridge, 1989.
L. Robbiano: Linear Algebra for everyone. ISBN: 978-88-470-1839-6 (online)
On successfully completing the module students will be able to:
1 demonstrate knowledge of the underlying concepts and principles associated with linear algebra;
2 demonstrate the capability to make sound judgements in accordance with the basic theories and concepts in the following areas, whilst demonstrating a reasonable level of skill in calculation and manipulation of the material: vector spaces, linear transformations, determinants, diagonalisation, bilinear forms, orthogonality, quadratic forms, applications including conics;
3 apply the underlying concepts and principles associated with linear algebra in several well-defined contexts, showing an ability to evaluate the appropriateness of different approaches to solving problems in this area;
4 make appropriate use of Maple.
The intended generic learning outcomes.
On successfully completing the module students will be able to demonstrate an increased ability to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation
9 work as a member of a team.