MX31410 - LINEAR ALGEBRA

Module Identifier	MX31410
Module Title	LINEAR ALGEBRA
Academic Year	2001/2002
Co-ordinator	Dr Robert Douglas
Semester	Semester 2
Pre-Requisite	MA11010
Mutually Exclusive	MA21410
Course delivery	Lecture	19 x 1 hour lectures
	Seminars / Tutorials	3 x 1 hour example classes
Assessment	Exam	2 Hours (written examination)	100%
	Resit assessment	2 Hours (written examination)	100%

General description

In this module the concept of a vector space is introduced. This develops some ideas which have occurred in the first year courses. It will be seen that superficially different problems in mathematics can be unified. For example, the solution of systems of linear equations and linear differential equations are essentially the same process and can be dealt with simultaneously in this context.

Aims

To develop some matrix theory techniques which have occurred in the first year courses in an abstract setting. To introduce the concepts of a vector space and a mapping between vector spaces. To develop further techniques for computation in vector spaces and to study some geometrical applications.

Learning outcomes

On completion of this module, a student should be able to:

determine whether given algebraic structures are vector spaces;
apply criteria for subspaces of a vector space;
determine bases for vector spaces;
prove propositions in the theory of vector spaces;
describe the concept of linear transformation;
calculate matrices representing linear transformations;
determine the rank and nullity of linear transformations and matrices;
perform calculations in inner product spaces;
diagonalise matrices, especially symmetric matrices, and draw inferences concerning the geometrical context.

Syllabus

1. VECTOR SPACES: Definition and examples, subspaces, spanning sets, linear independence, basis and dimensions.
2. LINEAR TRANSFORMATIONS:Definition and exmples, the matrix of a linear transformation, change of basis. The kernel and image of a linear transformation, rank and nullity. The dimension theorem.
3. INNER PRODUCT SPACES: Defrinition and examples. Orthogonality and Gram-Schmidt orthogonalisation process.
4. DIAGONALISATION OF MATRICES: Eigenvalues and eigenvectors, characteristic equation. Diagonalisation of symmetric matrices, quadratic forms. Applications to geometry, conics and quadrics.

Reading Lists

Books
** Recommended Text
H Anton & C Rorres. (2000) Elementary Linear Algebra, the applications version. 8th. J Wiley 0471170526
** Supplementary Text
T S Blyth and E F Robertson. (1998) Basic Linear Algebra. Springer 3540761225
R B J T Allenby. (1995) Linear Algebra. Edward Arnold 3540610441
Ph Gillett. (1975) Introduction to Linear Algebra. Houghton Mifflin Co. 0395185742
D H Griffel. Linear Algebra and its applications Vol. 1 & 2. Ellis Horwood 074580571X
T A Whitelaw. (1983) An Introduction to Linear Algebra. Blackie 021691437X
A O Morris. (1982) Linear Algebra - An Introduction. 2nd. Chapman & Hall 0412381001
S I Grossman. (1984) Elementary Linear Algebra. 2nd. Wadsworth 0534027385