MA21410 - LINEAR ALGEBRA

Module Identifier	MA21410
Module Title	LINEAR ALGEBRA
Academic Year	2001/2002
Co-ordinator	Dr Robert Douglas
Semester	Semester 2
Pre-Requisite	MA11010
Mutually Exclusive	MX31410
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 course. It will be seen that superficially different problems in mathematics can be unified. For example, the solution of systems of linear equations and linear diffential 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 show that this is the correct framework to consider linear problems in a unified way.

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.

Syllabus

1. VECTOR SPACES: Definition and examples, subspaces, spanning sets, linear independence, basis and dimensions.
2. LINEAR TRANSFORMATIONS: Definition and examples, 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: Definition and examples. Orthogonality and Gram-Schmidt orthogonalisation process.
4. DIAGONALISATION OF MATRICES: Eigenvalues and eigenvectors, characteristic equation. Diagonalisation of symmetric matrices, quadratic forms.

Reading Lists

Books
** Recommended Text
Howard Anton & Chris Rorres. (2000) Elementary Linear Algebra: Applications Version. 8th. J Wiley 0471170526
** Supplementary Text
T S Blyth and E F Robertson. (1998) Basic Linear Algebra. Springer 3540761225
Allenby, R B J T. (1995) Linear Algebra. Edward Arnold 0340610441
Whitelaw,T A. (1983) An Introduction to Linear Algebra. Blackie 021691437X
Morris, A O. (1982) Linear Algebra - An Introduction. 2nd. Chapman and Hall 0412381001
Grossman,S I. (1984) Elementary Linear Algebra. 2nd. Wadsworth 0534027385
Gillett, Ph. (1975) Introduction to Linear Algebra. Houghton Mifflin Co. 0395185742