Module Identifier MA21410  
Academic Year 2006/2007  
Co-ordinator Dr Robert J Douglas  
Semester Semester 2  
Other staff Dr Robert J Douglas  
Pre-Requisite MA11010  
Mutually Exclusive MX31410  
Course delivery Lecture   19 Hours. (19 x 1 hour lectures)  
  Seminars / Tutorials   3 Hours. (3 x 1 hour example classes)  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:
1. determine whether given algebraic structures are vector spaces;
2. apply criteria for subspaces of a vector space;
3. determine bases for vector spaces;
4. prove and apply propositions in the theory of vector spaces;
5. describe the concept of linear transformation;
6. calculate matrices representing linear transformations;
7. determine the rank and nullity of linear transformations and matrices;
8. perform calculations in inner product spaces;
9. diagonalise matrices, especially symmetric matrices.

Brief 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.


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.


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 matrices.

Reading Lists

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


This module is at CQFW Level 5