# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Seminars / Tutorials | 4 Hours. (4 x 1 hour example classes) |

Lecture | 18 Hours. (18 x 1 hour lectures) |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 100% |

Supplementary Exam | 2 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.

### 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.

### Content

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 List

**Recommended Text**

Howard Anton & Chris Rorres (2000) Elementary Linear Algebra: Applications Version 8th J Wiley Primo search

**Supplementary Text**

Allenby, R B J T (1995) Linear Algebra Edward Arnold Primo search Gillett, Ph (1975) Introduction to Linear Algebra Houghton Mifflin Co. Primo search Grossman,S I (1984) Elementary Linear Algebra 2nd Wadsworth Primo search Morris, A O (1982) Linear Algebra - An Introduction 2nd Chapman and Hall Primo search T S Blyth and E F Robertson (1998) Basic Linear Algebra Springer Primo search Whitelaw,T A (1983) An Introduction to Linear Algebra Blackie Primo search

### Notes

This module is at CQFW Level 5