Unit name | Linear Algebra 2 |
---|---|

Unit code | MATH21100 |

Credit points | 20 |

Level of study | I/5 |

Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |

Unit director | Professor. Rickard |

Open unit status | Not open |

Pre-requisites |
MATH11005 Linear Algebra and Geometry |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

This unit further develops the theory of vector spaces over arbitrary fields and linear maps between them. Topics include quotient spaces, dual spaces, determinants, and canonical forms of linear maps. The unit also introduces bilinear and quadratic forms, and touches on linear algebra over the ring of integers.

**Unit Description**

This unit continues the study of vector spaces over arbitrary fields begun in level C/4. Emphasis is on building insight into the concepts and reasoning clearly from basic definitions. Much of the unit is devoted to formulation and proof of the key results. The tools developed are essential in a variety of areas, both pure and applied, such as geometry, differential equations, group theory and functional analysis.

A major goal is to show that any linear operator on a vector space, even if it is not diagonalisable, has a certain canonical form, the "Jordan normal form". Another aim is to generalise inner products by defining and investigating bilinear and quadratic "forms" on vector spaces. There is also an introduction to linear algebra over the ring of integers, including the classification of finitely-generated abelian groups.

**Relation to Other Units**

This unit develops the linear algebra material from first year Linear Algebra and Geometry, giving a general and abstract treatment, using central algebraic structures, such as groups, rings, and fields. This material is an essential part of Pure Mathematics; it is a prerequisite for Representation Theory, and is relevant to other Pure Mathematics units at levels 3 and 4, particularly Functional Analysis.

Learning Objectives

Students will deepen their understanding of vector spaces and the natural maps between them. They will be able to state, use and prove fundamental results in linear algebra.

Transferable Skills

Assimilation of abstract ideas. Reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

**Recommended**

- Richard Kaye and Robert Wilson,
*Linear Algebra,*OUP, 1998 (Chapters 1,2,4,7-14) - Harvey E. Rose,
*Linear Algebra, a pure mathematical approach*, Birkhauser Verlag, 2002 (Chapters 1-4, and the first part of Chapter 7)

**Further**

- P. M. Cohn,
*Algebra, Volume 1*, Second Edition, Wiley, 1982 (A more advanced textbook; chapters 4, 5, 7, 8, 11) - Paul R. Halmos,
*Finite-dimensional Vector Spaces*, 2nd Edition Springer-Verlag, 1974 (An older classic; chapters I and II)