# Linear Algebra 2

## 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 & 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.

## Syllabus

Introduction on groups, rings, fields and permutations.

Review of vector spaces, bases and matrices. Determinants, rank, singularity, conjugacy.

Vector spaces: direct sums, projections, quotient spaces, dual spaces, annihilators, transpose.

Polynomial rings, Minimal polynomials, Cayley-Hamilton theorem, Spectral theorem, Jordan form.

Bilinear and quadratic forms. Hermitian, inner product and normed spaces.

Linear algebra over the integers, Smith normal form, finitely generated abelian groups.

## Reading and References

Students will be given printed notes, and there is no required textbook. The following books may be helpful, though they cover more material than in the course.

*Linear Algebra, a pure mathematical approach* by Harvey E. Rose, Birkhauser Verlag, 2002. (Chapters 1-4, and the first part of Chapter 7.)

*Linear Algebra* by Richard Kaye and Robert Wilson, OUP, 1998. (Chapters 1,2,4,7-14.)

Two other books which may be of interest to some are:

*Finite-dimensional Vector Spaces, 2nd Edition *by Paul R. Halmos, Springer-Verlag, 1974. (An older classic; chapters I and II.)

*Algebra, Volume 1, Second Edition *by P. M. Cohn, Wiley, 1982. (A more advanced textbook; chapters 4, 5, 7, 8, 11.)

**Unit code:** MATH21100

**Level of study:** I/5

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Dr Mark Hagen

## Pre-requisites

MATH11005 Linear Algebra and Geometry

## Co-requisites

None

## Methods of teaching

Lectures, problem classes, problems to be done by the students, and solutions to these problems.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.