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# 2014/15 Undergraduate Module Catalogue

## MATH2020 Algebraic Structures 1

### 15 creditsClass Size: 200

Module manager: Dr Peter Schuster
Email: P.Schuster@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2014/15

### Pre-requisite qualifications

(MATH1012 or MATH1060) and (MATH1025 or MATH1055) or equivalent

### This module is mutually exclusive with

 MATH2080 Further Linear Algebra MATH2200 Linear Algebra 2

This module is approved as a discovery module

### Module summary

Algebraic structures provide one way of encapsulating mathematical properties so that they can be more easily studied, and often lead to unifying diverse areas of mathematics. In this module we will study two such structures, those of groups and vector spaces.Group theory may be regarded as an abstract study of symmetry, and plays a central role in mathematics and its applications. For example, the degree of symmetry of a geometrical figure may be captured by the corresponding group, which tells us not just how many symmetries there are, but also precisely how they interact (the 'structure' of the group). This course treats the basic theory as far as Lagrange's theorem (the order of a subgroup divides the order of the group) and quotient groups.The other algebraic structure studied in this module is the notion of a vector space. Linear algebra is again an important topic in all areas of mathematics, and vector spaces are the natural setting for studying vectors and linear transformations in a coordinate-free way.

### Objectives

On completion of this module, students should be able to:
a) accurately reproduce appropriate definitions;
b) state the basic results about groups and vector spaces, and reproduce short proofs;
c) identify subgroups and orders of elements in the main examples of groups;
d) list the families of cosets of various groups of small order;
e) find bases of sums and intersections of subspaces and kernels and images of linear transformations;
f) represent a linear transformation by a matrix with respect to a given basis;
g) use eigenvectors to determine whether a matrix is diagonalizable.

### Syllabus

1. Definitions, basic properties and examples of groups; Subgroups and orders of elements; permutations and cycle notation; direct products.
2. Group homomorphisms, kernels and images; cyclic groups; cosets and Lagrange's Theorem; groups of prime order and groups of small order; normal subgroups, quotient groups and the First Isomorphism Theorem.
3. Definitions and examples of vector spaces over the real numbers, the complex numbers or the field with two elements; subspaces, the span of a subset; intersections, sums and direct sums of subspaces; revision of
linear independence and bases.
4. Linear transformations, kernels and images; classification of finite-dimensional vector spaces up to isomorphism; the matrix of a linear transformation, composition and inverse; the AP=PB theorem and similarity; revision of eigenvalues and eigenvectors, the characteristic polynomial and diagonalization; quotient vector spaces and the First Isomorphism Theorem; the Rank-Nullity Formula.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lectures 33 1.00 33.00 Private study hours 107.00 Total Contact hours 43.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Written, assessed work throughout the semester with feedback to students.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Written Work * 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated