# ALGEBRA - 2020/1

Module code: MAT1031

## Module Overview

This module combines an introduction to methods of proof with an overview of the basic mathematical entities and structures that will be encountered throughout the degree programme: integers, polynomials, complex numbers, vectors, matrices, groups.   These concepts are fundamental to subsequent modules including MAT1034 Linear Algebra and MAT2048 Groups and Rings

### Module provider

Mathematics

FISHER David (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 100

Lecture Hours: 44

Seminar Hours: 5

Tutorial Hours: 1

Semester 1

None

## Module content

Indicative content includes:

• Proof by deduction, induction, contraposition and contradiction.

• Prime numbers. Prime factorisation of integers.

• The Euclidean algorithm.  Greatest common divisor, lowest common multiple.

• Equivalence relations, congruences and modular arithmetic.

• Polynomials: definitions and basic properties.

• Complex numbers.  Modulus, argument, exponential form, De Moivre's theorem.

• Vectors in two and three dimensions. Scalar and vector products.

• Matrix algebra.  Properties of the transpose and the trace of a matrix.

• Permutations: definitions and basic properties.

• Determinants and inverse matrices.   Solution of simultaneous linear equations.

• Linear maps.  Eigenvalues and eigenvectors of 2 x 2 matrices.

• Introduction to groups and fields.

## Assessment pattern

Assessment type Unit of assessment Weighting
Examination 2 HOUR EXAMINATION 75
School-timetabled exam/test IN-SEMESTER TEST (50 MINUTES) 25

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

·         Ability to interpret and construct formal mathematical proofs.

·         Subject knowledge through the recall of key definitions and results.

·         Ability to apply the techniques learnt to unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

·         One two-hour examination at the end of Semester 1, worth 75% of the module mark.

·         One 50-minute in-semester test; worth 25% of the module mark.

Formative assessment and feedback

Students receive written comments on their marked coursework assignments.  Maple TA is used for part of the assignments and provides instant grading and immediate feedback.  Verbal feedback is provided in lectures, seminars and office hours.

## Module aims

• introduce the standard techniques of mathematical proof
• develop the theory and methods of a number of key algebraic systems
• develop confidence in algebraic manipulation and the selection of suitable techniques to solve problems

## Learning outcomes

 Attributes Developed 001 Understand and be able to formulate simple algebraic proofs, selecting an appropriate method. CT 002 Know properties of the integers and the integers modulo n. KC 003 Understand polynomials, complex numbers and vectors, and be able to solve problems involving them. KC 004 Understand matrices and determinants, and apply them in various contexts. KCT 005 Know and apply the concepts and notation associated with permutations, groups and fields. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to logical reasoning and methods of proof

• Experience of the methods used to interpret, understand and solve problems in elementary number theory and algebra

The learning and teaching methods include:

• Four 50-minute lectures per week for eleven weeks, some being used as tutorials and problem classes.

• Online notes supplemented by additional examples in lectures.

A 50-minute seminar in alternate weeks, with preparatory problem sheets which students are expected to attempt in advance.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT1031

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2020/1 academic year.