Global Opportunities

MAS2707 : Vector Spaces, Groups and Algorithms

Semesters
Semester 1 Credit Value: 10
Semester 2 Credit Value: 10
ECTS Credits: 10.0

Aims

To provide students with an introduction to modern abstract linear algebra. Building on their existing knowledge of matrix methods, students will experience the benefits of the abstract and rigorous mathematical theory of vector spaces for the deeper understanding of a mathematical subject.
To introduce the basic concepts, notation, and techniques of discrete mathematics, particularly group theory, graph theory and the theory of algorithms, and the use of these methods in the representation and solutions of problems coming from the real world as well as other parts of mathematics.

Module Summary

Linear algebra is a fundamental subject that pervades many areas of modern mathematics. On the one hand it is often convenient to replace a complicated problem by a linear approximation which is easier to solve. On the other hand, linear algebra has beautiful applications in coding theory, projective geometry, and many other areas of mathematics and statistics. Initially linear algebra aims to solve systems of linear equations. In the first-year courses this led naturally to matrix algebra. In the first half of this module abstraction and generalisation are pushed one level further with the formal introduction of vector spaces and linear maps as a replacement for real n-dimensional space and matrices, respectively. This allows us to consider analogous problems in different settings simultaneously and eventually makes explanations easier and faster. We will need to introduce notions of dimension and basis in this general setting. A guiding question is how to transform matrices (or linear maps) to a simple form in which essential properties can be immediately read off.
In the second half, groups will be introduced, and their basic properties will be studied via permutation groups, building on knowledge of mappings and permutations. The concept of an algorithm will be introduced and formally defined, with some discussion on how the complexity of an algorithm can be measured. The basic notation of graphs will be introduced, some well-known problems will be formulated in that language. Solutions to the problem will be discussed, as well as their complexities.

Outline Of Syllabus

Vector spaces, span and bases, linear maps and their properties. Eigenvectors and eigenvalues. Inner product spaces, change of basis, diagonalisation.

Permutations. Groups: definition, properties, examples. Symmetric, alternating and dihedral groups. Subgroups, conjugation, homomorphisms, isomorphisms, cosets, Lagrange's Theorem. Algorithms and growth of functions, comparison of algorithms, P vs NP. Basic concepts of graph theory: examples and problem-solving algorithms such as finite-state automata.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture401:0040:00Formal Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture41:004:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes – Synchronous On-Line
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesDrop-in/surgery101:0010:00Synchronous On-Line
Guided Independent StudyIndependent study1061:00106:00Preparation time for lectures, background reading, coursework review
Total200:00
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Problem-solving exercises
Prob solv exercises2M10Problem-solving exercises
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable