Mathematics

MTH2011 - Linear Algebra (2022)

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MODULE TITLELinear Algebra CREDIT VALUE15
MODULE CODEMTH2011 MODULE CONVENERProf Jan Sieber (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
Number of Students Taking Module (anticipated) 230
DESCRIPTION - summary of the module content
Abstract vector spaces are important objects in linear algebra, which has its origins in solving linear equations over a field such as the rational, real or complex numbers. The elements of a vector space can be somewhat abstract: for example, they can be  functions. However, it is precisely this abstraction that makes the theory of vector spaces such a powerful tool. They arise in almost every area of (pure and applied) mathematics and statistics. For example, PDEs (partial differential equations) of some types are just ODEs (ordinary differential equations) in vector spaces of functions, and numerical and data analysis methods consider vector spaces of increasing dimension to approximate function spaces. 
 
The material in this module underpins the study of many topics in pure and applied mathematics modules at levels 3 and M. 
 
Prerequisite module: MTH1001 (or equivalent).
AIMS - intentions of the module

This module aims to develop the theories and techniques of modern algebra, particularly in relation to vector spaces and inner product spaces.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 understand the relationship between linear maps and matrices, and how the properties of each influence the solvability of systems of linear equations;
2 comprehend algorithms for solving linear equations and finding eigenvalues and eigenvectors in rigorous and formal terms.

Discipline Specific Skills and Knowledge:
3 tackle problems in many branches of mathematics that are linearisable, using the core skills of solving linear systems;
4 understand fundamental concepts in linear algebra for subsequent studies in pure mathematics.

Personal and Key Transferable / Employment Skills and Knowledge:
5 appreciate that concrete problems often require abstract theories for their solution;
6 show the ability to monitor your own progress, to manage time, and to formulate and solve complex problems.
 

 

SYLLABUS PLAN - summary of the structure and academic content of the module

- vector spaces and subspaces
- linear independence, spanning sets;
- linear maps, matrices of linear maps, change of basis;
- kernel and image of linear maps;
- dimension of vector spaces;
- rank and nullity theorem;
- generalization of concepts and key results over arbitrary fields;
- characteristic and minimal polynomials; Cayley-Hamilton theorem; Jordan Canonical Form;
- normed and inner product spaces: bilinear forms and inner products; norms; Cauchy-Schwartz inequality; Gram-Schmidt;
- unitary matrices; self-adjoint operators, including the spectral theorem; diagonalisability; dual spaces and examples; adjoint maps.

 

 

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 38.00 Guided Independent Study 112.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS

Category

Hours of study time

Description

Scheduled learning and teaching activities

33

Lectures including example classes

Scheduled learning and teaching activities

5

Tutorials

Guided independent study

112

Lecture and assessment preparation; wider reading

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade

Form of Assessment

Size of Assessment (e.g. duration/length)

ILOs Assessed

Feedback Method

None

   

 

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT

Form of Assessment

% of Credit

Size of Assessment (e.g. duration/length)

ILOs Assessed

Feedback Method

Written Exam – closed book 

80%

2 hours (summer)

All

Written/verbal on request, SRS

Coursework Exercises 1 4% 10 hours All Annotated script and written/verbal feedback
Coursework Exercises 2 4% 10 hours All Annotated script and written/verbal feedback
Coursework Exercises 3 4% 10 hours All Annotated script and written/verbal feedback
Coursework Exercises 4 4% 10 hours All Annotated script and written/verbal feedback
Coursework Exercises 5 4% 10 hours All Annotated script and written/verbal feedback

 

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)

Original Form of Assessment

Form of Re-assessment

ILOs Re-assessed

Time Scale for Re-assessment

Written exam*

Written Exam (2 hours) (80%)

All

August Ref/Def period

Coursework Exercises 1* Coursework Exercises 1 (4%) All August Ref/Def period
Coursework Exercises 2* Coursework Exercises 2 (4%) All August Ref/Def period
Coursework Exercises 3* Coursework Exercises 3 (4%) All August Ref/Def period
Coursework Exercises 4* Coursework Exercises 4 (4%) All August Ref/Def period
CourseworkExercises  5* Coursework Exercises 5 (4%) All August Ref/Def period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

RE-ASSESSMENT NOTES
Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.   
    
Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Web based and Electronic Resources:

ELE: http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Axler, S. Linear Algebra Done Right 2nd Springer 1997 978-0387982588 [Library]
Set Cohn P.M. Elements of Linear Algebra 1st Chapman & Hall/CRC 1994 978-0412552809 [Library]
Set Griffel, D.H. Linear Algebra and Its Applications. Vol.1, A First Course Ellis Horwood Limited 1989 000-0-745-80571-X [Library]
Set Griffel D.H. Linear Algebra and Its Applications. Vol.2, More Advanced Ellis Horwood Limited 1989 000-0-470-21354-X [Library]
Set Cameron, P.J. Fields Introduction to Algebra Second Oxford Science Publications 2008 978-0-19-852793-0 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES MTH1001
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 5 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Wednesday 26 February 2020 LAST REVISION DATE Monday 30 May 2022
KEY WORDS SEARCH Vector spaces; linear maps; scalar products; orthogonal vectors; linear independence; spanning sets; subspaces; Jordan form; adjoint; dual; rings; groups; fields; isomorphism; irreducibility; characteristic polynomial.