# Mathematics

## ECM2712 - Linear Algebra (2015)

MODULE TITLE CREDIT VALUE Linear Algebra 15 ECM2712 Dr Gihan Marasingha (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 203
DESCRIPTION - summary of the module content

Building on knowledge from ECM1706 or its equivalent, this module will acquaint you with some fundamental notions of modern algebra, and provide you with a solid base for linear algebra. You will learn to use algorithms to solve linear equations. This is a necessary foundation for subsequent further modules, including Coding Theory. Topics will include linear maps; rings and fields; vector spaces and inner product spaces. Furthermore, you will learn that abstract theories are often required for the solution of concrete problems.

Prerequisite module: ECM1701 or equivalent

AIMS - intentions of the module

This module builds on the algorithmic knowledge of vectors and matrices and the introduction to abstract algebraic reasoning from study in stage one, developing this into some of the fundamental notions of modern algebra. After covering the concepts of rings and fields, the module will mainly focus on vector spaces and linear maps, giving a rigorous basis for linear algebra. The material of this module will be required in several subsequent modules.

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 demonstrate familiarity with the notions of rings and fields, and the most important examples of each;
2 understand the relationship between linear maps and matrices, and how the properties of each influence the solvability of systems of linear equations;
3 comprehend algorithms for solving linear equations and finding eigenvalues and eigenvectors in rigorous and formal terms.
Discipline Specific Skills and Knowledge:
4 tackle problems in many branches of mathematics that are linearisable, using the core skills of solving linear systems;
5 reveal sufficient knowledge of the fundamental algebraic concepts needed for subsequent studies in pure mathematics.
Personal and Key Transferable / Employment Skills and Knowledge:
6 appreciate that concrete problems often require abstract theories for their solution;
7 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

- rings and fields: review of field properties of Q, R and C; definitions of rings and fields; the fields Fp; other examples of rings, including Z and K[X] (for K a field); existence of greatest common divisors in Z and K[X]; extended Euclidean Algorithm;

- vector spaces: axioms of vector spaces; examples; subspaces, linear dependence and independence; spanning sets, bases and dimension; finite and infinite dimensional spaces;

- linear maps: definition and examples; image and kernel of a linear map and the dimension formula; isomorphisms; every finite dimensional vector space is isomorphic to a space of column vectors; linear maps and matrices; base change; the Jordan Canonical Form;

- inner product spaces: bilinear forms and inner products; norms; unitary matrices; normal matrices and diagonalisability.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 44 106 0
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 11 Tutorials
Guided independent study 106 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
Fortnightly exercise: four coursework sheets with assessed and non assessed exercises 60 per cent of the exercises are non-assessed All Exercises discussed in tutorials: solutions handed out.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 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 Two hours All Exam mark
Coursework – based on questions submitted for formative assessment 20   All Written

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
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

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

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

Other Resources: Details will be supplied at lectures