# Mathematics

## MTH2011 - Linear Algebra (2020)

MODULE TITLE CREDIT VALUE Linear Algebra 15 MTH2011 Prof Andreas Langer (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 (fields themselves are special types of ring). The elements of a vector space can be somewhat abstract: for example, they can be certain types of function. 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, and so their importance is hard to overstate. For example, familiarity with these objects will deepen understanding of PDEs (partial differential equations) and numerical analysis methods.

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 Guided Independent Study Placement / Study Abroad 38 112 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 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 Exercise sheets 5 x 10 hours All Exercises discussed in tutorials: tutor feedback.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 10 90 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 90% 2 hours All Written/verbal on request, SRS Coursework exercises 10% 2 assignments, 30 hours total 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 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

Web based and Electronic Resources:

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