Mathematics

ECM1701 - Vectors and Matrices (2015)

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MODULE TITLEVectors and Matrices CREDIT VALUE15
MODULE CODEECM1701 MODULE CONVENERDr Gihan Marasingha (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
Number of Students Taking Module (anticipated) 331
DESCRIPTION - summary of the module content

Vectors essentially underlie all areas of pure and applied mathematics, computer science and engineering. They are the fundamental ways of describing multi-dimensional objects and the natural language of information retrieval systems, computer aided design, 3D graphics, and pattern recognition. Matrices are operators that act on vectors. 


This introduction to vectors and matrices gives you an excellent foundation in and understanding of these concepts; aiming to familiarise you and build confidence in using and applying them to any computer science, mathematics and engineering theory and application. This foundation in the concepts of vectors and matrices, together with applications of geometry and the solution of systems of linear equations aims to prepare you for everything you are likely to encounter over the course of your chosen programme and discipline.


On completing this module you will be equipped with the skills to apply vectors and matrices to all areas of maths and computing, along with the ability to tackle a range of problems, and a sound understanding of the basic concepts of linear algebra. This course uses small tutorial-based group work (of up to 15 students per group) to encourage the practice of theory learned in large lectures.


Prerequisite module: ECM1706
 

AIMS - intentions of the module

The module aims to provide you with a foundation in the concepts of vectors and matrices, together with some applications both to geometry and to the solution of systems of linear equations. It is not essential to have previous knowledge of vectors and matrices. Lecturers will place an emphasis on both algorithmic aspects and theoretical development and will include proofs where appropriate.

 

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 perform accurate computations relating to matrices and column vectors with real or complex entries;
2 apply those concepts in tackling an appropriate range of problems;
3 discuss the abstract concepts of vector spaces, linear transformations, and other associated ideas;
4 apply the abstract concepts in 3 to concrete instances of vector spaces;
5 understand and be able to prove abstract theorems concerning vector spaces.
Discipline Specific Skills and Knowledge:
6 state and apply key definitions from this topic;
7 develop pure mathematical notions and skills including sets, functions, and mathematical proof.
Personal and Key Transferable/ Employment Skills and Knowledge:
8 reason using abstract ideas;
9 formulate and solve problems;
10 communicate reasoning and solutions effectively in writing;
11 use learning resources appropriately;
12 demonstrate self-management and time management skills;

 

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


- complex numbers, their arithmetic and geometry;

- extended example in two and three dimensions: vectors; matrices as linear transformations;

- eigenvalues and eigenvectors;

- systems of linear equations: matrix representation;

- elementary row operations;

- Gaussian elimination (including parameterisation of general solution);

- reduced echelon form;

- finding the inverse of a matrix by Gaussian elimination;

- criteria for invertibility of a matrix;

- matrices: addition/subtraction and multiplication of matrices;

- matrices as linear functions on sets of vectors; e.g. rotation and reflection matrices in two dimensions;

- algebra of matrices;

- transpose;

- symmetric matrices;

- inverse of a matrix;

- determinants: recursive definition of determinant;

- behaviour of determinants under elementary row/column operations;

- evaluation of determinants by Gaussian elimination;

- vectors: the notion of vector quantities;

- addition and scalar multiplication of vectors;

- simple geometric proofs;

- length of a vector;

- unit vectors, standard basis vectors;

- linear independence, bases and dimension;

- scalar product and vector product;

- triple products;

- eigenvectors and eigenvalues: definitions;

- characteristic polynomial;

- diagonalisation of matrices;

- examples in two and three dimensions (including cases with a repeated eigenvalue);

- abstract definition of vector space and linear maps with examples;

- the kernel and image of a linear map.

 

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 49.00 Guided Independent Study 101.00 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
Scheduled learning and teaching activities 5 Seminars
Scheduled learning and teaching activities 11 Tutorials
Guided independent study 101 Guided independent study

 

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 One sheet of problems 1-12 Three questions marked by tutors. Feedback given on all questions during tutorials
Mid-term test 40 minutes 1,2,6-12 Students mark each other's work, based on answer sheet and feedback session.
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 0 Written Exams 100 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written examination - closed book 100 2 hours 1-9 Available on request
         
         
         
         

 

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

 

RE-ASSESSMENT NOTES

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

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Thomas, G, Weir, M, Hass, J Thomas' Calculus 12th Pearson 2010 978-0321643636 [Library]
Set Lipschutz, S, Lipson, M Schaum's outlines: linear algebra 4th Mc-Graw-Hill 2008 978-0071543521 [Library]
Set Finney R.L., Maurice D., Weir M and Giordano F.R. Thomas' calculus based on the original work by George B. Thomas, Jr. 10th or later Addison-Wesley 2003 000-0-321-11636-4 [Library]
Set Allenby R.B. Linear Algebra, Modular Mathematics Arnold 1995 000-0-340-61044-1 [Library]
Set Hamilton A.G. Linear Algebra: an introduction with concurrent examples Cambridge University Press 1989 000-0-521-32517-X [Library]
Extended McGregor C., Nimmo J. & Stothers W. Fundamentals of University Mathematics 2nd Horwood, Chichester 2000 000-1-898-56310-1 [Library]
Extended Stewart J. Calculus 5th Brooks/Cole 2003 000-0-534-27408-0 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES ECM1706
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 4 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Friday 09 January 2015 LAST REVISION DATE Monday 12 January 2015
KEY WORDS SEARCH Vectors; matrices; Gaussian elimination; determinants; geometry; linear algebra.