Vectors and Matrices - 2019 entry

Mathematics is all about numbers and structures. Vectors are one of the most fundamental structures in Mathematics and, more generally, across science, engineering and business. They underlie all areas of mathematics, computer science and engineering. They are the fundamental to describing multi-dimensional objects and the natural language of information retrieval systems, computer aided design, 3D graphics, and pattern recognition. Matrices are the operators that transport vectors.

This introduction to vectors and matrices gives you an excellent foundation in and understanding of how vectors and matrices interact; aiming to familiarise you with, and build confidence in, using them in numerous applications across science, engineering and business. 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 with the ability to tackle a range of problems, as well as demonstrating a sound understanding of the basic concepts of Linear Algebra.

Parallel to the lectures and tutorials you will work on weekly exercise sheets with problems in the field of Linear Algebra. You will be asked to form study groups with fellow students to solve these problems. Starting from week two, you will be asked to prepare the solution for one of the problems from the previous week’s exercise sheet during one of the tutorial sessions and submit to the module convener (if you miss a session, you will have the chance to submit a solution in the following week). Each individual solution will be marked and you will receive feedback from the module convener. The problem mark counts 1/8 towards the overall Coursework mark.

This should encourage you to work out solutions for all problems from the weekly exercise sheets (this can happen in small groups, however you have to be able to reproduce the solution individually).

The aims are to provide a basic introduction to complex numbers and a foundation in the concepts of vectors and matrices, together with applications both to geometry and to the solution of systems of linear equations. No previous knowledge of vectors and matrices is assumed. There is an emphasis on algorithmic aspects rather than a rigorous theoretical development, but proofs will be included where appropriate.

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

Module Specific Skills and Knowledge:

1 demonstrate a sound understanding of the basic concepts of Linear Algebra;

2 apply those concepts in tackling an appropriate range of problems.

Discipline Specific Skills and Knowledge:

3 demonstrate a sufficient knowledge of those fundamental mathematical concepts, manipulations and results in Linear Algebra which are necessary to be able to progress to, and succeed in, further studies in all branches of the mathematical sciences;

4 understand how to read mathematical definitions, theorems and proofs; understand the logic behind different methods of proof, the mechanics of using them and gain experience in applying them; develop an appreciation of the role of rigorous proof in mathematics.

Personal and Key Transferable / Employment Skills and Knowledge:

5 reason using abstract ideas, formulate and solve problems and communicate reasoning and solutions effectively in writing and oral presentation;

6 work in groups to solve in-depth problems effectively; and learn to analyse/assess other solutions for problems;

7 acquire ability for self-criticism of your work;

8 demonstrate appropriate use of learning resources;

9 demonstrate self management and time management skills.

- “Calculus and Geometry” and “Vectors and matrices” fundamentals (see also ECM1901): numbers: The standard number systems N, Z, Q, R and C. Introduction to mathematical proofs [4 hours (+4 hours in “Calculus and Geometry”)];

- Systems of linear equations: Matrix representation; Examples, e.g. Population Projection Matrices and graphs; Elementary Row Operations; Gaussian elimination/Row reduction algorithm (including parameterisation of general solution), reduced echelon form and finding the inverse of a matrix by Gaussian elimination. [8 hours];

- Vectors and Vector Spaces: The notion of vector quantities; Addition and scalar multiplication of vectors; Simple geometric proofs; Representation of points in 2 or 3 dimensions as column vectors; Length of a vector; Unit vectors; Standard basis vectors; Linear independence; informal discussion of bases and dimension; Scalar product and vector/cross product [8 hours];

- “Calculus and Geometry” and “Vectors and matrices” revision and mastery (see also ECM1901): focus on rigorous analysis, mathematical proofs and in-depth problems [4 hours (+4 hours in “Calculus and Geometry”)];

- Matrix Algebra: Addition/subtraction and multiplication of matrices; Matrices as linear functions on sets of vectors; e.g. rotation and reflection matrices in 2 dimensions; Transpose; Symmetric matrices; Inverse of a matrix; Criteria for invertibility of a matrix [8 hours];

- Determinants: Recursive definition of determinant; Behaviour of determinants under Elementary Row/Column Operations [4 hours];

- Eigenvalues and eigenvectors: Definitions; Characteristic polynomial; Diagonalisation of matrices; Examples in 2 and 3 dimensions [4 hours];

- Revision and mastery [4 hours].

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.

Basic reading:

ELE: http://vle.exeter.ac.uk

Other Resources:

Reading list for this module: