# Linear Algebra: Introduction

## Overview

Linear algebra is the most central and fundamental part of mathematics.  Its only serious rival is the calculus. Its applications are legion--internal ones, to other parts of mathematics itself, and external ones, to problems arising outside mathematics. One cause of this importance is that so many non-linear transformations can be usefully approximated by linear ones and adequately understood by studying those approximations. Another is the comprehensiveness of our understanding of linear transformations and the matrices implementing them. Matrices are known to be reducible to special (canonical) forms whose behaviour is easily understood. Moreover, Linear Algebra has provided the inspiration and enlightening examples for much of advanced abstract algebra.

The course begins innocently enough by showing how any system of linear equations can be solved and by describing the set of all its solutions. Once this is well understood it functions as an underlying motif for the rest of the course, e.g. in the reductions which make the calculation of determinants numerically feasible, in computing orthogonal bases, in elucidating spectral theory with its eigenvalues and eigenvectors. This is the first course exploiting the simplifications available via linear changes of coordinates.

## Programme details

Course begins: 3 Oct 2024

Week 1:   Solving linear equations: Gaussian Elimination.

Week 2:   Matrix algebra.

Week 3:   Vector spaces.

Week 4:   LU Decomposition and related algorithms

Week 5: Numerical solution to systems of equations: Gauss-Jacobi and Gauss-Seidel techniques with possible coding.

Week 6: Determinants, Cramer's rule.

Week 7: Eigenvalues and Eigenvectors with applications.

Week 8 Applications of matrices to Computing and other disciplines.

Week 9: Solution to ODEs using a matrix approach. Use of eigenvalues and eigenvectors.

Week 10: Symmetric, Skew-Symmetric, and Orthogonal Matrices, Eigenbases. Diagonalization. Quadratic Forms

## Certification

Credit Application Transfer Scheme (CATS) points

To earn credit (CATS points) for your course you will need to register and pay an additional £30 fee for each course you enrol on. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online. If you do not register when you enrol, you have up until the course start date to register and pay the £30 fee.

Coursework is an integral part of all online courses and everyone enrolled will be expected to do coursework, but only those who have registered for credit will be awarded CATS points for completing work at the required standard. If you are enrolled on the Certificate of Higher Education, you need to indicate this on the enrolment form but there is no additional registration fee.

Digital credentials

All students who pass their final assignment, whether registered for credit or not, will be eligible for a digital Certificate of Completion. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so.

Please note that assignments are not graded but are marked either pass or fail.

## Fees

Description Costs
Course Fee £285.00
Take this course for CATS points £30.00

## Funding

If you are in receipt of a UK state benefit, you are a full-time student in the UK or a student on a low income, you may be eligible for a reduction of 50% of tuition fees. Please see the below link for full details:

Concessionary fees for short courses

## Tutor

### Dr Vasos Pavlika

Dr Vasos Pavlika is Associate Professor (Education) at University College London, he also teaches Mathematics and Statistics at the LSE (University of London), as well as Online at: SOAS, University of London (Mathematical Economics), Goldsmiths College (Computing and Data Science), University of London and the Open University (Applied Mathematics). He has been a lecturer in the Department for Continuing Education, Oxford since 2004. Vasos is the Director of Studies in the Physical Studies in the Department for Continuing Education, Oxford as well as a tutor in the Institute of Continuing Education at the University of Cambridge.

## Course aims

• Comfort with the language and notations of linear algebra.
• Comprehensive understanding of linear equations and their solutions.
• Mastery of basic matrix algebra.
• Knowledge of vector space basics:  linear combinations, spanning, bases.
• Ability to find the matrix which represents a given linear transformation with respect to a given basis.

## Teaching methods

Homework problems and lectures mixing theory, examples, and solutions to exercises.

## Learning outcomes

By the end of the course students will be expected to:

• know how to solve m linear equations in n unknowns and what the set of all solutions 'looks' like;
• be skilled at matrix arithmetic;
• be able to work out non-exotic examples and invoke appropriately, the standard theorems of basic linear algebra.

## Assessment methods

Students will be assessed on the basis of coursework to be completed at home and submitted electronically.

Coursework is an integral part of all weekly classes and everyone enrolled will be expected to do coursework in order to benefit fully from the course. Only those who have registered for credit will be awarded CATS points for completing work the required standard.

Students must submit a completed Declaration of Authorship form at the end of term when submitting your final piece of work. CATS points cannot be awarded without the aforementioned form - Declaration of Authorship form

## Application

We will close for enrolments 14 days prior to the start date to allow us to complete the course set up. We will email you at that time (14 days before the course begins) with further information and joining instructions. As always, students will want to check spam and junk folders during this period to ensure that these emails are received.

To earn credit (CATS points) for your course you will need to register and pay an additional £30 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

## Level and demands

Although this meaty course is far more substantial than the first course in algebra taught in the schools, school algebra is an adequate prerequisite.

Before attending this course, prospective students should know:

• how to graph y = 3x - 2;
• how to add, subtract, and multiply polynomials.

The Department's Weekly Classes are taught at FHEQ Level 4, i.e. first year undergraduate level, and you will be expected to engage in a significant amount of private study in preparation for the classes. This may take the form, for instance, of reading and analysing set texts, responding to questions or tasks, or preparing work to present in class.

Credit Accumulation and Transfer Scheme (CATS)

To earn credit (CATS points) you will need to register and pay an additional £30 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online. Students who register for CATS points will receive a Record of CATS points on successful completion of their course assessment.

Students who do not register for CATS points during the enrolment process can either register for CATS points prior to the start of their course or retrospectively from the January 1st after the current full academic year has been completed. If you are enrolled on the Certificate of Higher Education you need to indicate this on the enrolment form but there is no additional registration fee.