# Beginning Linear Algebra

## 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: 17 April 2024

Week 0:  Course orientation

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

## Digital Certification

To complete the course and receive a certificate, you will be required to attend and participate in at least 80% of the live sessions on the course and pass your final assignment. 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.

## Fees

Description Costs
Course Fee £257.00
Take this course for CATS points £10.00

## 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.

Guest lecturer Dr Bryan Cain is Professor Emeritus of Mathematics at Iowa State University, specializing in Linear Algebra, with 25 years experience teaching university level mathematics.

## 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.

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 7 days prior to the start date to allow us to complete the course set up. We will email you at that time (7 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 £10 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.

Students who register for CATS points will receive a Record of CATS points on successful completion of their course assessment.

To earn credit (CATS points) you will need to register and pay an additional £10 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

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 at the required standard.

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.

Most of the Department's weekly classes have 10 or 20 CATS points assigned to them. 10 CATS points at FHEQ Level 4 usually consist of ten 2-hour sessions. 20 CATS points at FHEQ Level 4 usually consist of twenty 2-hour sessions. It is expected that, for every 2 hours of tuition you are given, you will engage in eight hours of private study.

Credit Accumulation and Transfer Scheme (CATS)