Beginning Linear Algebra


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: 19 Jan 2022

Week 0:  An Introduction to Teams – Course orientation

Week 1:   Solving linear equations.

Week 2:   Matrix algebra.

Week 3:   Vector spaces.

Week 4:   Linear transformations.

Week 5:   Isomorphisms.

Week 6:   Change of basis.

Week 7:   Canonical forms.

Week 8:   Canonical forms (bis)

Week 9:   Inner products.

Week 10:  Miscellaneous.


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.


Description Costs
Course Fee £229.00
Take this course for CATS points £10.00


Prof Bryan Cain

Has more than 25 years experience teaching university-level mathematics.  Has lectured and/or taught and/or held a research appointment in 8 western European countries plus Russia, China, India, Japan, Israel, Canada, Iowa State University, University of Wisconsin, and MIT.

Course aims

and Course Objectives

1.  Comfort with the language and notations of linear algebra.

2.  Comprehensive understanding of linear equations and their solutions.

3.  Mastery of basic matrix algebra.

4.  Knowledge of vector space basics:  linear combinations, spanning, bases.

5.  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:

1. Know how to solve m linear equations in n unknowns and what the set of all solutions 'looks' like.

2. Be skilled at matrix arithmetic.

3. Be able to work out non-exotic examples and invoke appropriately, the standard theorems of basic linear algebra.

Assessment methods

Several tests designed to see that the basic facts are in the students' heads, not just in their notes.

Students must submit a completed Declaration of Authorship form covering the submitted tests. CATS points cannot be awarded without the aforementioned form.


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.

Please use the 'Book' or 'Apply' button on this page. Alternatively, please complete an application form.

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.

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)