Intermediate Linear Algebra


Linear Algebra is one of the most fundamental areas of Mathematics with applications in Geometry, Statistics, Applied Mathematics, Algebra, Analysis and indeed most topics within Mathematics. This course begins with the familiar example of solving linear equations, and gradually progresses into the abstract. You will be introduced to the central concept of a vector space, and the linear maps between them which ensure their structure is preserved. You will see proofs of two of the most famous results in Linear Algebra – the Spectral Theorem and Rank-Nullity Theorem, as well as an introduction to the idea of an Inner Product which is commonplace in Quantum Physics.

This is an ‘intermediate’ FHEQ level 4 course and therefore in order to get the most out of the teaching you should have some familiarity with Linear Algebra as a pre-requisite. Taking the 'Beginning Linear Algebra' course would be ample preparation.

The overall structure of the course follows the Undergraduate Mathematics Syllabus at the University of Oxford. Courses on the Weekly Oxford Worldwide programme consist of weekly live webinars with a tutor and weekly pre-recorded lectures.

*** Students should note that for this exceptional course, the pre-recorded lectures are already publicly available for free on YouTube ***

Programme details

Course Begins: 18 April

Week 0: Course orientation

Week 1: Solving a Linear System and Finding a Matrix Inverse via Elementary Row Operations

Week 2: The Determinant Function

Week 3: Eigenvalues and Eigenvectors

Week 4: Spectral Theorem

Week 5: Vector Spaces and Subspaces

Week 6: Basis, Spanning and Linear Independence

Week 7: Dimension Formula and Direct Sum

Week 8: Linear Transformations

Week 9: Rank Nullity Theorem

Week 10: Inner Product Spaces and the Gram-Schmidt Procedure     

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.


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


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


Dr Tom Crawford

Tom is the Public Engagement Lead at the Department and is also a Fellow by Special Election at St Edmund Hall where he teaches mathematics to the first and second year undergraduate students.

Alongside his teaching commitments, Tom runs an award-winning outreach programme through his website ’Tom Rocks Maths’ which hosts videos, podcasts, puzzles and articles that aim to make maths entertaining and understandable to all. Tom works with several partners including the BBC and the Numberphile YouTube channel – the largest maths channel on the platform with over pi-million subscribers. With over 20 million YouTube views, 2 TEDx talks, and guest lectures at the Royal Institution and New Scientist Live, Tom is well on his way to his goal of bringing maths to the masses.

Course aims

Develop a deeper knowledge of Linear Algebra with rigorous mathematical proofs. Follows OUDCE’s ‘Beginning Linear Algebra' course.


Course Objectives

Introduce abstract concepts through the vehicle of Linear Algebra;

Extend student’s knowledge beyond the basics of computation, to an understanding of theory and proof;

Develop the high-level analytical skills required of a Mathematician.

Teaching methods

The course will consist of the following:

- a weekly lecture video (50-60 minutes) to cover the core concepts of each topic

- guided reading of lecture notes, textbooks, sample exercises

- a weekly problem set

- a 1-hour weekly group tutorial to cover the solutions to the problem set and answer any questions about the content

Learning outcomes

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

- Utilise the tools of matrix algebra such as inverses, determinants, eigenvalues and eigenvectors to solve a variety of problems;

- Demonstrate an understanding of the structure of a Vector Space, the properties that follow, and their relationship to Linear Transformations.

- Explain the Spectral and Rank-Nullity Theorems and describe the key steps in their proofs.

Assessment methods

Weekly problem sets which will be used to determine the content of the tutorials. A 'mock' exam in week 6 as a formative assessment and practice for the final exam at the end of the course. The final exam will be untimed, open-book and will cover all topics in the course. It will determine the final grade.

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


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 Enrolment Form (Word) or Enrolment Form (Pdf).

Level and demands

Familiarity with the concepts of vectors, matrices, and some experience of mathematical proofs. For example, you may have taken a 'Linear Algebra' course already at university and want to extend your knowledge further; you may be aware of the applications of Linear Algebra but want to know more about where they come from and how to derive them from mathematical foundatiulns; or you may have taken the 'Beginning Linear Algebra' course at the Department for Continuing Education and wish to progress your studies.

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)