Linear Algebra: Intermediate



Overview

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 Linear Algebra: Introduction course would be ample preparation.

The overall structure of the course follows the Undergraduate Mathematics Syllabus at the University of Oxford. Courses on the short online course (Live) 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 ***

This course combines online study with a weekly 1-hour live webinar led by your tutor. Find out more about how our short online courses are taught.

Programme details

This course begins on the 15 Jan 2026 which is when course materials are made available to students. Students should study these materials in advance of the first live meeting which will be held on 22 Jan 2026, 7:30-8:30pm (UK time).

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: Vector Spaces and Subspaces

Week 5: Basis, Spanning and Linear Independence

Week 6: Dimension Formula and Direct Sum

Week 7: Linear Transformations

Week 8: Rank Nullity Theorem

Week 9: Inner Product Spaces and the Gram-Schmidt Procedure

Week 10: Spectral Theorem     

Certification

Credit Application Transfer Scheme (CATS) points 

Coursework is an integral part of all online courses and everyone enrolled will be expected to do coursework. All those enrolled on an online courses are registered for credit and will be awarded CATS points for completing work at the required standard.

See more information on CATS points

Digital credentials

All students who pass their final assignment 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 £360.00

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 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. This follows on from the Linear Algebra: Introduction course.
  • 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

Learning takes place on a weekly schedule. At the start of each weekly unit, students are provided with learning materials on our online platform, including one hour of pre-recorded video, often supplemented by guided readings and educational resources. These learning materials prepare students for a one-hour live webinar with an expert tutor at the end of each weekly unit which they attend in small groups. Webinars are held on Microsoft Teams, and provide the opportunity for students to respond to discussion prompts and ask questions. The blend of weekly learning materials that can be worked through flexibly, together with a live meeting with a tutor and their peers, maximise learning and engagement through interaction in a friendly, supportive environment.

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

You will be set independent formative and summative work for this course. Formative work will be submitted for informal assessment and feedback from your tutor, but has no impact on your final grade. The summative work will be formally assessed as pass or fail.

 

Application

Please use the 'Book' or 'Apply' button on this page. Alternatively, please complete an Enrolment form for short courses | Oxford University Department for Continuing Education

 

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 foundations; or you may have taken the 'Beginning Linear Algebra' course at the Department for Continuing Education and wish to progress your studies.

The Department's short online courses are taught at FHEQ Level 4, i.e. first year undergraduate level. FHEQ level 4 courses require approximately 10 hours study per week, therefore a total of about 100 study hours.

English Language Requirements

We do not insist that applicants hold an English language certification, but warn that they may be at a disadvantage if their language skills are not of a comparable level to those qualifications listed on our website. If you are confident in your proficiency, please feel free to enrol. For more information regarding English language requirements please follow this link: https://www.conted.ox.ac.uk/about/english-language-requirements

 

Terms & conditions for applicants and students

Information on financial support