Short course
Linear Algebra: Introduction
Course status:
Course ended
Location:
Online
Dates:
16/01/2026 - 27/03/2026
Study format:
Online - live
Fees:
£360.00
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.
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Programme details
This course begins on the 16 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 23 Jan 2026, 6:30-7:30pm (UK time).
Week 1:Â Â Solving linear equations: Gaussian Elimination.Â
Week 2:Â Â Matrix algebra.Â
Week 3:Â Â Vector spaces.Â
Week 4:Â Â LU Decomposition and related algorithmsÂ
Week 5: Numerical solution to systems of equations: Gauss-Jacobi and Gauss-Seidel techniques with possible coding.Â
Week 6: Determinants, Cramer’s rule. Â
Week 7: Eigenvalues and Eigenvectors with applications.Â
Week 8 Applications of matrices to Computing and other disciplines.Â
Week 9: Solution to ODEs using a matrix approach. Use of eigenvalues and eigenvectors. Â
Week 10: Symmetric, Skew-Symmetric, and Orthogonal Matrices, Eigenbases. Diagonalization. Quadratic Forms
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.
This course is offered at FHEQ Level 4 (i.e. first year undergraduate level), and you will be expected to engage in independent study in preparation for your assignments and for the weekly webinar. 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. Our 10-week Short Online Courses come with an expected total commitment of 100 study hours, including those spent in live webinars.
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
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:Â linear combinations, spanning, bases.
- Ability to find the matrix which represents a given linear transformation with respect to a given basis.
IT requirements
Any standard web browser can be used to access course materials on our virtual learning environment, but we recommend Google Chrome. We also recommend that students join the live webinars on Microsoft Teams using a laptop or desktop computer rather than a phone or tablet due to the limited functionality of the app on these devices.
Programme details
This course begins on the 16 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 23 Jan 2026, 6:30-7:30pm (UK time).
Week 1:Â Â Solving linear equations: Gaussian Elimination.Â
Week 2:Â Â Matrix algebra.Â
Week 3:Â Â Vector spaces.Â
Week 4:Â Â LU Decomposition and related algorithmsÂ
Week 5: Numerical solution to systems of equations: Gauss-Jacobi and Gauss-Seidel techniques with possible coding.Â
Week 6: Determinants, Cramer’s rule. Â
Week 7: Eigenvalues and Eigenvectors with applications.Â
Week 8 Applications of matrices to Computing and other disciplines.Â
Week 9: Solution to ODEs using a matrix approach. Use of eigenvalues and eigenvectors. Â
Week 10: Symmetric, Skew-Symmetric, and Orthogonal Matrices, Eigenbases. Diagonalization. Quadratic Forms
Teaching methods
This course takes place over 10 weeks, with a weekly learning schedule and weekly live webinar held on Microsoft Teams. Shortly before a course commences, students are provided with access to an online virtual learning environment, which houses the course content, including video lectures, complemented by readings or other study materials. Working through these materials over the course of the week will prepare students for a weekly 1-hour live webinar you will share with your expert tutor and fellow students. All courses are structured to amount to 100 study hours, so that on average, you should set aside 10 hours a week for study. Although the course finishes after 10 weeks, all learning materials remain available to all students for 12 months after the course has finished.
All courses are led by an expert tutor. Tutors guide students through the course materials as part of the live interactions during the weekly webinars. Tutors will also provide individualised feedback on your assignments. All online courses are taught in small student cohorts so that you and your peers will form a mutually supportive and vibrant learning community for the duration of the course. You will learn from your fellow students as well as from your tutor, and they will learn from you.
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
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.
Niccolò Salvatori
Niccolò Salvatori completed a Ph.D. in Pure Mathematics at KCL in 2017 on logarithmic structures of Topological Quantum Field Theories and has been teaching for the Department of Mathematics at LSE since 2016.
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.
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.
This course is offered at FHEQ Level 4 (i.e. first year undergraduate level), and you will be expected to engage in independent study in preparation for your assignments and for the weekly webinar. 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. Our 10-week Short Online Courses come with an expected total commitment of 100 study hours, including those spent in live webinars.
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
Fees
| Description | Costs |
|---|---|
| Course Fee | £360.00 |
Please use the ‘Book now’ button on this page. Alternatively, please complete an enrolment form.
