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Linear Algebra - MAT00026I

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  • Department: Mathematics
  • Module co-ordinator: Dr. Ian McIntosh
  • Credit value: 10 credits
  • Credit level: I
  • Academic year of delivery: 2021-22
    • See module specification for other years: 2022-23

Related modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching period
A Autumn Term 2021-22

Module aims

The Core modules in the second year of mathematics programmes cover material which is essential for accessing a wide range of topics later on. Such material underpins the development of mathematics across all three of the major streams we offer through our single-subject degrees, and it is also very important for many combined programmes. As well as covering fundamental material, these modules address applications and techniques which all students will subsequently be able to draw on in various contexts.


This module develops linear algebra through the study of general properties of linear systems of equations and linear transformations, moving on from the material developed in the first year to make the natural transition from linear properties in coordinate spaces to abstract linear algebra.

Module learning outcomes

Subject content

  • Linear systems of equations in n real or complex variables and their solutions: linear combinations, linear span, subspaces, independence and bases, dimension, existence and uniqueness of solutions to linear systems described through the kernel (null space) and image of a matrix, the Rank-Nullity Theorem.
  • Linear transformations: linearity of maps between coordinate spaces, composition as matrix multiplication, one-to-one and onto as consequences of nullity and rank, eigenvectors and eigenvalues, diagonalisability as a change of coordinates, diagonalisability criteria. Properties of trace and determinant.
  • Real and Hermitian inner products: real inner products and symmetric, positive definite matrices; Hermitian inner products and Hermitian symmetric positive definite matrices. Cauchy-Schwarz and triangle inequality. Orthonormal and unitary bases, orthogonal projection onto a subspace, Gram-Schmidt procedure. Diagonalisability of real and Hermitian symmetric matrices.
  • Abstract linear algebra: definition of a vector space. All concepts (linear combination, subspace, independence and bases, linear transformations, kernel and image, eigenvectors and eigenvalues) as above with coordinate space replaced by abstract vector space. Additional structure required for real and Hermitian inner products.

Academic and graduate skills

  • It is hard to overstate the importance of linear algebra in a mathematician’s toolkit. Techniques and results from linear algebra are used across the full spectrum of mathematics and its applications, both in an academic setting and in the wider world. To take an example, as well as having concrete applications in all three of our second year streams, the theory of eigenvalues and eigenvectors is essential in Google’s PageRank algorithm.


Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Linear Algebra
1.5 hours 100

Special assessment rules



Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Linear Algebra
1.5 hours 100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

  • R B J T Allenby, Linear Algebra, Arnold (S 2.897 ALL).
  • R Kaye and R Wilson, Linear Algebra, OUP (S 2.897 KAY).
  • D C Lay, Linear Algebra and its applications, Addison Wesley (S 2.897 LAY).
  • J. B. Fraleigh and R. A. Beauregard, Linear Algebra, Addison Wesley (S 2.897 FRA).
  • P. R. Halmos, Linear Algebra Problem Book, MAA ( S 2.897 HAL).

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.