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The University of Southampton

MATH1049 Linear Algebra II

Module Overview

Building on the intuitive understanding and calculation techniques from Linear Algebra I, this module introduces the concepts of vector spaces and linear maps in an abstract, axiomatic way. In particular, matrices are revisited as the representation of a linear map in a specific basis. We furthermore introduce the concept of bases of vector spaces and study diagonalisation of linear maps. We apply the abstract theory both in the context of Rn (as seen in Linear Algebra I) and in the context of function spaces; these are particularly important in the study of linear differential equations and hence for instance in physical sciences; for example we look at the derivative operator on the space of polynomial functions. One of the pre-requisites for MATH2003, MATH2014, MATH2045, MATH3033, MATH3076 and MATH3090

Aims and Objectives

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • Explain the axiomatic structures of abstract linear algebra and apply them in simple proofs
  • Apply concepts and theorems from linear algebra to vector spaces other than Rn, in particular function spaces
  • Find matrix representation of linear transformations on vectors spaces other than Rn.
  • Determine whether a linear transformation given by a matrix is diagonalisable.


• Basic introduction to groups: Q, R and C under addition, Q*, R* and C* under multiplication, matrix groups, cyclic groups, permutation groups, sign of a permutation. • Fields: R, Q, C, the field of two elements. • Definition of a vector space over K (where K is a field). • Examples of vector spaces including function spaces (functions from a set to K, differentiable functions, polynomials), subspaces. • Linear independence, spanning sets (generalisation of Linear Algebra I). • Basis and dimension. • Linear transformations, examples including differentiation. • Matrix representation of a linear transformation. • Image and kernel of a linear map, dimension theorem. • Isomorphism of vector spaces. • Determinants (axiomatic description, properties). • Eigenvalues, eigenvectors of linear transformations. • Diagonalisation, diagonalisability. • Cayley-Hamilton theorem.

Learning and Teaching

Teaching and learning methods

Lectures, problem classes, workshops, private study

Independent Study96
Total study time150

Resources & Reading list

Anthony Martin and Harvey Michele (2012). Linear Algebra Concepts and Methods. 

Any other book on Linear Algebra covering vector spaces other than R n can be used.. 

Robert Valenza (1993). Linear Algebra: An Introduction to Abstract Mathematics. 

Sheldon Axler (2015). Linear Algebra Done Right. 



MethodPercentage contribution
Coursework 40%
Written assessment 60%


MethodPercentage contribution
Written assessment 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Prerequisites: (MATH1006 OR MATH1008 OR MATH1059) AND MATH1048


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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