MATH1049 Linear Algebra II
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.
Aims and Objectives
This module aims to introduce the student to an abstract viewpoint on the concepts of linear algebra.
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 of polynomials. • 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
|Total study time||150|
Resources & Reading list
Sheldon Axler (2015). Linear Algebra Done Right.
Anthony Martin and Harvey Michele (2012). Linear Algebra Concepts and Methods.
Robert Valenza (1993). Linear Algebra: An Introduction to Abstract Mathematics.
Any other book on Linear Algebra covering vector spaces other than R n can be used..
Repeat type: Internal & External
Prerequisites: MATH1048 and (MATH1AAA or MATH1006 or MATH1008)
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 www.calendar.soton.ac.uk.