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
Prerequisites: (MATH1006 OR MATH1008 OR MATH1059) AND MATH1048
Aims and Objectives
Having successfully completed this module you will be able to:
- Find matrix representation of linear transformations on vectors spaces other than Rn.
- 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
- 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
|Total study time||150|
Resources & Reading list
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. Springer / Undergraduate Texts in Mathematics.
Anthony Martin and Harvey Michele (2012). Linear Algebra Concepts and Methods. Cambridge University Press.
Sheldon Axler (2015). Linear Algebra Done Right. Springer / Undergraduate Texts in Mathematics.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External