module specification

This module extends students’ knowledge of linear algebra and calculus, providing greater depth and rigour for these essential topics.

The aim of the first part of the module is to introduce students to the abstract algebraic structures of  vector spaces, developing on the material on linear algebra learnt at level 4. This primary example of algebraic structures has applications across science and engineering, and also provides a firm foundation of necessary basic algebraic notions for the student to further their study mathematical study. The aim of the second part of the module is to develop a rigorous approach to the analysis of functions of a real and a complex variable and to the analysis of the convergence or divergence of sequences and series.  Further topics in complex integration are also covered and students are introduced to applications of these topics to improper integration of functions of a real variable. An important technique in solving DEs by functions of a complex variable will be introduced.

Vector Spaces: Revision of matrices and determinants in  Rn and Cn  and application to systems of equations. Formal definition and further examples of vector spaces. Linear combination, spanning, independence. Subspaces. Bases and dimensions, change of basis. Quotient spaces. Linear transformations, matrix representation.  Kernel and image; first isomorphism theorem, rank/nullity. Eigenvalues and bases of eigenvectors, diagonalisation. Cayley-Hamilton.

Limits, continuity and differentiability of functions of a real variable. Mean Value Theorem, Taylor’s theorem with Lagrange remainder. Taylor and Maclaurin series of standard functions.
Limits, continuity and differentiability of functions of a complex variable. Exponential, trigonometric, hyperbolic and logarithmic functions. Cauchy Riemann equation. Contour integration.  Applications to integrals of real variables and to improper integrals.

Students’ blended learning is directed via face-to-face learning activities centred on lectures and seminars. There is full provision of documents related to the module in electronic format that can be accessed by students all the time (currently WebLearn). The documents include lecture notes, slides, guidance to further blended reading and relevant mathematical packages, and exercises and tests.
Teaching methods will include a range of the following: tutor led seminar discussions, student led discussions, small group discussions and exercises, individual exercises, lectures given by tutor. The teaching methods will support the main aim of encouraging independent lifelong learning. Tutorials will be student centred using carefully graduated exercises to build up student's confidence and self-esteem and will also provide the opportunity for students to reinforce learning and demonstrate their skills and receive individual advice from their tutor.
The tutorial sessions and workshops will also provide an ideal setting for students to meet up with their group members on regular basis and carry out their discussions and investigations.

On successful completion of this module, students should be able to:

LO1. Perform calculations with vectors and transformations using suitable matrix representations.
Construct homomorphisms vector spaces and determine their properties such as rank and nullity

LO2. Understand both the concrete examples and the concept of linear independence and be able
to apply them to find bases from given spanning sets.  Determine whether a transformation of  a vector spaces may be represented by a diagonal matrix with respect to an appropriate basis.

LO3. Relate the intuitive definitions of a limit, of continuity and of differentiability to more formal rigorous definitions of these concepts.  Determine whether given functions are continuous or differentiable over given interval

LO4. Apply Cauchy Riemann equations to decide whether a given function is differentiable.  Evaluate
contour integrals using parametrisation and apply the residue theorem to evaluate contour integrals.

There will be two progress tests, allowing students an early opportunity to demonstrate their understanding of selected of topics and receive feedback. Test 1 will assess LO1, LO2, test 2 will assess LO3, LO4. The final assessment will be an exam where students will be tested across the whole syllabus (LO1 to LO4).

1. J. Hefferon, online book Linear Algebra
2. R. Haggarty, Fundamentals of Mathematical Analysis, 1989, Addison-Wesley
3. R. Adams, A complete Course in Calculus, 1999, Addison-Wesley.
4. J. Stewart, Calculus; Concepts and Contexts, 1998, Brooks/Cole
5. R. Churchill and J. Brown, Complex Variables and Applications, 1990, McGraw Hill.
6. T. Osborne, Complex Variables and Their Applications, 1999, Addison-Wesley