# Further mathematics for economists MT3095

This course provides students with the mathematical techniques and methods which find application in economics and related areas, and enables students to understand why, and in what circumstances, these techniques work.

### Topics covered

Linear algebra:

• Vector spaces, linear independence and dependence, bases and dimension, rank and nullity of a matrix
• Linear mappings, their rank and nullity, their matrix representation, and change of basis
• Eigenvalues and eigenvectors
• Diagonalisation of matrices, with applications to systems of difference and differential equations (including stability)
• Quadratic forms and orthogonal diagonalisation. Inner product spaces, norms, orthogonality and orthonormalisation.

Functions and mathematical analysis:

• Sets and functions
• Supremum and infinum of bounded sets
• Limits of sequences in R and Rm
• Limits and continuity of functions
• Open subsets and closed subsets of Rm
• Compact subsets of Rm
• Convex sets, convex and concave functions
• The Jacobian derivative
• The Edgeworth Box and contract curves.

Optimisation:

• Inconstrained optimisation and the second-order conditions
• Constrained optimisation and the Kuhn-Tucker theorem.
• Envelope Theorems
• Theory of linear programming (computational methods will not be included)
• Duality, with applications
• Basic Game Theory.

### Learning outcomes

If you complete the course successfully, you should be able to:

• Use the concepts, terminology, methods and conventions covered in the unit to solve mathematical problems in this subject.
• Demonstrate an understanding of the underlying principles of the subject.
• Solve unseen mathematical problems involving understanding of these concepts and application of these methods.
• Prove statements and to formulate precise mathematical arguments.

### Assessment

Unseen written exam (3 hrs).