Mathematical Structures - entry

A key success of mathematics throughout its history has been its ability to unify and generalise disparate situations exhibiting similar mathematical properties through the use of abstract algebraic structures. In this module, you will explore key developments along that journey, including the theory of groups and vector spaces, and you will learn how to develop proofs and present your reasoning clearly.

The purpose of this module is to provide you with an introduction to axiomatic reasoning in mathematics, particularly in relation to the perspective adopted by modern algebra and analysis. The building blocks of mathematics will be developed, from sets and functions through to proving key properties of the standard number systems.  We will introduce and explore the abstract definition of a group, and rigorously prove standard results in the theory of groups, before progressing to consider vector spaces, both in the abstract and with a specific focus on finite-dimensional vector spaces over the real and complex numbers.  The ideas and techniques of this module are essential to the further development of these themes in the two second-year streams Analysis and Algebra, and subsequent pure mathematics modules in years 3 and 4.

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

Module Specific Skills and Knowledge:
1 read, write and evaluate expressions in formal logic relating to a wide variety of mathematical contexts;
2 use accurately the abstract language of sets, relations, functions and their mathematical properties;
3 identify and use common methods of proof and understand their foundations in the logical and axiomatic basis of modern mathematics.
4 state and apply properties of familiar number systems (N, Z, Z/nZ, Q, R, C) and the logical relationships between these properties;
5 recall key definitions, theorems and proofs in the theory of groups and vector spaces;
Discipline Specific Skills and Knowledge:
6 evaluate the importance of abstract algebraic structures in unifying and generalising disparate situations exhibiting similar mathematical properties;
7 explore open-ended problems independently and clearly state their findings with appropriate justification.
Personal and Key Transferable/ Employment Skills and  Knowledge:
8 formulate and express precise and rigorous arguments, based on explicitly stated assumptions;
9 reason using abstract ideas and communicate reasoning effectively in writing;
10 use learning resources appropriately;
11 exhibit self management and time management skills.

Sets; relations; functions; countability; logic; proof.
Primes; elementary number theory.
Groups; examples; basic proofs; homomorphisms & isomorphisms; Cayley's Theorem.
Topology of the real and complex numbers; limits of sequences; power series; radius of convergence.
Vector spaces; linear independence; spanning; bases; linear maps; isomorphisms; n-dimensional spaces over C (resp. R) are isomorphic to C^n (resp. R^n).

The module mark is calculated solely from the mark on the referred/deferred exam. For referred candidates, this mark is capped at 40%. 

ELE – http://vle.exeter.ac.uk

Reading list for this module: