Representations of matroids

The concept of matroids was originally introduced by Whitney and Van
der Waerden in the 1930's to generalise the notion of linear dependence in
a vector space; certain axioms satisfied by this relation were observed to
be satisfied by other types of ’ dependence’ relations, such as algebraic
dependence and ’ cycle’ dependence in a graph. Consequently a matroid was
defined to be a set with an abstract dependence relation satisfying these
axioms. One of the most natural questions to ask is whether every such
’ matroid' is representable in the obvious sense in a vector space. The
answer is of course no (otherwise matroid theory would be equivalent to
linear algebra) although in the early years of the subject examples of
non-representable matroids were not easily obtainable. In this thesis we
continue the work of Inglcton (in [20]) and Vamos (in [35,36]) on the
representation problem, buiding up to an algebraic treatment in the
important last chapter.