  We introduce a noncommutative binary operation on matroids, called free
product. We show that this operation respects matroid duality, and has the
property that, given only the cardinalities, an ordered pair of matroids may be
recovered, up to isomorphism, from its free product. We use these results to
give a short proof of Welsh's 1969 conjecture, which provides a progressive
lower bound for the number of isomorphism classes of matroids on an n-element
set.
