 
Summary: LARGE SPACES OF MATRICES
OF BOUNDED RANK
By M. D. ATKINSON and S. LLOYD
[Received 2nd November 1979] .
IN THIS paper we consider subspaces X of M^*, the space of all m x n
matrices with entries in some given field, with the property that each
matrix of X has rank at most r. In [2] Flanders showed that such spaces
necessarily have dimension at most max (mr, nr) and he determined the
spaces of precisely this dimension. We shall extend this work by classify
ing the spaces of dimension slightly lower than this upper bound. Our
results depend on the (often unstated) assumption that the ground field
has at least r +1 elements but, unlike Flanders, we do not need to exclude
the characteristic 2 case.
If every matrix in the space X has rank at most r the same is clearly
true of the space PXQ = {PXQ: X e X} where P, Q are nonsingular
mxtn, nxn matrices respectively. This equivalent space PXQ can also
be derived from X by performing row and column operations to all
matrices of X simultaneously. A wide class of examples is provided by
spaces equivalent to subspaces of the space 9i(p, q) of all matrices of the
form I I where A is a p x q matrix and p + q = r. These examples we
