 
Summary: SPACES OF MATRICES OF BOUNDED RANK
By M. D. ATKINSON and N. M. STEPHENS
[Received 28 October 1977]
IN this paper we shall consider matrices over a field F and shall prove the
following result:
THEOREM. Let M be a 2dimensional space o/mxn matrices with the
property that rank (X) *£ k < \F\ for every XeM. Then there exist two
integers r, s, O^r, s *£fcwith r + s = k, and two nonsingular matrices P, Q
such that, for all XeM, PXQ has the form
Notice that a matrix of the above form necessarily has rank at most k
and so, apart from the restriction \F\ >fc,our theorem essentially charac
terises such 2dimensional subspaces. We may interpret the matrices as
the matrices of linear transformations or of bilinear forms and this gives
the following two equivalent forms of the theorem valid for finite dimen
sional vector spaces:
COROLLARY 1. Let Jibe a 2dimensional space of linear transformations
from a vector space U to a space V such that rank (X)s£ k < \F\ for every
XeM. Then there exist subspaces U0^U, V0^V such that
[U: L/0] + dim {V0)=k and L/0X=£ Vo for every XeM.
COROLLARY 2. Let M be a 2dimensional space of bilinear forms on
