Linear rank one preservers between spaces of matrices with zero trace

Let X, Y be a pair of vector spaces over a field F associated with a bilinear form (,) such that (x,y) = 0 for all y in Y, implies that x = 0. Let (X circle times Y)(0) be the subspace of X circle times Y spanned by all decomposable elements x circle times y with (x,y) = 0. Let U, V be any two vector spaces over F. In this note, we study linear mappings from (X circle times Y)(0) to U circle times V that send nonzero decomposable elements to nonzero decomposable elements and some of its consequences. (C) 2009 Elsevier Inc. All rights reserved.