  Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the
$G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$
matrices is determined by a matroid. To do this, we split the natural
surjective map from the $G$ equvariant Chow ring of the space of matrices to
the torus equivariant Chow ring of the Grassmannian. The splitting takes the
class of a Schubert variety to the corresponding factorial Schur polynomial,
and also has the property that the class of a subvariety of the Grassmannian is
mapped to the class of the closure of those matrices whose row span is in the
variety.
