A property of the inverse of a subspace of a finite field

We prove a geometric property of the set A−1A−1 of inverses of the nonzero elements of an FqFq-subspace A   of a finite field involving the size of its intersection with two-dimensional FqFq-subspaces. We give some applications, including a new upper bound on |A−1∩B||A−1∩B| when A and B   are FqFq-subspaces of different dimension of a finite field, satisfying a suitable assumption.