I've been up way too long, so pardon me if this doesn't make sense, but.. Let V and W be vector spaces. Let T and U be linear transformations from V to W. Consider the set of all x in V such that T(x) = U(x) 1.) I think that this is a subspace of V. 2.) Can I say anything about its dimension? The dimension is at most V, when the two transformations are the same. What is the minimum dimension in terms of dimensions for V, W, null spaces, or ranges? Is it AT LEAST the smaller of the two nullities?