Let V1, V2, W1 and W2 be vector spaces over a field F. Let T ∈L(V1, V2), U1 ∈ L(W1, V1) and
U2 ∈ L(V2,W2). Suppose that nullity(T) is finite and U1 and U2 are isomorphisms. (For full
marks, do not assume that the vector spaces Vj and Wj are finite-dimensional.) Prove that
nullity(U1TU2) = nullity(T).