let g be the inverse function of f (not necessarily a bijective inverse function)
If S is any subset of W then the pre-image of S is
g(S) = {v ε V: f(v) ε S}
Suppose that U is a subspace of W.
Prove that g(U) is a subspace of V.
Also prove that
dim( g(U) )= dim( U intersect Im(f) ) +...