>>851115Since S is injective and surjective, it's bijective and hence has a unique (two-sided) inverse S^{-1}. By composing S^{-1} on the left to both sides of id_W = S;T, we see that S^{-1} = T.
With that, it suffices for you to prove that if a linear function (T) is invertible, then its inverse (S) will also be linear, which is now straightforward since you're also allowed to use T;S = id_V.
