Suppose that V is a vector space, and M is a subspace of V . A transformation $P : V \to V$ is called the projection of V onto M if
(i) there exists a subspace N such that every vector v ∈ V can be written uniquely as $v = x + y$ for some $x ∈ M$ and $y ∈ N$; and
(ii) P is given by $P(x + y) = x$, for all $x ∈ M$ and $y ∈ N$.
Question: Suppose that $P : V \to V$ is a linear transformation. Prove that P is a projection onto some subspace of V if and only if $P^2 = P$
So I can't seem to prove the reverse direction where I assume P^2 = P. Like what do we need to prove in order to show that it is a projection??