Let U and W denote subspaces of a vector space V. (a) If V = U

Question:

Let U and W denote subspaces of a vector space V.
(a) If V = U ⊕ W, define T : V→ V by T(v) = w where v is written (uniquely) as v = u + w with u in U and w in W. Show that 7 is a linear transformation, U = ker T,W= im T, and T2 = T.
(b) Conversely, if T : V → V is a linear transformation such that T2 = T, show that V = ker T ⊕ im T. [Hint: v - 7(v) lies in ker T for all v in V.]