This question copied from "Linear Algebra - Friedberg". Can anyone explain the procedure, i.e. the strategy, of how to prove the statement you're asked to prove.
The question is:
Let W1 and W2 be subspaces of a vector space V . Prove that V is the direct sum of W1 and W2 if and only if each vector in V can be uniquely written as x1 + x2 where x1 ∈ W1 and x2 ∈ W2.
My swing at it: $V = W_1 \oplus W_2 \ \ \ \ \ <=> \ \ \ \ \ V = \{x_1 + x_2: x_1 \in W_1, x_2 \in W_2\}$
I don't know how to proceed. In reality, the answer seems so obvious to me, I just don't know how to put it down on paper.