Subspace as a representative system of the quotient space

Question

Can someone help me with the following problem from Linear Algebra: Let $\Bbb K$ be a field and $V$ a vectorspace over $\Bbb K$ and $U$,$W$ two subspaces of $V$. Now I want to show:

$$W \text{ is a representative system of V/U} \Leftrightarrow W \text{ is the direct sum of U and W}$$ Where $V/U$ is the quotient space.

Answer 1

$W $ is a representative sytem for $V/U$ if

1. $ \bigcup_{w \in W} w + U = V$
2. For all $w_1,w_2 \in W $ $(w_1 + U = w_2 + U \iff w_1 = w_2)$

Condition 1 is equivalent to $ V = U + W$, and condition 2 is equivalent to $V = U \oplus W$.