sub spaces of vector spaces problems

Question

Let $W, U_1, U_2$ be sub spaces of linear spaces $V$. It is said that $$\ V = U_1 \oplus U_2 $$

1. Prove that if $\ U_1 \subseteq W$ , then $W=U_1 \oplus(U_2\cap W) .$
2. Give an example of a linear space V and sub spaces $U_1, U_2, W $ so that $V = U_1\oplus U_2$ and $ W \neq(U_1 \cap W) \oplus (U_2\cap W). $
3. $v_1,v_2,\dots,v_k,u,w$ are vectors in linear space $V$. It is said that $x_1 v_1+x_2v_2+...+x_k v_k = u$ has a single solution and that $x_1 v_1+x_2v_2+...+x_k v_k = w$ has no solution. Find the dimension of $\operatorname{SP}\left\lbrace v_1,v_2,\dots,v_k,w\right\rbrace$.

I'm having a real hard time and I hope that by helping and showing me the correct way I'll be able to learn and become better.

I understand that if $W,U_1,U_2$ are subspaces of V, and that V=U1⊕U2 and that W should be $U_1$ + $W-U_2$ (since V=U1⊕U2 and that U1⊆W) but i don't know how to explain it correctly mathmatically. i don't know how to prove that in a correct manner, that's why i asked for help.