Let $V$ be a finite dimensional vector space over a field $F$ and $W_1,W_2$ be two subsapces then $(W_1+W_2)^0=W_1^0\cap W_2^0.$ Let $V$ be a finite dimensional vector space over a field $F$ and $W_1,W_2$ be two subsapces then 

$(W_1+W_2)^0=W_1^0\cap W_2^0.$


Hints:
$$f\in(W_1+W_2)^0\implies f(x)=0\;\;\forall\,x\in W_1+W_2\;,\;\;\text{and since}\;\;W_1\,,\,W_2\subseteq W_1+W_2\;\ldots$$
On the other hand
$$f\in W_1^0\cap W_2^0\implies f(x)=0\;\forall\,x\in W_1\;\text{and also}\;f(y)=0\;\forall\,y\in W_2,\; \text{and since}$$
$$\forall\,v\in W_1+W_2\;\;\exists \,x\in W_1\,,\,y\in W_2\;\;s.t.\;\;v=x+y\;\ldots$$
BTW, no finite dimension assumption needed here. It is needed in other inclusion, though...
