Regarding vector spaces If A and B are n dimensional vector spaces 
1) Is A+B a vector space?
2) Is A and B a vector space?
 A: The set $\{(w,v)\mid w\in A, v\in B\}$ is a vector space of dimension $2n$.
Hereafter, suppose that $A$ and $B$ are two subspaces of a third vector space. 
The set $\{w+v\mid w\in A, v\in B\}$ is also a vector space, but its dimension can vary depending on how much $A$ and $B$ overlap, but for sure its dimension is at least $n$.
Finally, $A\cap B$ is also a vector space, and again its dimension can vary, but its dimension can be no more than $n$. This was the best interpretation of "A and B" in your post that I could see. However, the intersection does not make any sense unless $A$ and $B$ are subsets of a common vector space.
A: yes
$\forall \alpha_1= ,\alpha_2 \in A+B$and for each $c\in F$ exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $\alpha_1=a_1+b_1$and$\alpha_2=a_2+b_2$
$c\alpha_1+\alpha_2=c(a_1+b_1)+(a_2+b_2)=ca_1+a_2+cb_1+b_2\in A+B$
($ca_1+a_2\in A$ $cb_1+b_2\in B$ becuase A and B are vector spaces.)
$\forall \alpha_1$ and $\alpha_2 \in A\cap B$ and $c\in F$ 
$\alpha_1,\alpha_2\in A$ so $c\alpha_1+\alpha_2 \in A$ 
$\alpha_1,\alpha_2\in B$ so $c\alpha_1+\alpha_2 \in B$
so : $c\alpha_1+\alpha_2 \in A\cap B$
