(a) Suppose U and W are subspaces of a vector space V. Prove that U \cap W = \{v : v \in U \text{

a) Suppose we have two spaces subspaces U and W of V To prove U ${\displaystyle \cap }$ W is a subspace of V, it is sufficient to prove that any linear combination also belongs to U ${\displaystyle \cap }$ W.
Let us choose two vectors u and w belong to U ${\displaystyle \cap }$ W.
i.e. u,w ${\displaystyle \in }$ U ${\displaystyle \cap }$ W
i.e. u,w ${\displaystyle \in }$ U and u,w ${\displaystyle \in }$ W Since U and W are vector spaces and u,w ${\displaystyle \in }$ U and u,w ${\displaystyle \in }$ W,so their linear combination is also belong to that spaces.
i.e. ${\displaystyle \alpha u+\beta v\in }$ U and ${\displaystyle \alpha u+\beta v\in }$ W where ${\displaystyle \alpha ,\beta }$ are any scaler come from underlying scaler field.
i.e. ${\displaystyle \alpha u+\beta v\in }$ U ${\displaystyle \cap }$ W It shows that any linear combination of the vectors u and w also belongs to U ${\displaystyle \cap }$ W.
This prove that U ${\displaystyle \cap }$ W is the subspace of V. (proved)