I am asked to prove
$L(S\cap T) \subseteq L(S)\cap L(T)$
I am at a complete loss as to where to even start. I understand that they are not equal because for example part of the basis for $S$ could be $e_i$ and for $T$ could be $2e_i$, but how do I prove the above statement? I am not looking for a complete proof but merely a hint to get me started.
Should I maybe write the elements of the two sets as linear combinations of their basis components? Where would I go from there though?
Well, let $V$ be a vector space over $K$ and let $A,B$ be subsets of $V$. If $A\subseteq B$, then $L(A)\subseteq L(B)$.
In your case, $S\cap T\subseteq S$ and so $L(S\cap T)\subseteq L(S)$. Similarly, $L(S\cap T)\subseteq L(T)$. Therefore, $L(S\cap T)\subseteq L(S)\cap L(T)$.