Let $U$ and $V$ be any subspaces of a vector space $W$. Show that $U \cap V$ is a subspace of $W$.
Proof. We must show that $U\cap V$ is non-empty, closed under + and closed under scalar *.
- Since $U$ and $V$ are subspaces $0 \in U \land 0 \in V \implies 0 \in U \cap V$
- Let $x$ and $y$ be any elements of $U \cap V$
consider $x+y$: (I need help here)
- Let $x$ be any element in $U \cap V$ and let $a$ be any field element
consider $a\cdot x$ (I need help here)
I drew a venn-diagram and this property does not seem intuitive at all.