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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 *.

  1. Since $U$ and $V$ are subspaces $0 \in U \land 0 \in V \implies 0 \in U \cap V$
  2. Let $x$ and $y$ be any elements of $U \cap V$

consider $x+y$: (I need help here)

  1. 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.

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up vote 2 down vote accepted

For the second part, note that $x\in U$ and $y\in U$; as $U$ is a subspace of $W$, $x+y\in U$. By the same argument, $x+y\in V$, as $V$ is a subspace of $W$.

Do the same for $a\cdot x$.

We notice that we can generalize it to an arbitrary intersection of subspaces (not necessarily a finite one).

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