Let $V$ be a vector space and let $U, W$ be subspaces. The [[intersection]] $U \cap W$, the set of elements which lie both in $U$ and $W$, is a [[subspace]] of both. - The sum of two vectors from the intersection behaves the same as the sum of vectors from $U$, so it is found in $U$. The same can be said for $W$. - Scaled versions of vectors from $U$ or $W$ would still be in their respective sets, so if a vector is in both, then its scaled brethren are also in the intersection - Both $U$ and $W$ must have contained a zero element, so that 0 must be present in the intersection