Intersection of subspaces

Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be linear subspaces of $V$. Then $U \cap W$ is a linear subspace of $V$. It is the largest subspace of $V$ that is contained in both $U$ and $W$. More generally, the intersection of each nonempty family of subspaces of $V$ is a subspace of $V$.

Remarks

The union $U \cup W$ is a subspace only when $U \subseteq W$ or $W \subseteq U$. The smallest subspace that contains $U \cup W$ is the sum $U + W$. In $\R^3$, two different planes through the origin meet in a line through the origin.

Explanations

L1 human

A vector that lies in two subspaces stays in both of them when we add or scale. So the common part of two subspaces is again a subspace. Two planes through the origin in space cross in a line, which is a subspace.