---
tag: 0V33SPH7RR
kind: statement
role: proposition
name: Intersection of subspaces
msc: [15A03]
uses: [0EPTJA5QV4]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_subspace
    locator: Operations and relations on subspaces
---

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] 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|0JMTY8MF54]] $U + W$.
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.