---
tag: 0EPTJA5QV4
kind: definition
name: Linear subspace
msc: [15A03]
defines: 00YC1D652K
uses: [05EGHEQZAW]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_subspace
    locator: Definition
---

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *linear subspace* of $V$ is a subset $W \subseteq V$ with these three properties:

1. $0 \in W$.
2. If $u, v \in W$, then $u + v \in W$.
3. If $a \in F$ and $v \in W$, then $av \in W$.

## Remarks

With the operations of $V$, a subspace $W$ is itself a vector space over $F$.
The sets $\{0\}$ and $V$ are subspaces of $V$.
The lines through the origin are the one-dimensional subspaces of $\R^2$; a line that misses the origin is not a subspace.
The kernel and the image of a linear map are subspaces.

## Explanations

### L1 human

A subspace is a smaller vector space inside a bigger one.
We cannot leave it by adding its vectors or by scaling them.
In space, the subspaces are the origin, the lines and planes through the origin, and the whole space.