---
tag: 0ZERJX6PH6
kind: statement
role: theorem
name: Universal property of the quotient space
msc: [15A04]
uses: [0J0SK5FF4M, 0RJF668P1T]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field, let $V$ and $U$ be vector spaces over $F$, and let $W$ be a subspace of $V$.
Let $\pi \colon V \to V / W$ be the [[quotient map|0N4F5FMQZW]].
Let $T \colon V \to U$ be a [[linear map|0NRYHJPQG2]] with $W \subseteq \ker T$.
Then there is exactly one linear map $\bar T \colon V / W \to U$ with $\bar T \circ \pi = T$.
It is given by $\bar T(v + W) = T(v)$.

## Remarks

The condition $W \subseteq \ker T$ makes $\bar T$ well defined.
The kernel of $\bar T$ is $\ker T / W$, and its image is $\im T$.
For $W = \ker T$, the map $\bar T$ is injective, which gives the [[first isomorphism theorem|0MTBGVZ4XJ]].

## Explanations

### L1 human

A linear map that kills every vector of $W$ cannot tell apart two vectors that differ by an element of $W$.
So it makes sense on the cosets, and it becomes a map from the quotient space.
This property describes the quotient space completely.