Universal property of the quotient space

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. Let $T \colon V \to U$ be a linear map 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.

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.