= Quotient map of vector spaces
{title2=$\pi:V\to V/U$}
The <quotient map of vector spaces> $\pi:V\to V/U$ sends a vector $v$ to its coset $v+U$, where $U$ is a <linear subspace>. It is a <surjective> <linear map> with kernel $U$: two vectors have the same image exactly when their difference lies in $U$.
For $V=\mathbb R^2$ and $U=\operatorname{span}(1,0)$, the quotient retains only the second coordinate, so every horizontal line represents one quotient vector. More generally, a <linear map> $f:V\to W$ vanishing on $U$ factors uniquely as $f=\overline f\circ\pi$. The formula $\overline f(v+U)=f(v)$ is well defined because changing the representative by an element of $U$ leaves its value unchanged.
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