Quotient map of vector spaces ($\pi :V\to V/U$)

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 $U=\mathrm{span}(1,0)$, 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.