= Projection
{disambiguate=mathematics}
{title2=$p^2=p$}
{wiki}
A <projection (mathematics)> is an <idempotent> <self-map> $p:X\to X$: applying it twice has the same effect as applying it once, $p\circ p=p$. Equivalently, it fixes every point of its <image of a function>. The map $p(x,y)=(x,0)$ on the real plane is an example.
A coordinate <projection map> $X\times Y\to X$ has different source and target, so it is not literally an <idempotent> <self-map>. After choosing $y_0\in Y$ and identifying $X$ with $X\times\{y_0\}$, its lifted form $(x,y)\mapsto(x,y_0)$ is a <projection (mathematics)>. A <linear projection> additionally respects <vector addition> and <scalar multiplication>.
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