Projection (mathematics)

A projection 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. A linear projection additionally respects vector addition and scalar multiplication.

In mathematics, "projection" can refer to several concepts depending on the context, but it typically involves the idea of reducing a higher-dimensional object to a lower-dimensional representation or mapping points from one space to another. Here are some common interpretations of projection: 1. **Linear Projection in Linear Algebra**: In the context of vector spaces, a projection refers to a linear transformation that maps a vector onto a subspace.