linear projection

The transformation P is the orthogonal projection onto the line m.

In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself such that ${\displaystyle P^{2}=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.