# Projection (linear algebra)

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.[1] 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.