Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself (an endomorphism) such that ${\displaystyle P\circ P=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any vector, it gives the same result as if it were applied once (i.e. ${\displaystyle P}$ is idempotent). It leaves its image unchanged.^[1] 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.