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