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Projection (linear algebra)
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{{Short description|Idempotent linear transformation from a vector space to itself}} {{redirect|Orthogonal projection|the technical drawing concept|Orthographic projection|a concrete discussion of orthogonal projections in finite-dimensional linear spaces |Vector projection}} [[File:Orthogonal projection.svg|frame|right|The transformation ''P'' is the orthogonal projection [[surjective function|onto]] the [[line (geometry)|line]] ''m''.]] In [[linear algebra]] and [[functional analysis]], a '''projection''' is a [[linear transformation]] <math>P</math> from a [[vector space]] to itself (an [[endomorphism]]) such that <math>P\circ P=P</math>. That is, whenever <math>P</math> is applied twice to any vector, it gives the same result as if it were applied once (i.e. <math>P</math> is [[idempotent]]). It leaves its [[image (mathematics)|image]] unchanged.<ref>Meyer, pp 386+387</ref> 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 [[point (geometry)|point]]s in the object.
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