Projection (linear algebra)

<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a projection is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> 
  
    
      
        P
      
    
    {\displaystyle P}
  
 from a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> to itself (an <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a>) 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 <a href="/facts/Idempotent/khbSpexv">idempotent</a>). It leaves its <a href="/facts/Image_(mathematics)/KANefMAl">image</a> unchanged. This definition of "projection" formalizes and generalizes the idea of <a href="/facts/Graphical_projection/FATfGH2W">graphical projection</a>.  One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> in the object.
</p>

Projection (linear algebra) open-in-new

Projection (linear algebra)