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Projection operators

  1. Mar 21, 2008 #1
    Supposing we have a vector space [tex]V[/tex] and a subspace [tex]V_1\subset V[/tex].
    Suppose further that we have two different direct sum decompositions of the total space [tex]V=V_1\oplus V_2[/tex] and [tex]V_1\oplus V_2'[/tex]. Given the linear projection operators [tex]P_1, P_2, P_1', P_2'[/tex] onto these decompositions, we have that [tex]P_2\circ P_1=P_2\circ P_1'=0[/tex]. But then we have that [tex]P_2(v)=P_2(P_1'+P_2')(v)=P_2 (P_2'(v))[/tex]. Now, for [tex]v\notin V_1[/tex], given any [tex]w\notin V_1[/tex], we can find a decomposition such that [tex]P_2'(v)=w[/tex].
    This gives the apparently wrong result that for any [tex]v,w\notin V_1, ~P_2(v)=P_2(w)[/tex]. Can anyone clarify the mistake?
  2. jcsd
  3. Mar 21, 2008 #2


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    Can you elaborate on this?
  4. Mar 21, 2008 #3
    Of course, I can't, because it's not true. Only if [tex]v-w\in V_1[/tex]. I don't know from where I got that idea...
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