Projection operators

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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?
 

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morphism
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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].
Can you elaborate on this?
 
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...
Thanks
 

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