Projection Operators on Vector Spaces: Clarifying Mistakes

[HenryGomes]

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

 

[morphism]

Now, for v\notin V_1, given any w\notin V_1, we can find a decomposition such that P_2'(v)=w.

Can you elaborate on this?

 

[HenryGomes]

Of course, I can't, because it's not true. Only if v-w\in V_1. I don't know from where I got that idea...
Thanks