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