I'm sort of confused by this statement in a text I'm reading:
Given a projector $P \in \mathbb{C}^{m \times m}$ define $\text{range}(P) = S_1$ and null$(P)= S_2$. The projector and its complement can be seen as the unique solution to the problem
- Given $v$, find vectors $v_1 \in S_1$ and $v_2 \in S_2$ such that $v_1 + v_2 = v$ The projection $Pv$ gives $v_1$, and the complementary projection $(I-P)v$ gives $v_2$.
These vectors are unique because all solutions must be of the form
$$(Pv + v_3) + ((I-P)v-v_3) = v$$ where it is clear that $v_3$ must be in both $S_1$ and $S_2$, i.e., $v_3 = 0$.
To me it appears $v_3$ could be any vector since the form above is simply the addition and subtraction of $v_3$. Why does that form make clear that $v_3$ must be zero? For example let $v_3$ be any nonzero vector then
$$(Pv + v_3) + ((I-P)v-v_3) = Pv + (I-P)v = v_1 + v_2 = v$$
$$\begin{align}& \underbrace{Pv+v_3}_{\in S_1}+\underbrace{(I-P)v-v_3}_{\in S_2}=v \\ &\implies v_3 \in S_1 \cap S_2 \\ &\implies v_3 = 0 \qquad \qquad \text{since } S_1 \cap S_2 = 0 \text{ for complementary subspaces}\end{align}$$