Conditions that we can get from $P^2=P$ for $P \in L(V)$ where $V$ is finite-dimensional inner product space.

Question

This must imply $V=N(P) \oplus R(P)$.

Also, I've learned that $V=N(P) \oplus R(P)$ implies that $u-Pu$ can represent any element in $N(P)$.

However, I don't understand why this is possible to represent any vector in $N(P)$.

Doesn't this have to be $a-Pb$ for any $a,b \in V$? Why can they share the same vector $u$?