Exercise 12

Prove that the map $p\colon V\to V$ is the projection into $M$ along $N$ if and only if the map $\operatorname{id}-p$ is the projection into $N$ along $M$. (Recall the definition of the difference of two maps: $(\operatorname{id}-p)\,(\vec{v})=\operatorname{id}(\vec{v})-p(\vec{v})=\vec{v}-p(\vec{v})$.)