In this exercise, it will help to draw pictures! Suppose that w is a unit vector in $\mathbb R^d$, so that $w \cdot w = w^T w$ = 1.
(a) Consider the $d \times d$ matrix $P_w = w w^T$. By considering its action on a general vector$v \in \mathbb R^d$, interpret $P_w$ as the rank-$1$ projection onto direction $w$. Show that $P_w$ is symmetric and idempotent (i.e., $P^2 w = Pw$).
(b) Next consider the orthogonal projection $P_\perp = I − P_w$. Show that this is also symmetric and idempotent, and interpret $P_\perp$ geometrically. What is its rank? [Hint: for any $v$, check that we can write $v = P_{wv} + P_{\perp v} $.]
I'm currently stuck on part b by interpreting $P_\perp$ geometrically and finding its rank. I know the projection is a projector onto $(n-1)$-dimensional subspace and the rank is $n-1$?
In addition, the hint is quite obvious but I don't know what can I observe from it by writing $v = P_{wv} + P_{\perp v}$. Stuck on this last part for a while, any response will be helpful, thanks.