How to prove that proj(proj(b onto a) onto a) = proj(b onto a)? How to prove that proj(proj(b onto a) onto a) = proj(b onto a)?
It makes perfect sense conceptually, but I keep going in circles when I try to prove it mathematically. Any help would be appreciated.

Note that a projection $P$ satisfies $P^2 = P$. You are just applying a projection twice, so that's the same thing as applying it once. 
