(Quadratic Forms) Let $\Omega$ be a variance matrix and W a full-column rank matrix such that $WW' = \Omega$.

Let $W^+ = (W'W)^{-1}W'$ be the Moore-Penrose generalized inverse of W. Show

(a) $P_W = WW^+$ and $W^+y$ is the coefficient vector of W for the orthogonal projection of y onto Col(W),

(b) the quadratic form $y'\Omega^-y = (W^+y)'W'y$ is the squared length of the coefficient vector of W for the orthogonal projection of y onto Col(W),

(c) $W^+\Omega(W^+)' = I$, and

(d) $W'\Omega^-W = I$ for all generalized inverses $\Omega^-$.