(a) Choose c so that Q is an orthogonal matrix:
Q=c\left[ \begin{matrix} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end{matrix} \right] .
Project b = (1, 1, 1, 1) onto the first column. Then project b onto the plane of the first two columns.
(a) c = \frac{1}{2} normalizes all the orthogonal columns to have unit length
(b) The projection
({a}^{T}b/{a}^{T}a)a of b = (1, 1, 1, 1)
onto the first column is
{p}_{1} = \frac{1}{2} (-1, 1, 1, 1)(Check e = 0.) To project onto the plane, add
{p}_{2} = \frac{1}{2} (1, -1, 1, 1) to get (0, 0, 1, 1).