For \(y \in \mathbb{R}^{n}\), a decomposition of the total sum of squares

\[
\|y\|^{2}=y^{\prime} y=y^{\prime} A_{1} y+\cdots+y^{\prime} A_{k} y
\]

is called orthogonal if each \(A_{r}\) is an orthogonal projection \(A_{r}=A_{r}^{2}=A_{r}^{\prime}\). This implies \(\sum A_{r}=I_{n}\) and \(A_{r} A_{s}=0\) for \(r eq s\). Let \(\mathbf{1}_{n}\) be the vector in \(\mathbb{R}^{n}\) with unit components, and let \(J_{n}=\mathbf{1}_{n} \mathbf{1}_{n}^{\prime} / n\). Show that \(J_{n}\) is a projection matrix of rank one, and that \(I_{n}-J_{n}\) is a projection of rank- \(n-1\).