>For example, consider the subspace of $\mathbf{R}^2$ determined by vectors of the form $(2a, a)^{\prime}$. It is not difficult to see that the orthogonal complement of this subspace consists of vectors of the form $(b, 2b)^{\prime}$. The perpendicular projection operator onto the $(2a, a)^{\prime}$ subspace is $$M = \begin{bmatrix}
0.8 & 0.4 \\
0.4 & 0.2\end{bmatrix}$$