Math 6010 • Linear Models

(20 points) Let
\[
v = (1, -1, 0)^{\top}, \qquad \Omega = \operatorname{Span}\{v\} \subseteq \mathbb{R}^{3}.
\]
\begin{enumerate}
\item Write the projection matrix \(P_{\Omega}\) onto \(\Omega\).
\item Verify that \(P_{\Omega}\) is a projection matrix by checking that it is symmetric and idempotent. (To
avoid a long calculation, you can write out these calculations in terms of \(v\).)
\item Determine the eigenvalues of \(P_{\Omega}\) and their multiplicities. What is \(\operatorname{rank}(P_{\Omega})\)?
\item Find an orthogonal basis \(\{b_{1}, b_{2}\}\) for \(\Omega^{\perp}\) and verify that \(P_{\Omega}b_{i} = 0\) for \(i = 1, 2\).
\end{enumerate}