span class="arithmatex">\(\textbf{x}_k\)</span> onto the column space of <span class="arithmatex">\(\textbf{X}_1\)</span>, denoted by <span class="arithmatex">\(\mathcal{P}_{\textbf{X}_1}(\textbf{x}_k)\)</span>, is</p>
<div class="arithmatex">\[\begin{equation}
    \mathcal{P}_{\textbf{X}_1}(\textbf{x}_k) = \sum_{j=1}^q(\textbf{x}_k^T\textbf{q}_j) \textbf{q}_j.\nonumber
\end{equation}\]</div>
<p>Define</p>
<div class="arithmatex">\[\begin{eqnarray}
    \textbf{r}_k &amp;=&amp; \textbf{x}_k - \mathcal{P}_{\t