extbf{X}_1}(\textbf{x}_k),\nonumber\\
    \textbf{q}_k &amp;=&amp; \textbf{r}_k/\|\textbf{r}_k\|.\nonumber 
\end{eqnarray}\]</div>
<p>If we add <span class="arithmatex">\(\textbf{x}_k\)</span> into <span class="arithmatex">\(\textbf{X}_1\)</span>, the newly fitted value, which is the projection of <span class="arithmatex">\(\textbf{y}\)</span> onto newly spanned space by <span class="arithmatex">\(\textbf{X}_1\)</span> and <span class="arithmatex">\(\textbf{x}_k\)</span>, becomes</p>
<div class="arithmatex"