ils>
<details class="done"><summary>Soln. 3.9</summary><p>Without loss of generality we assume <span class="arithmatex">\(\textbf{X}_1=(\textbf{x}_1, ...,\textbf{x}_q)\in\mathbb{R}^{N\times q}\)</span>. Write the QR decomposition for <span class="arithmatex">\(\textbf{X}_1\)</span> as</p>
<div class="arithmatex">\[\begin{equation}
        \textbf{X}_1 = \textbf{Q}_1\textbf{R}_1,\nonumber
\end{equation}\]</div>
<p>where <span class="arithmatex">\(\textbf{Q}_1=(\textbf{q}_1, ..., \textbf{q}_q)\)</span> is a <