X}_j-\bb{X}^T\bb{X})\hat\beta.\non
\end{eqnarray}\]</div>
<p>Note that <span class="arithmatex">\(\bb{X}_j\)</span> has the same elements as <span class="arithmatex">\(\bb{X}\)</span> except in their <span class="arithmatex">\(j\)</span>-th column, thus we can rewrite </p>
<div class="arithmatex">\[\begin{equation}
    2\bb{Y}^T(\bb{X}-\bb{X}_j)\hat\beta=2\hat\beta_j\bb{Y}^T(x_j-x^\ast_j),\non
\end{equation}\]</div>
<p>where <span class="arithmatex">\(x_j^\ast\)</span> and <span class="arithmatex">\(x_j\)</