pan class="arithmatex">\(\bb{X}\)</span> to <span class="arithmatex">\(\boldsymbol{\hat Y}\)</span> via linear regression. In detail, let <span class="arithmatex">\(\boldsymbol{\hat Y} = \bb{X}(\bb{X}^T\bb{X})^{-1}\bb{X}^T\bb{Y} = \bb{X}\hat B\)</span> where <span class="arithmatex">\(\bb{Y}\)</span> is the indicator response matrix. Similarly for any input <span class="arithmatex">\(x\in\mathbb{R}^p\)</span>, we get a transformed vector <span class="arithmatex">\(\hat y = \hat B^Tx\in \mathbb{R}^K\)</span>