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In the least squares regression model, we have

$$\hat{Y} = X^T \hat{B}$$

where $$\hat{B} \in \mathbb{R}^{pxn}, X^T \in \mathbb{R}^{nxk}$$

How is it that $(X, \hat{Y}) \in \mathbb{R}^n$ referred to as the n-dimensional input-output space? I understand that the inputs are n-vectors, but the components of y aren't necessarily unless $p = n$, so curious as to why this is referred to as the input-output space (machine learning context)

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