n-dimensional input-outspace 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)

First of all, there is a small mistake in the formula. Here, $p=k$ necessarily, in order that you can write the the matrix product $X^{T}\hat{B}$
Then, if you pay attention, in the way you wrote it, $Y$ will indeed be a n-dimensional vector.
Having said that, it is true that in general, in the least square regression problem, generally the input does not have the same dimension as the output.
