In the $(p + 1)$-dimensional input-output space, $(X, \hat{Y})$ represents a hyperplane? I am currently studying the textbook The Elements of Statistical Learning, second edition, by Hastie, Tibshirani, and Friedman. Chapter 2.3.1 Linear Models and Least Squares says the following:

The linear model has been a mainstay of statistics for the past 30 years and remains one of our most important tools. Given a vector of inputs $X^T = (X_1, X_2, \dots, X_p)$, we predict the output $Y$ via the model
$$\hat{Y} = \hat{\beta}_0 + \sum_{j = 1}^p X_j \hat{\beta}_j . \tag{2.1}$$
The term $\hat{\beta}_0$ is the intercept, also known as the bias in machine learning. Often it is convenient to include the constant variable $1$ in $X$, include $\hat{\beta}_0$ in the vector of coefficients $\hat{\beta}$, and then write the linear model in vector form as inner product
$$\hat{Y} = X^T \hat{\beta}, \tag{2.2}$$
where $X^T$ denotes vector or matrix transpose ($X$ being a column vector). Here we are modelling a single output, so $\hat{Y}$ is a scalar; in general $\hat{Y}$ can be a $K$-vector, in which case $\beta$ would be a $p \times K$ matrix of coefficients. In the $(p + 1)$-dimensional input-output space, $(X, \hat{Y})$ represents a hyperplane.

I'm struggling to make sense of this. If $X^T$ is a vector or matrix transpose and $\hat{Y}$ is a $K$-vector, then what exactly are the dimensions of $X^T$, $\hat{Y}$, and $\hat{\beta}$ for this all to make mathematical sense? Furthermore, how does the $\beta$ relate to this (this might be a typo by the authors)? And lastly, how does this then show that, in the $(p + 1)$-dimensional input-output space, $(X, \hat{Y})$ represents a hyperplane?
 A: You have to think that $(X, \hat{Y}) $ is a vector with $p+K$ components, given by $ (X_1,\ldots, X_p, \hat{Y}_1, \ldots, \hat{Y}_K ) $.
In general, let $B$ be the $(p+K) \times K$ matrix which consists of two blocks, one over the other:

*

*The upper block is the $p\times K$ matrix $\hat{\beta}$;

*The lower block is the $K \times K$ matrix given by $-Id$.

Let me denote by $\hat{\beta}^i$ the $i$-th column of $\hat{\beta}$. Note that the $i$-th column of $B$ is then given by $(\hat{\beta}^i, -e_i) $; here I am using the same convention of $(X, \hat{Y}) $ by concatening two vectors one after another (the first with $p$ components and the second with $K$ components).
Let's see what does the equation $(X, \hat{Y})^T B = 0$ means.  The output is a vector with $K$ components. The $i$-th component is the scalar product of the the input-output with the $i$-th column of $B$, that is
$$ 0= (X, \hat{Y}) \cdot (\hat{\beta}^i, -e_i) = (X, \hat{\beta}^i) + (\hat{Y}, -e_i) $$
Which is equivalent to
$$ \hat{Y}_i =  (X, \hat{\beta}^i) $$
In the same spirit of above, note that the RHS is the i-th component of $X^T \hat{\beta}$! This means our equation is equivalent to
$$ \hat{Y} = X^T \hat{\beta}$$
Which is the original equation! All this stuff was a fancy way to demonstrate you that, whenever you have some variables constrained by linear equations, you can always put your equations into the form $(variables) ^T (matrix) = 0$, and this is what happening in your case.
In the pleasant case in which you have just one equation ($K$=1), the matrix $B$ will have just one column; in other words $B$ is a vector, and the $B$ equation reads as
$$ (X, \hat{Y}) \cdot B =0$$
Such an equation describes exactly the hyperplane perpendicular to the vector $B$! Think about this geometrically: if I ask you what is the locus of points in $\mathbb{R}^3$ perpendicular to the vector $(0, 0,1) $, this will be the $x-y$ hyperplane.
