# Representation (Notation) of the Design matrix for Linear Regressions

In the section on linear regressions $\mathbf{Y} = \mathbb{X}\beta + \epsilon$, my textbook represents the design matrix as

$\mathbb{X} = \begin{bmatrix} \mathbf{x}_{1}^T \\ \vdots \\ \mathbf{x}_{n}^T \\ \end{bmatrix} = \begin{bmatrix} x_{11} & \dots & x_{1p} \\ \vdots \\ x_{n1} & \dots & x_{np} \end{bmatrix} \in \mathbb{R}^{n \times p}$

I realise that the $T$ in the vector means transpose, but since the transpose operator is on each individual element $\mathbf{x}$ rather than the entire vector itself, what is its function here? Comparing the vector and the matrix, it seems like the transpose operator is doing nothing here; the matrix looks as it would were the $T$ not present. Or am I misunderstanding something?

I would greatly appreciate it if people could please take the time to clarify this.

Each $\mathbf{x}_i$ is a $p \times 1$ matrix: $$\mathbf{x}_1 = \begin{bmatrix}{x_{11} \\ \ \ \vdots \\ x_{1p}}\end{bmatrix}, \ldots,\ \mathbf{x}_n = \begin{bmatrix}{x_{n1} \\ \ \ \vdots \\ x_{np}}\end{bmatrix}.$$ So each $\mathbf{x}_i^T$ is a $1 \times p$ matrix: $$\mathbf{x}_1^T = \begin{bmatrix}{x_{11} \ \ldots \ x_{1p}}\end{bmatrix}, \ldots,\ \mathbf{x}_n^T = \begin{bmatrix}{x_{n1} \ \ldots \ x_{np}}\end{bmatrix}.$$ The notation $$\begin{bmatrix} \mathbf{x}_{1}^T \\ \vdots \\ \mathbf{x}_{n}^T \\ \end{bmatrix}$$ means that the $1 \times p$ matrices are stacked on top of each other, giving the $n \times p$ matrix $$\begin{bmatrix} x_{11} & \dots & x_{1p} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{np} \end{bmatrix}.$$
The issue here is only about notation. Usually, in textbooks the vector symbol $\boldsymbol{x}$ represents the column vector of dimension $p \times 1$. By using the transpose, you are effectively considering the design matrix of dimension $n \times p$.
The initial notational convention here is that by $\mathbb x$ we mean a column vector.
Further, in regression analysis (and in most but not all places/fields), and when using matrix algebra notation we use $\mathbb x_i$ to represent the observation vector (i.e. one value from each regressors in the same cross section or time period).
Therefore, $\mathbb x_i^T$ represents the full first observation as a row.