Linear Regression - Vector Notation

Question

I'm a bit confused about the vector notation of the linear regression vector notation. We have this:

$X : n\times p$ matrix of data we have obtained;

$\beta: p\times 1$ matrix of coefficients

I understand the use of these in matrix notation, however when it gets to vector notation we have:

$$Y_i = x_i^T\beta + \epsilon_i$$

What is confusing me is the dimensions of the $x_i^T$ and the $\beta$. I understand what $x_i^T$ is (i.e. the vector of predictors for observation $i$), but to me the dimensions of $x_i^T$ are $p \times 1$, which would be incompatible to multiply with $\beta$. I know they should be (and presumably are) $1 \times p$, but to me the inclusion of the transpose makes them $p \times 1$.

If anyone could shed light on what I'm missing out on here, that'd be great.

Answer 1

The matrix $\boldsymbol{X}$ is defined as $$ \boldsymbol{X}= \begin{pmatrix} \boldsymbol{x}_1^T\\ \vdots\\ \boldsymbol{x}_n^T\\ \end{pmatrix}, $$ so that, indeed, $\boldsymbol{x}_i^T$ is $(1\times p)$.