I'm currently studying principal component analysis, and I'd like some clarification with regards to the notation used for population and sample principal component scores/vectors. The population principal component scores/vectors are defined as follows:
For $\mathbf{X} \sim (\mathbf{\mu}, \Sigma)$ and $\Sigma = \Gamma \Lambda \Gamma^T$ with $r$ the rank of $\Sigma$, write $\mathbf{X}_{\text{cent}} = \mathbf{X} - \mathbf{\mu}$ for the centered vector.
The principal component score $W_k$ corresponding to the index $k \le r$ is the projection of $\mathbf{X}_{\text{cent}}$ in the direction of $\mathbf{\eta}_k$: $$W_k = \mathbf{\eta}^T_k \mathbf{X}_{\text{cent}} = \mathbf{\eta}^T_k ( \mathbf{X} - \mathbf{\mu})$$ And for $\kappa \le r$, the $\kappa$-dimensional principal component vector $\mathbf{W}^{(\kappa)}$ is $$\mathbf{W}^{(\kappa)} = \begin{bmatrix} W_1 \\ \vdots \\ W_\kappa \end{bmatrix} = \Gamma^T_\kappa \mathbf{X}_{\text{cent}} = [\mathbf{\eta}_1 \dots \mathbf{\eta}_\kappa]^T \mathbf{X}_\text{cent}$$ $\mathbf{W}^{(\kappa)} = [ W_1 \dots W_\kappa]^T$ is the $\kappa$-dimensional principal component vector.
The sample principal component scores/vectors are defined as follows:
For data $\mathbb{X} \sim \text{Sam}(\overline{\mathbf{X}}, S)$ with $S = \hat{\Gamma} \hat{\Lambda} \hat{\Gamma}^T$, the $k$th sample principal component score $\mathbf{W}_{\bullet k}$ is the row vector $$\mathbf{W}_{\bullet k} = \mathbf{\hat{\eta}}_k^T \mathbb{X}_{\text{cent}} = \mathbf{\hat{\eta}}_k^T (\mathbb{X} - \overline{\mathbf{X}})$$ And for $\kappa \le r$ the principal component data $\mathbb{W}^{(\kappa)}$ consist of the first $\kappa$ principal component vectors $\mathbf{W}_{\bullet k}$ with $k \le \kappa$: $$\mathbb{W}^{(\kappa)} = \begin{bmatrix} \mathbf{W}_{\bullet 1} \\ \vdots \\ \mathbf{W}_{\bullet k} \end{bmatrix} = \mathbf{\hat{\Gamma}}^T_\kappa \mathbb{X}_{\text{cent}}$$
The introduction of the $\mathbf{W}_{\bullet k}$ confused me. Whereas the $W_k$ have a single dimension (they seem to just be scalars), the $\mathbf{W}_{\bullet k}$ seem to have two dimensions (including the dot $\bullet$, which I assume means something about across all rows). How do the dimensions of the $\mathbf{W}_{\bullet k}$ relate to its population analogue $W_k$?
EDIT:
With regards to Javier TG's answer, if $\mathbf{X}$ has dimensions $d \times 1$, $\mathbb{X}$ has dimensions $d \times n$, and $\mathbb{W}^{(\kappa)}$ has dimensions $k \times n$, then $\mathbf{W}_{\bullet k}$ would have dimensions $1 \times n$, no? So the dot $\bullet$ could mean that it is over all rows, and therefore $\bullet k$ means that it is over all rows but only column $k$. I mean, it even said that "... $\mathbb{W}^{(\kappa)}$ consist of the first $\kappa$ principal component vectors $\mathbf{W}_{\bullet k}$ ...", so $\mathbf{W}_{\bullet k}$ can't be a scalar, right?