Suppose I have $m$ column vectors $x_i \in X \subset \mathbb{R}^n$, $i \in \{1,\ldots,n\}$. If I construct a matrix $\mathbf{x}= [x_1^T ; \ldots ; x_m^T]$, then what set does $\mathbf{x}$ live in terms of X?
If I consider $\mathbf{x}$ as a stacked vector $[x_1 ; \ldots; x_m]$, I could say that $\mathbf{x} \in X \times ... \times X$ (m times), but in this case I'm not so sure I can use the Cartesian product anymore since $\mathbf{x}$ is not a vector space.