Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases

Let $\mathcal B=(b_0,\ldots,b_{n-1})$ be an ordered basis of $V$. The unique coordinates $v=\sum_{j<n}x_jb_j$ form the coordinate column $[v]_{\mathcal B}\in M_{n\times1}(F)$, whose $j$-th entry is $x_j$.

Let $\mathcal C=(c_0,\ldots,c_{m-1})$ be an ordered basis of $W$ and let $T:V\to W$ be linear. The matrix of $T$ relative to $\mathcal B$ and $\mathcal C$ is the matrix $[T]_{\mathcal B}^{\mathcal C}\in M_{m\times n}(F)$ whose $j$-th column is $[T(b_j)]_{\mathcal C}$. Equivalently, if $T(b_j)=\sum_{i<m}t_{ij}c_i$, then $([T]_{\mathcal B}^{\mathcal C})_{ij}=t_{ij}$.

For an empty ordered basis, these definitions give the unique coordinate column or matrix of the corresponding zero-sized shape.