A basis $\mathcal{B}$ for an $n$-dimensional vector space $V$ makes $V$ look like ${\bf R}^n$. A basis $\mathcal{D}$ for an $m$-dimensional vector space $W$ makes $W$ look like ${\bf R}^m$. Together, they make a linear transformation $L: V \to W$ look like a linear transformation from ${\bf R}^n$ to ${\bf R}^n$. In other words, like an $m \times n$ matrix. This is called the matrix of the linear transformation $L$ (with respect to the bases $\mathcal{B}$ and $\mathcal{D}$), and is written $[L]_{\mathcal{DB}}$. This matrix converts of coordinates of the input to the coordinates of the output: $$ [L({\bf v})]_{\mathcal{D}} = [L]_{\mathcal{DB}} [{\bf v}]_{\mathcal{B}}.$$

A special case is where $V=W$, in which case the linear transformation $L: V \to V$ is called an operator on $V$. If we also pick $\mathcal{D}=\mathcal{B}$, using the same coordinates for input and output, then we write $[L]_{\mathcal{B}}$ as shorthand for $[L]_{\mathcal{BB}}$.