$\large \begin{align} T_{\mathcal S} &: \vec x_{\mathcal S} \mapsto \vec y_{\mathcal S}\\ T_\mathcal{B} &= \mathcal{C_{S\to B}} \circ T_{\mathcal S} \circ \mathcal{B\to \mathcal S} \end{align} $ >[!example] Let $\small\mathcal B= \ca{\mat{1\\4},\mat{-1\\3}}$ be a [[Basis]] of $\Rn 3$. $\begin{align} T &: \R^n \to \Rn 3 \\ T &: \mat{\vec x}_{S} \mapsto \mat{3x_{1} + x_{2}\\x_{1}}_{\mathcal S} \end{align} $ $ $