invertible transformation

A linear map $T$ with an $S$ that perfectly undoes it, $S(T(\mathbf x))=\mathbf x$ .

$T:\mathbb R^n\to\mathbb R^n$ is invertible if a transformation $S$ exists with $S(T(\mathbf x))=\mathbf x$ and $T(S(\mathbf y))=\mathbf y$ for all inputs; $S$ , written $T^{-1}$ , is unique and automatically linear. $T$ is invertible exactly when it is both one-to-one and onto, exactly when its standard matrix $A$ is invertible ( $\det A\ne 0$ , full rank), and then $T^{-1}(\mathbf y)=A^{-1}\mathbf y$ .