Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. If there is a function $\ltdefn{S}{V}{U}$ such that \begin{align*} \compose{S}{T}&=I_U & \compose{T}{S}&=I_V \end{align*} then $T$ is invertible. In this case, we call $S$ the inverse of $T$ and write $S=\ltinverse{T}$.