Suppose $\ltdefn{T}{U}{V}$ is a linear transformation and $\vect{v}\in V$. If the preimage $\preimage{T}{\vect{v}}$ is nonempty, and $\vect{u}\in\preimage{T}{\vect{v}}$ then \begin{equation*} \preimage{T}{\vect{v}}= \setparts{\vect{u}+\vect{z}}{\vect{z}\in\krn{T}} =\vect{u}+\krn{T} \end{equation*}