\begin{definition}[Definition:Set of All Linear Transformations]
Let $R$ be a ring.
Let $G$ and $H$ be $R$-modules.
Then $\map {\LL_R} {G, H}$ is the '''set of all linear transformations''' from $G$ to $H$:
:$\map {\LL_R} {G, H} := \set {\phi: G \to H: \phi \mbox{ is a linear transformation} }$
If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\map \LL {G, H}$.
\end{definition}