# Definition:Set of All Linear Transformations/Linear Operators

## Definition

Let $R$ be a ring.

Let $G$ be an $R$-module.

The set of all linear operators on $G$ is denoted:

$\map {\LL_R} G := \set {\phi: G \to G: \phi \text{ is a linear operator} }$

If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\map \LL G$.

## Also denoted as

The usual notation for the set of linear operators uses $\mathscr L$ out of the mathscript font, whose $\LaTeX$ code is \mathscr L, but this does not render well on many versions of $\LaTeX$.

When this page was written, that font was unavailable. It is still a future possibility that we change to use $\mathscr L$.

The set of all linear operators can also be denoted as $\map {\mathrm {Hom}_R} G$, or $\map {\mathrm {Hom} } G$ if $R$ is understood.