# Definition:Set of All Linear Transformations/Linear Operators

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## 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.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations

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- 1969: M.F. Atiyah and I.G. MacDonald:
*Introduction to Commutative Algebra*: $\S 2$: Modules: Modules and Module Homomorphisms