# Definition:Set of All Linear Transformations

## Contents

## Definition

Let $R$ be a ring.

Let $G$ and $H$ be $R$-modules.

Then $\operatorname{Hom}_R \left({G, H}\right)$ is the **set of all linear transformations** from $G$ to $H$:

- $\operatorname{Hom}_R \left({G, H}\right) := \left\{{\phi: G \to H: \phi \mbox{ is a linear transformation}}\right\}$

If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\operatorname{Hom} \left({G, H}\right)$.

Similarly, $\operatorname{Hom}_R \left({G}\right)$ is the set of all linear operators on $G$:

- $\operatorname{Hom}_R \left({G}\right) := \left\{{\phi: G \to G: \phi \text{ is a linear operator}}\right\}$

Again, this can also be written $\operatorname{Hom} \left({G}\right)$.

## Also denoted as

The **set of all linear transformations** can also be denoted as $\mathcal L_R \left({G, H}\right)$.

The usual notation for the set of linear transformations uses $\mathscr L$ out of the **mathscript** font, whose $\LaTeX$ code is `\mathscr L`, but this does not render 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 $\mathscr L$.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$ - 1969: M.F. Atiyah and I.G. MacDonald:
*Introduction to Commutative Algebra*: $\S 2$: Modules: Modules and Module Homomorphisms