# Definition:Set of All Linear Transformations

## 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 $\mathscr L$.