Definition:Set of All Linear Transformations

Definition
Let $R$ be a ring. Let $G$ and $H$ be $R$-modules.

Then $\map {\mathrm {Hom}_R} {G, H}$ is the set of all linear transformations from $G$ to $H$:


 * $\map {\mathrm {Hom}_R} {G, H} := \set {\phi: G \to H: \phi \mbox{ is a linear transformation} }$

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

Similarly, $\map {\mathrm {Hom}_R} G$ is the set of all linear operators on $G$:


 * $\map {\mathrm {Hom}_R} G := \set {\phi: G \to G: \phi \text{ is a linear operator} }$

Again, this can also be written $\map {\mathrm {Hom} } G$.

Specific Instances
Specific instantiations of this concept to particular modules are as follows:

Also denoted as
The set of all linear transformations can also be denoted as $\map {\LL_R} {G, H}$.

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

 * Definition:Endomorphism Ring of Module