# 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:

### Vector Space

Let $K$ be a field.

Let $X, Y$ be vector spaces over $K$.

Then $\map {\LL} {X, Y}$ is the **set of all linear transformations** from $X$ to $Y$:

- $\map {\LL} {X, Y} := \set {\phi: X \to Y: \phi \mbox{ is a linear transformation} }$

## 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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations - 1969: M.F. Atiyah and I.G. MacDonald:
*Introduction to Commutative Algebra*: $\S 2$: Modules: Modules and Module Homomorphisms