Definition:Set of All Linear Transformations

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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