Definition:Set of All Linear Transformations

Definition

Let:

$(1) \quad \left({G, +_G, \circ}\right)_R$

$(2) \quad \left({H, +_H, \circ}\right)_R$

be $R$-modules.

Then $\mathcal L_R \left({G, H}\right)$ is the set of all linear transformations from $G$ to $H$:

$\mathcal L_R \left({G, H}\right) := \left\{{\phi: G \to H: \phi \mbox{ is a linear transformation}}\right\}$

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

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

$\mathcal L_R \left({G}\right) := \left\{{\phi: G \to G: \phi \text{ is a linear operator}}\right\}$

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

Note

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 $\mathscr L$.

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$