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 $$H$$:


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

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 $$G$$:


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

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

Note
The usual notation for the set of linear transformations involves use of the mathscript font, whose LaTeX code is \mathscr L, but this does not render on ProofWiki.