Definition:Set of All Linear Transformations

Definition
Let:
 * 1) $$\left({G, +_G: \circ}\right)_R$$ and
 * 2) $$\left({H, +_H: \circ}\right)_R$$

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.