Symbols:L

The Set of All Linear Transformations

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 $\mathcal L \left({G}\right)$.

Their $\LaTeX$ codes are as follows:

• $\mathcal L_R \left({G, H}\right)$ : \mathcal L_R \left({G, H}\right)
• $\mathcal L \left({G, H}\right)$ : \mathcal L \left({G, H}\right)
• $\mathcal L_R \left({G}\right)$ : \mathcal L_R \left({G}\right)
• $\mathcal L \left({G}\right)$ : \mathcal L \left({G}\right)

Note

The usual notation for the set of linear transformations involves use of the mathscript font, that is: $\mathscr L$, whose $\LaTeX$ code is \mathscr L, but this does not render in many versions of $\LaTeX$.

Since this site migrated to MathJax, it has become possible to use the $\mathscr L$. However, this has not yet been attempted.