Linear Combination of Laplace Transforms

Theorem
Let $\mathcal L$ be the Laplace Transform.

Let $f, g$ be a functions such that $\mathcal L f$ and $\mathcal Lg$ exist.

Let $\lambda \in \C$ or $\R$ be constant.

Then $\mathcal L$ is a linear operator on $f$ and $g$:


 * $\mathcal L \left \{{\lambda f\left({t}\right) + g\left({t}\right)}\right\} = \lambda \mathcal L \{ {f\left({t}\right)}\} + \mathcal L \left\{ {g\left({t}\right)}\right\}$

everywhere all the above expressions are defined.