Inverse Laplace Transform of zero

Question

How to prove that the inverse Laplace Transform of zero is zero itself?

$$\mathscr{L}^{-1}\{0\}=0$$

I know that the inverse Laplace Transform of a constant is Dirac's Delta. But I think that that applies only to positive constants.

Thanks

Answer 1

If $\mathcal{L}(f) = F$, then $\mathcal{L}^{-1}(F) =f$. $\mathcal{L}(0) = 0$ because $\mathcal{L}$ is a linear operator. Or you can actually compute $\mathcal{L}(0)$ using the definition.