I am in an introductory linear algebra course and have already taken differential equations.
In my textbook (David Lay's Linear Algebra and it's Applications) Lay states that to show something is a linear transformation you must show two properties.
$\mathcal{L}\{f(x)+g(x)\}=\mathcal{L}\{f(x)\}+\mathcal{L}\{g(x)\}$
and
$c*\mathcal{L}\{f(x)\}=\mathcal{L}\{c*f(x)\}$
I feel like I've shown this to be true on my own but I have no way of checking. My assumption is that it has to be a linear transformation, if it's being used to solve linear ordinary differential equations but I haven't seen this question asked and I can't find any literature on it. I've been able to find out that it is a linear operator which appears to be equivalent statement to saying that it is a linear transformation but I'm unsure.
Just unrolling the definition: $\mathcal L(f+g)(s)=\int_\mathbb R e^{-st}\left(f(t)+g(t)\right)dt=\int_\mathbb R e^{-st}f(t)dt+\int_\mathbb Re^{-st}g(t)dt=\mathcal Lf(s)+\mathcal Lg(s)$.
And
$$ \mathcal L(cf)(s)=\int_\mathbb Re^{-st} (cf(t))dt=c\int_\mathbb Re^{-st} (f(t))dt=c\mathcal L(f)(s). $$