Linearity

The Laplace transform is a linear operator. To show this, let $w(t)$ denote a linear combination of signals $x(t)$ and $y(t)$ ,

$$w(t) = \alpha x(t) + \beta y(t),$$

where $\alpha$ and $\beta$ are real or complex constants. Then we have

$$\begin{eqnarray*} W(s) &\isdef & {\cal L}_s\{w\} \isdef {\cal L}_s\{\alpha x(t) + \beta y(t)\}\\ &\isdef & \int_0^\infty \left[\alpha x(t) + \beta y(t)\right] e^{-st} dt\\ &=& \alpha \int_0^\infty x(t) e^{-st} dt + \beta \int_0^\infty y(t) e^{-st} dt\\ &\isdef & \alpha X(s) + \beta Y(s). \end{eqnarray*}$$

Thus, linearity of the Laplace transform follows immediately from the linearity of integration.