My textbook is trying to prove the linearity of Laplace transforms.
The third equation of the proof has a "$*$" symbol and a negative sign come out from no where? Is this an error/typo? I think it should be
$\int_0^\infty \left( aF_1 e^{-st} + bF_2 e^{-st}\right) \ dt = a\int_0^\infty F_1 e^{-st} \ dt + b\int_0^\infty F_2 e^{-st} \ dt $
I would also be thankful for someone to check that I understand the second part with the exponential orders:
I think this part is showing that $F_1$ and $F_2$ are of exponential order, yes?
$|aF_1 + bF_2| \le |a||F_1| + |b||F_2|$ By the properties of normed vector spaces that multiplying a vector by a positive number changes its length without changing its direction, so $||\alpha \mathbf{x}|| = |\alpha| ||\mathbf{x}||$ for any scalar $\alpha$.
$|a||F_1| + |b||F_2| \le (|a|M_1 + |b|M_2)e^{\alpha_3 t}$, where $\alpha_3 = \max\{\alpha_1, \alpha_2 \}$.
Because we had that $|F_1| \le M_1 e^{\alpha_1 t}$ and $|F_2| \le M_2e^{\alpha_2 t}$
Thank you for all help.