Proving the property of Linearity in Fourier Transform:
$ ax(t)+by(t) $
is linearly related to
$ aX(j\omega)+bY(j\omega) $
Proof:
$ \mathcal{F}(ax(t))=aX(j\omega) $ and $ \mathcal{F}(by(t))=bY(j\omega) $ for $ ax(t)=2\delta(t) $ and $ bx(t)=3\delta(t-1) $ we find that
$ \mathcal{F}(2\delta(t)=2 $ and $ \mathcal{F}(3\delta(t-1)= 3e^{-j\omega} $ and that $ aX+bY = 2+3e^{-j \omega} $ Summing the two prior to transform we derive $ \mathcal{F}(2\delta(t) + 3\delta(t-1)) = \int_{-\infty}^{+\infty}2\delta(t)+3\delta(t-1)e^{j\omega t}d\omega = 2+3e^{-j \omega} $
