Proving that the Continuous-Time Fourier Transform demonstrates linearity
Property:
$ F(a x(t) + b y(t)) = a X(jw) + b Y(jw) $
Derivation:
$ F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}[a x(t)+b y(t)] e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = \int\limits_{-\infty}^{\infty}a x(t) e^{(jwt)}dt + \int\limits_{-\infty}^{\infty}b y(t) e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = a \int\limits_{-\infty}^{\infty}x(t) e^{(jwt)}dt + b \int\limits_{-\infty}^{\infty}y(t) e^{(-jwt)}dt $
$ F(a x(t) + b y(t)) = a X(jw) + b Y(jw) $ (definition of linearity)
