Linearity
A system Y is linear if for any signals x1(t) and x2(t) x1(t) passed through Y yeilds y1(t) , x2(t) passed through Y yeilds y2(t) and x1(t) + x2(t) passed through Y yeilds y1(t) + y2(t).
Example
for Example the system y(t) = 3*x(t)
x1(t) = sin(t)+8*pi*t
x2(t) = cos(2*t) + 27*t
x1(t) + x2(t) = sin(t) + cos(2*t) + 27*t + 8*pi*t
passing x1(t) through y(t) yeilds y1(t) = 3*sin(t) + 24*pi*t
passing x2(t) through y(t) yeilds y2(t) =3*cos(2*t) + 81*t
y1(t) + y2(t) = 3*cos(2*t) + 3*sin(t)+ 24*pi*t + 81*t
passing x1(t) + x2(t) through y(t) yeilds 3*cos(2*t) + 3*sin(t)+ 24*pi*t + 81*t
Non-Linear System
y(t) = x(t) + 2
x1(t) = sin(t)
x2(t) = cos(t)
x1(t) + x2(t) = sin(t) + cos(t)
passing x1(t) through y(t) yeilds y1(t) = sin(t) + 2
passing x2(t) through y(t) yeilds y2(t) = cos(t) + 2
y1(t) + y2(t) = sin(t) + cos(t) + 4
passing x1(t) + x2(t) through y(t) yeilds sin(t) + cos(t) + 2
these are not equal, thus the system is not linear
