x t ax1 t bx2 t Prove that y t 2 pi x t is linearSolution

x (t) = ax_1 (t) + bx_2 (t) Prove that y (t) = 2 pi x (t) is linear.

Solution

Given Y(t)=2*pi*X(t);

For input X₁(t) output Y₁(t)=2*pi*X₁(t);

For input X2(t) output Y2(t)=2*pi*X2(t);

For input X₃(t) =a*X₁(t)+b*X₂(t) output

  Y₃(t)=2*pi*X₃(t);

Y₃(t)=2*pi*(a*X₁(t)+b*X₂(t)) ;

Y₃(t)=2*pi*a*X₁(t)+2*pi*b*X₂(t);

Y₃(t)=aY₁(t)+b*Y₂(t) ;

The linear combination of inputs gives linear combination of its individual outputs Hence system is linear