A linear function
we have seen is a function whose graph lies on a straight line, and which can be described by giving its slope and its y intercept
Linearity
If both system yield the same output function, this is called a linear system.
Prove
y(t)=2x(t)
[1]
x1(t)--->[system]---->y1(t)=2x1(t)---->*a---(1) a*2*x1(t)
x2(t)--->[system]---->y2(t)=2x2(t)---->*b---(2) b*2*x2(t)
(1)+(2)= 2ax1(t)+2bx2(t)
[2]
x1(t)--->*a---(3) a*x1(t)
x2(t)--->*b---(4) b*x2(t)
(3)+(4)=a*x1(t)+b*x2(t) ---->[system]---->2(a*x1(t)+b*x2(t))=2ax1(t)+2bx1(t)
The results of [1] and [2] are the same. Thus, this is linear system.
y(t)=x(t)^2
[3]
x1(t)--->[system]---->y1(t)=x1(t)^2---->*a---(5) a*x1(t)^2
x2(t)--->[system]---->y2(t)=x2(t)^2---->*b---(6) b*x2(t)^2
(5)+(6)= a*x1(t)^2+b*x2(t)^2
[4]
x1(t)--->*a---(7) a*x1(t)
x2(t)--->*b---(8) b*x2(t)
(7)+(8)=a*x1(t)+b*x2(t) ---->[system]---->(a*x1(t)+b*x2(t))^2
The results of [3] and [4] are not the same. Thus, this is non-linear system.
