### Is Delay a Linear System

There are two key points in this question:

Definition of linear systemMathematical expression of a delay system

1) Any system $f(x(t))$ satisfying following two properties is a linear system:

$ f(\alpha x(t)) = \alpha f(x(t)) $$ f(x_1(t) + x_2(t)) = f(x_1(t)) + f(x_2(t)) $ 2) A pure unit system is $f(x(t)) = x(t)$, delay is just make the output delay by $d$, so delay is $f(x(t)) = x(t-d)$.

Let's verify the first property,

$f(\alpha x(t)) = \alpha x(t-d) = \alpha f(x(t)) $

For the second property,

$ f(x_1(t) + x_2(t)) = x_1(t-d) + x_2(t-d) = f(x_1(t)) + f(x_2(t)) $

Therefore,

Definition of linear systemMathematical expression of a delay system

1) Any system $f(x(t))$ satisfying following two properties is a linear system:

$ f(\alpha x(t)) = \alpha f(x(t)) $$ f(x_1(t) + x_2(t)) = f(x_1(t)) + f(x_2(t)) $ 2) A pure unit system is $f(x(t)) = x(t)$, delay is just make the output delay by $d$, so delay is $f(x(t)) = x(t-d)$.

Let's verify the first property,

$f(\alpha x(t)) = \alpha x(t-d) = \alpha f(x(t)) $

For the second property,

$ f(x_1(t) + x_2(t)) = x_1(t-d) + x_2(t-d) = f(x_1(t)) + f(x_2(t)) $

Therefore,

**delay is a linear system.**