Linearity and nonlinearity of a system

Question

Is $y(t)=\sin(t)$ and $y(t)=\cos(t) \, u(t)$ linear or nonlinear system? İn many resources about linear systems, the relation between $y(t)$ and $u(t)$ is examined. For the first system, there is no $u(t)$ part hence should I think $u(t)=1$ for the first system? Thanks.

Answer 1

A linear function $y(t)$ must satisfy two properties:

1. Additivity: $y(t_1+t_2) = y(t_1) + y(t_2)$
2. Homogeneity: $y(\alpha t) = \alpha y(t)$

Since, $sin(t_1+t_2) \neq sin(t_1) + sin(t_2)$ and $cos(t_1+t_2) \neq cos(t_1) + cos(t_2)$, then $sin(t)$ and $cos(t)$ violate the additivity property and are therefore not linear.

However, the term linear in your case is used to denote how the input $u(t)$ affects an output $y(t)$. So in your example $y(t) = cos(t) u(t)$, we can think of $y$ as having two arguments: $t$ and $u$, such that $y = y(t,u)$. Now we can see that $y(t,u)$ is a linear function with respect to $u$ by looking at the additivity and homogeneity properties:

1. $y(t,u_1+u_2) = cos(t)(u_1+u_2) = cos(t) u_1 + cos(t) u_2 = y(t,u_1) + y(t,u_2)$
2. $y(t,\alpha u) = cos(t)\alpha u = \alpha cos(t)u = \alpha y(t,u)$