Definition of a Linear System
A Linear System is a system that satisfies the properties of Superposition and Scaling.
Example of a Linear System
Inputs:
- $ x_1(t) = 24t\! $
- $ x_2(t) = 13t\! $
Outputs:
- $ y_1(t) = 3(x_1(t)) = 3(24t)\! $
- $ y_2(t) = 3(x_2(t)) = 3(13t)\! $
- $ \alpha y_1(t) + \beta y_2(t) = \alpha*3(24t) + \beta*3(13t)\! $
- $ \alpha y_1(t) + \beta y_2(t) = 3[\alpha(24t) + \beta(13t)]\! $
- $ \alpha y_1(t) + \beta y_2(t) = 3[\alpha x_1(t) + \beta x_2(t)]\! $
Example of a Non-Linear System
- $ x_1(t) = 9t\! $
- $ x_2(t) = t/5\! $
- $ y_1(t) = e^{3x_1(t)} = e^{27t} $
- $ y_2(t) = e^{3x_2(t)} = e^{3t/5} $
- $ y(t) = e^{27t + 3t/5} = e^{27t}e^{3t/5}\! $
- $ e^{27t}e^{3t/5} \neq e^{27t} + e^{3t/5}\! $
