Linear Systems

Q1: What is a linear system?
A1: Consider a system that maps:
$x_1 \to y_1$
$x_2 \to y_2$
The system is linear if it satisfies:
1. Additivity: The output of a sum of inputs is the sum of each input's individual outputs
$x_1 + x_2 \to y_1 + y_2$
2. Homogeneity: The output of a scaled input is the output of that original input scaled by the same factor
$ax_1 \to ay_1$
$bx_2 \to by_2$
where a and b are constants

Q2: Can the two conditions of A1 be encapsulated more succinctly?
A2: Yes. The more succinct restatement is as follows:
Consider a system that maps:
$x_1 \to y_1$
$x_2 \to y_2$
The system is linear if it satisfies additivity and homogeneity:
$ax_1 + bx_2 \to ay_1 + by_2$

Q3: Why care about linear systems?
A3: Systems that are linear allow us to use the technique of superposition.