Definition
A linear system is one that satisfies both superposition and homogeneity, also called scaling. Superposition means that the system passes the following test: $ f(x+y)=f(x)+f(y) $. Scaling means that system passes the following test: $ f(ax)=af(x) $.
Example of a Linear System
Let's take the system $ y(t)=9x(t) $.
Let's also say that $ x1(t)=3t $ and $ x2(t)=7t $
Now for the proof...
$ x3(t)=3t+7t $
Put it in the system ----> $ y3(t)=9(3t+7t)=90t $
Now check to see if it works the other direction...
$ y1=9(3t)=27t $
$ y2=9(7t)=63t $
$ y3=y1+y2=63t+27t=90t $
The system checks out, so it is a linear system.
Example of a Non-Linear System
Now let's take a similar system and see if it is linear or not. Let's take $ y(t)=9x(t)+1 $.
Let's also say that $ x1(t)=3t $ and $ x2(t)=7t $
$ x3=3t+7t=10 $
Let's put that into the system...
$ y3=9(10t)+1=90t+1 $
$ y1=9(3t)+1=27t+1 $
$ y2=9(7t)+1=63t+1 $
$ y3=y1+y2=90t+2 $
Note how there are two different results for y3 depending on which step was taken first. This is indicative of a non-linear system, so this system is non-linear.
