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== Example of a Linear System == | == Example of a Linear System == | ||
| − | + | Let's take the system <math>y(t)=9x(t)</math>.<br> | |
| + | Let's also say that <math>x1(t)=3t)</math> and <math>x12(t)=7t)</math><br> | ||
| + | Now for the proof...<br> | ||
| + | <math>x3(t)=3t+7t</math><br> | ||
| + | Put it in the system ----> <math>y3(t)=9(3t+7t)=90t</math><br> | ||
| + | Now check to see if it works the other direction...<br> | ||
| + | <math>y1=9(3t)=27t</math><br> | ||
| + | <math>y1=9(7t)=63t</math><br> | ||
| + | <math>y3=y1+y2=63t+27t=90t</math><br> | ||
| + | The system checks out, so it is a linear system. | ||
== Example of a Non-Linear System == | == Example of a Non-Linear System == | ||
Revision as of 06:00, 11 September 2008
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 $ x12(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 $
$ y1=9(7t)=63t $
$ y3=y1+y2=63t+27t=90t $
The system checks out, so it is a linear system.
