(New page: == Definition == A linear system is one that satisfies both superposition and homogeneity. Superposition means that the system passes the following test: <math>f(x+y)=f(x)+f(y)</math>. Hom...) |
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== Definition == | == Definition == | ||
| − | A linear system is one that satisfies both superposition and homogeneity. Superposition means that the system passes the following test: <math>f(x+y)=f(x)+f(y)</math>. | + | A linear system is one that satisfies both superposition and homogeneity, also called scaling. Superposition means that the system passes the following test: <math>f(x+y)=f(x)+f(y)</math>. Scaling means that system passes the following test: <math>f(ax)=af(x)</math>. |
| − | + | ||
| − | + | ||
== 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>x2(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>y2=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 == | ||
| + | Now let's take a similar system and see if it is linear or not. Let's take <math>y(t)=9x(t)+1</math>.<br> | ||
| + | Let's also say that <math>x1(t)=3t</math> and <math>x2(t)=7t</math><br> | ||
| + | <math>x3=3t+7t=10</math><br> | ||
| + | Let's put that into the system...<br> | ||
| + | <math>y3=9(10t)+1=90t+1</math><br> | ||
| + | <math>y1=9(3t)+1=27t+1</math><br> | ||
| + | <math>y2=9(7t)+1=63t+1</math><br> | ||
| + | <math>y3=y1+y2=90t+2</math><br> | ||
| + | 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. | ||
Latest revision as of 07:10, 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 $ 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.
