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== Example ==
 
== Example ==
<pre>
+
A)LINEAR
Let:
+
  Let:
     x1(t)=t, x2(t)=2*t;
+
     x1(t)=t, x2(t)=2t; 
     y1(t)=2*x1(t)= 2*t, y2(t)=3*x2(t)= 6*t;
+
    System: y(t)=3*x(t)
     a=2, b=3;
+
     Thus, y1(t)=3t,y2(t)=6t
  so, a*x1(t)+b*x2(t)=
+
  So say a,b are any non-zero constant
</pre>
+
    a*x1(t)->system->3at
 +
                        +    --->Output= 3at+6bt    -----(1)
 +
    b*x2(t)->system->6bt 
 +
 
 +
 
 +
    a*x1(t)+b*x2(t)=at+2bt->system->Output=3*(at+2bt)= 3at+6bt ----------(2)
 +
 
 +
    (1)=(2),so linear.
 +
 
 +
B)NON-LINEAR
 +
  Let:
 +
    x1(t)=t, x2(t)=2t;  
 +
     System: y(t)=x(t)^2
 +
    Thus, y1(t)=t^2,y2(t)=4t^2
 +
 
 +
  So say a,b are any non-zero constant
 +
    a*x1(t)->system->a*t^2
 +
                            +    --->Output= a*t^2+4b*t^2    -----(3)
 +
    b*x2(t)->system->b*4t^2
 +
 
 +
    a*x1(t)+b*x2(t)=at+2bt->system->Output=(at+2bt)^2 ----------(4)
 +
 
 +
    (3)!=(4),so non-linear.

Latest revision as of 17:02, 12 September 2008

Linear System

A system is called "Linear" if for any constants a,b and for any inputs x1(t),x2(t),(x1[n],x2[n]) yielding output y1(t),y2(t),respectively, the respond to a*x1(t)+b*x2(t) is a*y1(t)+b*y2(t)

Example

A)LINEAR

 Let:
    x1(t)=t, x2(t)=2t;   
    System: y(t)=3*x(t)
    Thus, y1(t)=3t,y2(t)=6t
 So say a,b are any non-zero constant
    a*x1(t)->system->3at
                       +    --->Output= 3at+6bt     -----(1)
    b*x2(t)->system->6bt  


    a*x1(t)+b*x2(t)=at+2bt->system->Output=3*(at+2bt)= 3at+6bt ----------(2)
    (1)=(2),so linear.

B)NON-LINEAR

 Let:
    x1(t)=t, x2(t)=2t;   
    System: y(t)=x(t)^2
    Thus, y1(t)=t^2,y2(t)=4t^2
 So say a,b are any non-zero constant
    a*x1(t)->system->a*t^2
                           +    --->Output= a*t^2+4b*t^2     -----(3)
    b*x2(t)->system->b*4t^2
    a*x1(t)+b*x2(t)=at+2bt->system->Output=(at+2bt)^2 ----------(4)
    (3)!=(4),so non-linear.

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

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