(Example)
(Example)
Line 7: Line 7:
  
 
== Example ==
 
== Example ==
<pre>
 
 
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*y1(t)=3at
 +
              +  ---->Output= 3at+6bt ----------(2)
 +
  b*y2(t)=6bt
 +
 
 +
(1)=(2),so linear

Revision as of 16:56, 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

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*y1(t)=3at
             +  ---->Output= 3at+6bt ----------(2)
  b*y2(t)=6bt

(1)=(2),so linear

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett