(prove)
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== prove ==
+
== Prove ==
 +
 
 +
[1]
 +
y(t)=2x(t)
 +
 
 +
x1(t)--->[system]---->y1(t)=2x1(t)---->*a  ---(1) a*2*x1(t)
 +
 
 +
x2(t)--->[system]---->y2(t)=2x2(t)---->*b  ---(2) b*2*x2(t)
 +
 
 +
(1)+(2)= 2ax1(t)+2bx2(t)
 +
 
 +
 
 +
 
 +
[2]
 +
x1(t)--->*a --- (3) a*x1(t)
 +
x2(t)--->*b --- (4) b*x2(t)
 +
 
 +
(3)+(4)=a*x1(t)+b*x2(t) ---->[system]---->2(a*x1(t)+b*x2(t))=2ax1(t)+2bx1(t)
 +
 
 +
The results of [1] and [2] are the same. Thus, this is linear system.

Revision as of 03:57, 12 September 2008

A linear function

we have seen is a function whose graph lies on a straight line, and which can be described by giving its slope and its y intercept

Linearity

If both system yield the same output function, this is called a linear system.


Prove

[1] y(t)=2x(t)

x1(t)--->[system]---->y1(t)=2x1(t)---->*a ---(1) a*2*x1(t)

x2(t)--->[system]---->y2(t)=2x2(t)---->*b ---(2) b*2*x2(t)

(1)+(2)= 2ax1(t)+2bx2(t)


[2] x1(t)--->*a --- (3) a*x1(t) x2(t)--->*b --- (4) b*x2(t)

(3)+(4)=a*x1(t)+b*x2(t) ---->[system]---->2(a*x1(t)+b*x2(t))=2ax1(t)+2bx1(t)

The results of [1] and [2] are the same. Thus, this is linear system.

Alumni Liaison

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

Dr. Paul Garrett