(Example of a Linear System)
Line 17: Line 17:
 
::<math>\,x2(t) = cos(t)</math>
 
::<math>\,x2(t) = cos(t)</math>
  
::<math>\,y1(t) = C\left\{x1(t)\right\} = C(sin(t))</math>
+
::<math>\,y1(t) = \pi\left\{x1(t)\right\} = \pi(sin(t))</math>
::<math>\,y2(t) = C\left\{x2(t)\right\} = C(cos(t))</math>
+
::<math>\,y2(t) = \pi\left\{x2(t)\right\} = \pi(cos(t))</math>
  
::<math>\,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\}</math>
+
::<math>\,ay1(t) + by2(t) = a*\pi*sin(t) + b*\pi*cos(t) = \pi\left\{asin(t) + bcos(t)\right\}  
 +
= \pi\left\{ax1(t) + bx2(t)\right\}</math>
  
  
Thus, <math>\,y(t) = Cx(t)</math> is a linear system.
+
Thus, <math>\,y(t) = \pi(x(t))</math> is a linear system.

Revision as of 04:48, 9 September 2008

Linearity

A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems:

Suppose there are two inputs
$ \,x1(t) $
$ \,x2(t) $
with outputs
$ \,y1(t) = C\left\{x1(t)\right\} $
$ \,y2(t) = C\left\{x2(t)\right\} $
A linear system must satisfy the condition
$ \,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\} $

Example of a Linear System

$ \,x1(t) = sin(t) $
$ \,x2(t) = cos(t) $
$ \,y1(t) = \pi\left\{x1(t)\right\} = \pi(sin(t)) $
$ \,y2(t) = \pi\left\{x2(t)\right\} = \pi(cos(t)) $
$ \,ay1(t) + by2(t) = a*\pi*sin(t) + b*\pi*cos(t) = \pi\left\{asin(t) + bcos(t)\right\} = \pi\left\{ax1(t) + bx2(t)\right\} $


Thus, $ \,y(t) = \pi(x(t)) $ is a linear system.

Alumni Liaison

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

Dr. Paul Garrett