Linearity

A system is linear if the responses to inputs multiplied by constants are the original responses multiplied by the same constants.

In symbols:
If
$ y1(t)+y2(t)\, $ are the responses to inputs $ x1(t)+x2(t)\, $
Then
$ a*x1(t)+b*x2(t)\, $ should yield $ a*y1(t)+b*y2(t)\, $

Example

A very simple linear system is $ y(t) = t*x(t)\, $
If $ x(t) = t^2\, $, then $ y(t) = t^3\, $
Multiplying $ x(t)\, $ by a constant $ a\, $ yields $ y(t) = t*x(t) = t*a*t^2 = a*t^3\, $
This fulfills the requirements that $ a*x(t)\, $ yields $ a*y(t)\, $

In the same sense, you can prove that $ y(t) = x(t)^2\, $ is not linear. Multiplying $ x(t)\, $ by $ a\, $ would yield $ y(t) = a^2*t^2\, $

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang