There are three definitions we discussed in class for linearity.

Definition 1

A system is called linear if for any constants $ a,b\in $  all complex numbers and for any input signals x1(t),x2(t) with response y1(t),y2(t), respectively, the system's response to ax1(t) + bx2(t) is ay1(t) + by2(t). 

Definition 2


$ x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_1(t) $

$ x_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_2(t) $


$ ax_1(t) + bx_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow ay_1(t) + by_2(t) $

for any $ a,b\in $  all complex numbers, any x1(t),x2(t) then we say the system is linear.

Definition 3



Linearity can be used to simplify the Fourier transform.  Integration and differentiation are also linear.  Once a non-linear system is made linear, complex systems are easier to model mathematically.  True linear systems are virtually unknown in the real world, but over a small range of variables, systems can be modeled as linear.  

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Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal