Linearity
Suppose the output of a system is y1(t) when the input is x1(t), and the output is y2(t) when the input is x2(t). We say that the system is linear if and only if the output is :
\({y\left(t\right)=ay}_1\left(t\right)+by_2(t)\) (1)
when the input is a linear combination:
\({x\left(t\right)=ax}_1\left(t\right)+bx_2(t)\) (2)
We call this property the superposition property :
\(x\left(t\right)=\sum_{i}{a_ix_i(t)} matches the answer y\left(t\right)=\sum_{i}{a_iy_i(t)}\)
For example
\(y\left(t\right)=Rx(t)\)
This equation can represent an output, which is the voltage across a resistor, given an input current x(t). For input x1(t), the output is Rx1(t), and, for input x2(t), and the output is Rx2(t).
What is the output for \(ax_1\left(t\right)+bx_2(t)\) ? The output is R times the input, or \(R\left[ax_1\left(t\right)+bx_2(t)\right]\) which we can express as:
\(R\left[ax_1\left(t\right)+bx_2(t)\right]=aRx_1\left(t\right)+bRx_2\left(t\right)=ay_1\left(t\right)+by_2(t)\)
So, indeed, it is a linear system
Example (Counter-example)
\(y\left(t\right)=x^2(t)\)
This equation can represent an output, which is the current of a photodiode given the incident photons x(t). For input x1(t), the output is x12(t), and, for input x2(t), the output is x22(t).
What is the output for \(ax_1\left(t\right)+bx_2(t)\) ? The output is the square of the input, so:
\(y\left(t\right)=\left[ax_1\left(t\right)+bx_2\left(t\right)\right]^2=a^2x_1^2\left(t\right)+b^2x_2^2\left(t\right)+2abx_1\left(t\right)x_2\left(t\right)\neq ay_1\left(t\right)+by_2\left(t\right)\)
This system is not linear system.