Linearity

Linearity is a property of linear systems (including linear circuits). What this means is that outputs are linear functions of the inputs, i.e., for some input $x$ and output $y$, $y=kx$.

Two important properties of linear systems are homogeneity and additivity, respectively:

$$f(Kx)=Kf(x)$$ $$f(x_1+x_2)=f(x_1)+f(x_2)$$

In other words, if the input is a linear combination of different signals, $T$ can be applied to each component and the results are added (i.e., superposition). Superposition also extends to discrete and continuous sums (integrals), i.e.:

$$T\left\{\sum _{j=1}^n{\alpha }_j{x}_j\right\}=\sum _{j=1}^n{\alpha }_{j}T\{x_j\}$$ $$T\{\int \alpha (\tau )x_{\tau }d\tau \}=\int \alpha (\tau )T\{x_{\tau }\}d\tau$$

What this means is that systems that incorporate integral and derivative components (i.e., capacitors or inductors) are still linear.

Analysis

In circuit analysis, linearity allows us to “work backwards” to solve circuits, with the unit output method. At a terminal, we assume either $i=1A$ or $v=1V$. Then, we use other analysis methods and work backwards to solve for the resulting source value as a linear function of the assumed output. This linear function allows us to find the true output.

To prove the linearity of a system, we need to prove the additivity property holds true; i.e., we take some arbitrary $x_1,x_2$ and outputs $f_1,f_2$, then prove that the final $f(x_1+x_2)$ is a linear combination of the outputs. The homogeneity property is used in our final verification step. Note that a proof doesn’t affect the system, only the input.

One helpful property of linear systems is that their state-space model reduces down to a set of vector/matrix equations. Note that if a ODE is non-linear, it has no transfer function.