A system is \emph{linear} if a system defined by $y = F(x)$ such that
$$x \to \fbox{F} \to y $$
satisfies two properties. The first property is \emph{homogeneity}, where $\alpha y = F(\alpha x)$ such that
$$\alpha x \to \fbox{F} \to \alpha y $$
where $\alpha \in \C$. The second property is \emph{additivity}, where $y_1 = F(x_1)$ and $y_2 = F(x_2)$ such that
$$x_1 + x_2 \to \fbox{F} \to y_1 + y_2. $$
As a result, these two properties together form \emph{superposition}, where we have
$$\alpha_1 x_1 + \alpha_2 x_2 \to \fbox{F} \to \alpha_1 y_1 + \alpha_2 y_2 $$


Lets look at some examples:

\begin{problem}

Consider a circuit where the voltage across the circuit is given by Ohm's Law, $V=IR$, where $I$ is the current, and $R$ is the resistance.

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%\includegraphics{images/circuits/resistor.ps}
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\begin{enumerate}
\item If the resistance is constant, is $y(t)=Rx(t)$ linear?
\item What if $R$ varies with time, $y(t)=R(t)x(t)$?
\end{enumerate}
\end{problem}

\begin{solution}
Let $\hat{x} = \alpha x$., then $\hat{y} = R\hat{x} = \alpha R x = \alpha y$. Let $\tilde{x} = x_1 +  x_2$, then $\tilde{y}=R\tilde{x}=Rx_1+Rx_2=y_1+y_2$. Therefore the system is linear. Since we never used the constancy of $R$, we can observe that in both cases we have linearity.
\end{solution}



\begin{problem}
Are the following signals linear?
\begin{enumerate}
\item $y(t) = x^2(t)$
\item $y(t) = \Re(x(t))$
\item $y(t)=\log _i e$
\end{enumerate}
\end{problem}

\begin{solution}

\begin{enumerate}
None of the systems were linear.
\item Not linear. Let $\hat{x} = \alpha x(t)$, then $\hat{y} = \left( \alpha x(t) \right)^2= \alpha^2 x^2(t) \neq \alpha y(t)$
\item Not linear. Let $\hat{x} = \alpha x(t)$, then $\hat{y} = \Re(\alpha x(t))$ where $\alpha \in \C$. When $\alpha = i$, then we have $\hat{y} = \Re(i a(t) - b(t))=-b(t)$, and if $\alpha = 1$, then $\hat{y} = \Re( a(t) +i b(t))=a(t)$. In both cases, $\hat{y} \neq \alpha y(t)$.
\item Not linear, but a fun problem to solve for. Let $\log _i e=y$, then $i^y = e \Rightarrow e^{i \frac{\pi}{2}y} = e \Rightarrow  i \frac{\pi}{2}y = 1 \Rightarrow y = \frac{2}{\pi i}$
\end{enumerate}
\end{solution}


Linearity