Textbook question regarding LTI System I was posed the following homework problem:


  2.10 The following input-output pairs have been observed during the operation of a time-invariant system:
  \begin{align}
x_1(n)&amp;=\{\underset{\uparrow}{1}, 0, 2\}\overset{\mathcal T}{\longleftrightarrow} y_1(n)=\{\underset{\uparrow}{0}, 1, 2\}\\
x_2(n)&amp;=\{\underset{\uparrow}{0}, 0, 3\}\overset{\mathcal T}{\longleftrightarrow} y_2(n)=\{\underset{\uparrow}{0}, 1, 0, 2\}\\
x_3(n)&amp;=\{\underset{\uparrow}{0}, 0, 0,1\}\overset{\mathcal T}{\longleftrightarrow} y_3(n)=\{1,\underset{\uparrow}{2}, 1\}\\
\end{align}
  Can you draw any conclusions regarding the linearity of the system ? What is the impulse response of the system ?


I thought that I had a decent grasp on linearity and time-invariance, but I am having trouble relating them to a specific problem.  I know that linearity means that the system satisfies the superposition principle - and that the weighted sum of the input signals is equal to the weighted sum to each output signal.  

How do I apply that to this problem?  I have the solution below, but I am unsure how they get there:


  2.10
  
  The system is nonlinear. This is evident form observation of the pairs
  $$x_3(n)\leftrightarrow y_3 (n)\text{ and }x_2(n)\leftrightarrow y_2(n).$$
  If the system were linear, $y_2(n)$ would be of the form
  $$y_2(n)=\{3, 6, 3\}$$
  because the system is time-invariant. However, this is not the case.
This looks like a very contrived question, but the idea is simple: if the system were linear, then, since $x_2[n]=3x_3[n+1]$, then $y_2[n]$ should be $3y_3[n+1]$. The impulse response can be obtained by time-shifting $x_3$ so that the $1$ is at $n=0$, and then time-shift $y_3$ the same amount.