I've become very confused whilst doing a few practice questions for a digital signal processing exam. Whilst a linear system can be defined as:

$\displaystyle x_{1}[n] \rightarrow y_{1}[n]$

$\displaystyle x_{2}[n] \rightarrow y_{2}[n]$

$\displaystyle a.x_{1}[n] + b.x_{2} \rightarrow a.y_{1}[n] + b.y_{2} $

The more I think about it the more I don't really get how any system won't be linear with those conditions. Maybe I'm just not thinking of the right systems.

Anyway, here are the four systems I have to analyse:

$\displaystyle y[n] = n^{3}.x[n]$

$\displaystyle y[n] = x[n^{2}]$

$\displaystyle y[n] = x[n] . x[n]$

$\displaystyle y[n] = A.x[n] + B$

As far as I can see all equations can fit into the linear proof, but this just doesn't seem to make sense. Can someone help out and give a worked example of one which is and one which isn't linear time invariant, as I can't seem to find anything of that sort in my notes or on the internet. Thanks.