How does one understand graphically which map is linear and which one is not?I list 2 conditions below which I intuitively think to be both necessary and sufficient graphically for a map to be linear.I will restrict myself to transformations from $\Bbb R^n$ to $\Bbb R^m$,where $n,m = 1,2,3$.Then I think that the following checking will do:
1.Grid lines will remain parallel and equally spaced.
2.Origin remains fixed at origin.
3.Any line will remain a line or squish into a point. I think checking these and only these conditions will guarantee that the map is linear.If I am wrong then give a counterexample and if correct,then give me a formal proof.