Why can the transformation derived from a list of points and a list of their transformed counterparts not be affine or linear?

Some context (original question below): I wanted to know if there's a nice concise formula to calculate the transformation based on a list of points and another list of the transformed points. This is all 2D or $\mathbb{R}^2$.

By that I mean some matrix equation that has a matrix that contains the given values, so that one can invert this matrix to solve for the transformation matrix or its components.

The question I link to below has the very same goal and especially a nice answer that I was looking for, but it does not create a linear or affine transform.

In his answer to this question bubba makes the following statement:

The transformation can not be linear or affine, it has to be a "perspective" transform.

Why is that? What if I want to find the affine or linear transformation and not the perspective/nonlinear one?

I'm not sure about this, but I guess that if $c_0 = 0$ and $c_1 = 0$, then the perspective transformation will be linear. Would that help me to find the linear or affine transform of points?

• It might be good to edit in more of the context so that your question does not depend on those links never breaking. – Mark S. Jun 5 '15 at 13:13