In a given affine coordinates, an affine transformation is given by:
\begin{cases} x'=3x+y \\ y'=-4x-y \end{cases}
How can I see if it preserves distance? Is it correct to see that the distance between 2 points $(1,0),(0,1)$ is $\sqrt{2}$ and after applying this affine transfromation these two points will become $(3,-4)$ and $(1,-1)$ and the distance between them is $\neq{\sqrt{2}}$?
Yes, that would disprove that it is distance preserving.
Remark:
To prove that distance is not preserved, a counter example suffices.
To prove that distance is preserved, a general proof is required.