Regular $n$-gons and Linear Tranformations

Question

Let $P$ be a regular $n$-gon and let $Q$ be an another convex polygon with the same number of vertices. Is there always a linear transformation $M\colon \mathbb{R}^2 \to \mathbb{R}^2$ such that $M(P) = Q$?

Answer 1

No. Linear transformations take pairs of parallel lines to pairs of parallel lines; a linear transformation of a square is always a parallelogram