Centroids under Linear Transformations

Question

Let $M \in \mathbb{R}^{m\times m}$ be an invertible matrix, and let $c\in\mathbb{R}^{m}$ be a vector. Define $\Omega \triangleq \{x\in\mathbb{R}^{m}: M(x-c) \in \Delta\}$, where $\Delta$ is some region in $\mathbb{R}^{m}$. Let $a$ be the centroid of $\Omega$, and $b$ be the centroid of $\Delta$. Then is it true that $M(a-c)=b?$.

Answer 1

Let $b$ by any point and $\Delta = \{b\}$ be a one point set. Let $a$ be the unique solution to $ M(x-c) = b.$ Thus, $\Omega = \{a\}$ and $M(a-c)=b.$ The centroid of a one point set is the point itself. If we replace $\Delta$ with any set with centroid $b$, then the corresponding set $\Omega$ has centroid $a$ since the map $f(x)=y$ given by $M(x-a)=y-b$ is an affine map and centroids are preserved by affine maps.