If we transform a set of unit vectors in $R^n$ using some $n \times n$ permutation matrix, is the centroid also transformed by the permutation matrix?

Question

Given a set of unit vectors $V$ in $R^n$, the set's centroid $c^V$, and some $n \times n$ permutation matrix $P$, we define $V'=\{Pv~|~\forall v \in V\}$.

Is it true that $c^{V'} = Pc^V$?

If so, can we generalize $P$ to any $n \times n$ matrix $A$?

Answer 1

$P$ performs a linear transformation. The centroid is a linear combination of the points. Linear transformations preserve linear combinations, by definition. Hence the centroid also transforms via $P$.