I am looking at $Iso(\mathbb{R}^2)$ and I wonder if $T_{v}A=T_{Av}$ (where $T$ is a translation and $A\in O(2)$ is a rotation matrix).
I tried imagining this in $\mathbb{R}^2$ and had difficulty to verify this, what about the more general case (any natural $n>1$) ?
I could use some help with this (I do hope it won't require an nxn matrix and a direct verification...)