# Does $T_{v}A=T_{Av}$ where $T$ is a translation and $A\in O(2)$ is a rotation matrix?

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...)

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Look at what happens to an element $x \in \mathbf R^n$ under both maps: $T_v(A(x)) = Ax + v$ and $T_{Av}(x) = x + Av$. What is true is that $A \circ T_v \circ A^{-1} = T_{Av}$. Indeed, $\operatorname{Isom} \mathbf R^n$ is the semidirect product $\mathbf R^n \rtimes O_n$ and this calculation shows that the requisite action of $O_n$ on $\mathbf R^n$ is the obvious one.