Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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...)

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

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.

share|improve this answer
    
Thanks, I understand what you did there –  Belgi Apr 28 '12 at 0:54
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.