suppose that R is a reflection matrix ($R^TR = I , det(R) = -1$) and $T_0$ is a vector and we have these equations:
$$ R' = R_0 R R_0^T ,\, T' = (I - R_0RR_0^T)T_0 \quad s.t \quad R_0^TR_0 = I , det(R_0) = 1 $$ $T_0$ is vector. now I proved part of it that it says $R'T' = -T'$ , $\hat{T'}$ is skew-symmetric matrix of $T'$ vector . now I can't prove that $R'\hat{T'} = \hat{T'}R' = \hat{T'}$ and then I want to prove let $n = \frac{\hat{T'}}{\|\hat{T'}\|}$ then $R' = I - 2nn^T$
Can any one help me please? I need to prove this but I don't know how to handle $R'\hat{T'}$ ??