Quote by Math Amateur
Thanks Fredrik
Definitely dealihng with case [itex] \mathbb{R^2}[/itex]
Will just reflect on what you wrote!
Still puzzling a bit about showing that [itex] T e_2 = \pm (  \nu , \mu) [/itex]
Peter

You know that Te
_{2} is the second column of T, that ##(\mu,\nu)## is the first column, and that the columns are orthonormal. So you only have to prove that if ##(\alpha,\beta)\cdot (\mu,\nu)=0## and ##(\alpha,\beta)=1##, then ##(\alpha,\beta)=\pm(\nu,\mu)##.