I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra.

Some terminology from Grove and Benson:

V is a real Euclidean vector space

A transformation of V is understood to be a linear transformation

The group of all orthogonal transformations of V will be denoted O(V)

Then in chapter 2, Grove and Benson write the following:

If T [TEX]\inEX] O(V), then T is completely determined by its action on the basis vectors e_1 = (1,0) and e_2 = (0,1).

If T e_1 = ( \mu , \nu ), then {\mu}^2 + {\nu}^2 = 1 and T e_2 = \pm ( - \nu , \mu)

Can someone please help by proving why the last statement is true?