I am seeking to understand Finite Reflection Groups and am reading Grove and Benson (G&B) : Finite Reflection Groups

Grove and Benson, in Section 21 Orthogonal Transformations in Two Dimensions define T as a linear transformation belonging to the groups of all orthogonal transformations .

G&B point out that the vector is an eigenvector having eigenvalue 1 for T and that similarly is an eigenvector with eigenvalue -1 and[see attachment]G&B then state that if , then and T sends x to its mirror image with respect to the line L (see Figure 2.2(b) in attachement)

The transformation T is called the refection through L or the reflection along .

G&B then say "observe that for all "

Can someone help me show (explicitly and formally) that for all ?

And further (and possibly more important) can someone help me get a geometric sense of what the formula above means?ie why do G&B highlight this particular relationship?

Would very much appreciate such help

Peter