I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups

On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement:

It is easy to verify (Exercise 2.1) that the vector $\displaystyle x_1 = (cos \ \theta /2, sin \ \theta /2 ) $ is an eigenvector having eigenvalue 1 for T, so that the line

$\displaystyle L = \{ \lambda x_1 : \lambda \in \mathbb{R} \} $ is left pointwise fixed by T.

I am struggling to see why it follows that L above is left pointwise fixed by T (whatever that means exactly! - can someone please clarify this matter?).

Can someone please help - I am hoping to be able to formally and explicitly justify the statement.

The preamble to the above statement is given in the attachment, including the definition of T

Notes (see attachment)

1. T belongs to the group of all orthogonal transformatios, $\displaystyle O ( \mathbb{R} ) $.

2. Det T = -1

For other details see attachment

Peter