# If T is a reflection then $Ker (T-Id) = Im (T-Id)^{\perp}$

Let V be a finite dimensional space with inner product . If T is an orthogonal transformation and a reflection then $$Ker (T-Id) = Im (T-Id)^{\perp}$$, where $$Id$$ denotes the identity matrix

I know that $$det(T) = -1$$ because T is a reflection ,that $$T^2 = Id$$ and that at least one eigenvalue of T is $$-1$$ , but I am not finding a way to proceed.

Any hints ?

• Yes it is the orthogonal complement. There is an inner product on V, I forgot to mention sorry – math.pr Oct 10 '18 at 10:15
• I have already edited it. – math.pr Oct 10 '18 at 11:37
• You can try to apply the result in this question. – Arnaud D. Oct 10 '18 at 11:48