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 ?

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

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