Orthogonal matrix and subspace

Question

I am facing some difficulties proving the below statement:

Let B be an NxN orthogonal matrix, and U is a subspace of Rn.

I want to prove that :

for all u ∈ U , w ∈ U⊥:
Bu ⊥ Bw

I would appriciate your help!

Answer 1

Denote by $\langle\rangle\;$ the inner product in that space, then:

$$\langle Bu,Bw\rangle=\langle u,B^*Bw\rangle$$

and now just remember (1) what being "an orthogonal matrix" means, and (2) where did you take $\;u,\,w\;$ from...