Question: let v be a finite dimensional inner product space and...

Let V be a finite dimensional inner product space and W ⊂ V a subspace. For every v ∈ V there is a unique decomposition v = w + w ′ with w ∈ W and w ′ ∈ W⊥. Define a map T : V → V by Tv = w − w ′ . Prove that T is a unitary and self-adjoint operator.