Let V be an inner product space and let W be a subspace of V . The orthogonal complement of W in V is the set W = v V | hv, wi = 0 for all w W ....

Let V be an inner product space and let W be a subspace of V . The orthogonal complement of W in V is the set W⊥ = v ∈ V | hv, wi = 0 for all w ∈ W .

Prove the following:

a) W⊥ is a subspace of V .

b) W⊥ ∩ W = {0}. That is, the only vector they have in common is the zero vector.

Hint: For the second statement, assume that there is a vector W not equal to 0 that is common to both. Show that this leads to something that we know to be impossible, and therefore such a W cannot exist. This method is called a proof by contradiction.