(3 points) V is an inner product space with an inner product . v is a vector in V . Show that v = {w V |w v = 0} is a subspace of V . (4 points) a.

1. (3 points) V is an inner product space with an inner product ∗. v is a vector in V . Show that v ⊥ = {w ∈ V |w ∗ v = 0} is a subspace of V .

2. (4 points) a. Prove that A ∗ B = T race(AB) defines an inner product on the space S n of symmetric n × n matrices. b. Find an orthonormal basis for S n in this inner product.