Let V be an inner product space of dimension n Let S be an o

Let V be an inner product space, of dimension n. Let S be an orthonormal basis of V. Show that (u, v) = [u]_S middot [v]_S for all u, v elementof V. Recall, [u]_S elementof R^n denotes the coordinate vector of u.

Solution

Let S = {e₁,e₂,…,e_n} be an orthonormal basis for V and let u=a₁e₁+a₂e₂+…+a_n e_n and v = b₁e₁+b₂e₂ +…+b_ne_n where a_i s and b_j s are scalars. Then [u]s = (a₁,a₂, …,a_n) and [v]s = (b₁,b₂, …,b_n). Further, <u,v>= (a₁e₁+a₂e₂+…+a_n e_n).( b₁e₁+b₂e₂ +…+b_ne_n) = a₁b₁ +a₂b₂ +…+a_nb_n as e_i² = 1 for 1 I n. Also, [u]s . [v]s = (a₁,a₂, …,a_n) . ((b₁,b₂, …,b_n) = a₁b₁ +a₂b₂ +…+a_nb_n . Hence, < u, v> = [u]s . [v]s