How to solve (b) (c) and (d)?

Problem 1. Let V be a vector space with ordered basis B = (u], ..., un), where n E N. (a) Let (-, .) be any inner product on V, and let z, y E V. Prove if (1, U) = (y, v) for every DEV, then i = y. (b) Prove that there exists a unique inner product (., .) on V relative to which B is orthonormal. (c) Let T : V - V be a linear transformation, and let , be the B-matrix of T. Prove that (w, T(U)) = [wBT]slug for all wvEV, where (-, .) is the inner product from part (b). (d) Prove that there is a unique linear transformation S: V - V such that: (S(w), U) = (w,T(v)) for all w, DEV, where (., .) is the inner product from part (b). Hint: Use the formula from part (c) to come up with a formula for the B-matrix of S. For uniqueness, part (a) is helpful