Exercise 8. Let W be a subspace of C" of dimension k. Show that W possesses a basis {b1,..., bx} which is orthonormal with respect to the Hermitian inner product (i.e. b; b; = 0 for i + j, and bib; = 1 for each i). Hint: Start with any basis {c1,...,Ck} of W, and adapt it to a basis of mutually orthogonal vectors using some version of the Gram-Schmidt orthogonalizaton process. Write b = c and show that some linear combination b2 of b and c2 is orthogonal to b1. Then show that some linear combination b2 of bi, b2 and c3 is orthogonal to both b and b2. Build up a basis this way - once you have a basis whose elements are all orthogonal to each other, you can scale the elements to ensure b b = 1 for each i.