Let B = {b1, b2,... bn} be an ordered basis for vector space V. Show that every inner-product on V can be computed using the values ai, j = (bj|bi) i = 1,..., n, j = 1,..., n. Hint: Take x, y V and write them as linear combinations of bi's (i = 1,..., n). Form the inner prod (x|y) and the coordinates matrices of x and y in the ordered basis B, [x] B and [y] B. Show that (x|y) = [y] B A [x] B, where n times n matrix A = [ai, j] consists of the inner products of the vectors in ordered basis B.
Show transcribed image text Let B = {b1, b2,... bn} be an ordered basis for vector space V. Show that every inner-product on V can be computed using the values ai, j = (bj|bi) i = 1,..., n, j = 1,..., n. Hint: Take x, y V and write them as linear combinations of bi's (i = 1,..., n). Form the inner prod (x|y) and the coordinates matrices of x and y in the ordered basis B, [x] B and [y] B. Show that (x|y) = [y] B A [x] B, where n times n matrix A = [ai, j] consists of the inner products of the vectors in ordered basis B.