In this exercise, we show that any inner product (, ) on Rn can be reduced to the dot product when expressed in a suitably

In this exercise, we show that any inner product (ˆ™, ˆ™) on Rn can be reduced to the dot product when expressed in a suitably adapted basis.
(a) Specifically, prove that there exists a basis v1,..., v" of Rn such that

where c = (c1, c2,... , cn)T are the coordinates of x and d = (d1, d2, ..., dn)T those of y with respect to the basis. Is the basis uniquely determined?
(b) Find bases that reduce the following inner products to the dot product on R2:
(i) (v, w) = 2v1w1 + 3v2w2,
(ii) (v, w) = v1w1 - v1w2 - v2w1 + 3v2w2.

(x. y) = c, d, = c d i=1