Question: (a) Show that v1,... , vn form an orthonormal basis of Rn for the inner product (v, w) = vT K w for K >

(a) Show that v1,... , vn form an orthonormal basis of Rn for the inner product (v, w) = vT K w for K > 0 if and only if AT KA = I where A = (v1, v2 . . . vn).
(b) Prove that any basis of Rn is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined?
(c) Find the inner product on R2 that makes v, = (1, 1)T, v2 = (2, 3)T into an orthonormal basis.
(d) Find the inner product on R3 that makes v, = (1, 1, 1)T, v2 = (1, 1, 2)T, v3 = (1,2, 3)T into an orthonormal basis.

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a The i j entry of A T KA is v T i Kv j v i v j Thus A T KA I if and only if and so the vectors ... View full answer

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