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Theorem: Let V be a finite dimensional inner product space with basis B = {b1, . . . bn} and an inner product denoted by (, )v. Then there exists a symmetric matrix B such that for all vectors U, we V, (v, W)v = [UB B [w]B. The matrix B = In if and only if B is orthonormal. (1) Let TB : V - R" be the coordinate isomorphism with respect to the basis B. Define the map (, ) Rn : Rn X Rn - R by (, 9) Rn := (TB (x), TB (y))v. Prove that (, )R is an inner product on Rn. (2) Show that there exists a symmetric matrix B such that for all vectors U, we V, (7, W)v = [UBB [w]Bf (3) What is the ij-th entry of B in terms of the inner product of the vectors in B. (4) Prove that (U, W)v = [U]B . [wlB if and only if B is an orthonormal basis