1) Let V be a real inner product space of dimension n. Let B ? V be a basis. Prove the following statement: ?x, y? = [x]_B · [y]_B for all x, y ? V if and only if B is orthonormal.
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We want to show that (3,1) = [#ls ' [ilu for all €,@ € V. Let € = ∑_{i=1}^n a_i b_i and @ = ∑_{i=1}^n c_i b_i be any two vectors in V, where a_i, c_i are scalars and b_i are the basis vectors in B. Then, we have: (€,@) = (∑_{i=1}^n a_i b_i, ∑_{j=1}^n c_j b_j) = Show more…
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