Can anyone help me with this problem please?
Let {v_1,v_2,...v_k} be an orthogonal set in V, and let a_1,a_2,...,a_k be scalars. Prove that (the norm of the summation of a_i*v_i)^2 is equal to the summation of l a_i l^2* (norm of v_i)^2
i runs from 1 to k.
This should be . The two sums are independent, so should have different dummy variables. However, when i≠j. So the only terms that survive are those for which j=i, and the rest of kalagota's solution is correct:
The only other comment is that if the scalar field is the complex numbers then when comes out of the right side of the inner product it becomes (the complex conjugate). But that's okay, because we then get , as required.