I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = \left(\sum_{i=1}^n|a_i|^2\right)\left(\sum_{i=1}^n|b_i|^2\right) - \sum_{1\leq i Thanks!!
What are applications of Lagrange's identity?
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