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Lagrange's identity

In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:
${\\displaystyle {\egin{aligned}\\left(\\sum _{k=1}^{n}a_{k}^{2}\ ight)\\left(\\sum _{k=1}^{n}b_{k}^{2}\ ight)-\\left(\\sum _{k=1}^{n}a_{k}b_{k}\ ight)^{2}&=\\sum _{i=1}^{n-1}\\sum _{j=i+1}^{n}\\left(a_{i}b_{j}-a_{j}b_{i}\ ight)^{2}\\\\&\\left(={\ rac {1}{2}}\\sum _{i=1}^{n}\\sum _{j=1,j\ eq i}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ ight),\\end{aligned}}}$
which applies to any two sets {a₁, a₂, ..., a_n} and {b₁, b₂, ..., b_n} of real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lagrange%27s_identity)

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