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We give a short proof of the Cauchy-Binet determinantal formula usingmultilinear algebra by first generalizing it to an identity {\em not} involvingdeterminants. By extending the formula to abstract Hilbert spaces we obtain, asa corollary, a generalization of the classical Parseval identity.

A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity