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Definition: Let V be a real vector space. A bilinear form on V is a real-valued function of two vector variables - a map VxV -> R. Given a pair v, w of vectors, the form returns a real number that will usually be denoted by <v,w>. A bilinear form is required to be linear in each variable: <rv1,w1> = r<v1,w1> and <v1+v2,w1> = <v1,w1> + <v2,w1>; <v1,rw1> = r<v1,w1> and <v1,w1+w2> = <v1,w1> + <v1,w2>.
Source: Algebra, second edition by Michael Artin