A symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function formula_1 that maps every pair formula_2 of elements of the vector space formula_3 to the underlying field such that formula_4 for every formula_5 and formula_6 in formula_3. They are also referred to more briefly as just symmetric forms when "bilinear" is understood.