A
symmetric bilinear form on a
vector space is a
linear 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
that maps every pair of elements of the vector space
to the underlying field such that for every
and
in
. They are also referred to more briefly as just
symmetric forms when "bilinear" is understood.