The definition works without any changes if instead of vector spaces we use modules over a commutative
ring *R*. It also can be easily generalized to n-ary functions, where the proper term is *multilinear*.

For the case of a non-commutative base ring *R* and
a right module *M _{R}* and a left
module

*B*(*mr*,*n*) =*B*(*m*,*rn*)

One often thinks of a bilinear operator as a generalized "multiplication" which satisfies the distributive law.

- Matrix multiplication is a bilinear map M(
*m*,*n*) x M(*n*,*p*)`->`M(*m*,*p*). - If a vector space
*V*over the real numbers**R**carries an inner product, then the inner product is a bilinear operator*V*x*V*`->`**R**. - In general, for a vector space
*V*over a field*F*, a**bilinear form**on*V*is the same as a bilinear operator*V*x*V*`->`*F*. - If
*V*is a vector space with dual space*V**, then the application operator,*b*(*f*,*v*) =*f*(*v*) is a bilinear operator from*V*x^{*}*V*to the base field. - Let
*V*and*W*be vector spaces over the same base field*F*. If*f*is a member of*V*and^{*}*g*a member of*W*, then^{*}*b*(*v*,*w*) =*f*(*v*)*g*(*w*) defines a bilinear operator*V*x*W*`->`*F*. - The cross product in
**R**^{3}is a bilinear operator**R**^{3}x**R**^{3}`->`**R**^{3}. - Let
*B*:*V*x*W*->*X*be a bilinear operator, and*L*:*U*->*W*be a linear operator, then (*v*,*u*) ->*B*(*v*,*Lu*) is a bilinear operator on*V*x*U* - The operator
*B*:*V*x*W*->*X*where*B*(*v*,*w*) = 0 for all*v*in*V*and*w*in*W*is bilinear