Bilinear operator

In mathematics, given three vector spaces V, W and X over the same base field F, a bilinear operator is a function B: VxW -> X such that for any w in W, v |-> B(v, w) is a linear operator from V to X, and for any v in V, w |-> B(v, w) is a linear operator from W to X. In other words, if we hold the first entry the bilinear operator fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.

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 _RN, we can define a bilinear operator B: MxN -> T, where T is a commutative group, such that for any n in N, m |-> B(m, n) is a group homomorphism, and for any m in M, n |-> B(m, n) is a group homomorphism, and which also satisfies

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

for all m in M, n in N and r in R.

Examples

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^*xV 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 R³ is a bilinear operator R³ x R³ -> R³.
Let B: VxW->X be a bilinear operator, and L: U->W be a linear operator, then (v, u) -> B(v, Lu) is a bilinear operator on VxU
The operator B: VxW -> X where B(v, w) = 0 for all v in V and w in W is bilinear