Bilinear operator

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.

The definition works without any changes for modules over a commutative ring R and can easily be generalized to the multi-linear case.

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.

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Examples

• If a vector space over the real numbers, R, has an inner product, then it is a bilinear operator.
• The application operator, b(f, v) = f(v) is a bilinear operator from V^*xV.
• Let f be a member of V^* and g a member of W^*. Then b(v, w) = f(v)g(w) is a bilinear operator.
• In R², assign to each two vectors the signed area of the parallelogram they define.
• 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 is bilinear