[Home]Bilinear operator

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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 MR 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]?.


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Last edited October 5, 2001 1:30 am by AxelBoldt (diff)
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