Bilinear mapping

bilinear function

A mapping from the product of a left unitary -module and of right unitary -module into an -bimodule , satisfying the conditions

where are arbitrarily chosen elements, and and are rings with a unit element. The tensor product over has the natural structure of an -bimodule. Let be a canonical mapping; any bilinear mapping will then induce a homomorphism of -bimodules for which . If and is commutative, then the set of all bilinear mappings is an -module with respect to the pointwise defined operations of addition and multiplication with elements in , while the correspondence establishes a canonical isomorphism between the -module and the -module of all linear mappings from into .

Let and be free modules with bases , and , , respectively. A bilinear mapping is fully determined by specifying for all , , since for any finite subsets , , the following formula is valid:

(*)

Conversely, after the elements , have been chosen arbitrarily, formula (*), where , defines a bilinear mapping from into . If and are finite, the matrix is said to be the matrix of with respect to the given bases.

Let a bilinear mapping be given. Two elements , are said to be orthogonal with respect to if . Two subsets and are said to be orthogonal with respect to if any is orthogonal to any . If is a submodule in , then

which is a submodule of , is called the orthogonal submodule or the orthogonal complement to . The orthogonal complement of the submodule in is defined in a similar way. The mapping is said to be right-degenerate (left-degenerate) if (). The submodules and are called, respectively, the left and right kernels of the bilinear mapping . If and , then is said to be non-degenerate; otherwise it is said to be degenerate. The mapping is said to be a zero mapping if and .

Let , be a set of left -modules, let , be a set of right -modules, let be a bilinear mapping from into , let be the direct sum of the -modules , and let be the direct sum of the -modules . The mapping , defined by the rule

is a bilinear mapping and is said to be the direct sum of the mappings . This is an orthogonal sum, i.e. the submodule is orthogonal to the submodule with respect to if .

The bilinear mapping is non-degenerate if and only if is non-degenerate for all . Moreover, if is non-degenerate then one has

If , a bilinear mapping is called a bilinear form.

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2]	S. Lang, "Algebra" , Addison-Wesley (1974)