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
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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
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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
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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
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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) |
Bilinear mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_mapping&oldid=30648