# Bilinear mapping

In the mathematical sub-area of linear algebra and related areas, the bilinear mappings generalize the most diverse concepts of products (in the sense of a multiplication ). The bilinearity corresponds to the distributive law

${\ displaystyle a \ cdot (b + c) = a \ cdot b + a \ cdot c}$

with normal multiplication.

## definition

A -bilinear mapping is a 2- multilinear mapping , that is, a mapping ${\ displaystyle K}$

${\ displaystyle f \ colon E \ times F \ to G}$, where , and are three modules or vector spaces above the (same) ring or body ,${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle G}$${\ displaystyle K}$${\ displaystyle K}$ ${\ displaystyle K}$

such that for each (solid selected) from${\ displaystyle y}$${\ displaystyle F}$

${\ displaystyle x \ mapsto f (x, y)}$

a - is linear mapping , and for each off${\ displaystyle K}$ ${\ displaystyle E \ to G}$${\ displaystyle x}$${\ displaystyle E}$

${\ displaystyle y \ mapsto f (x, y)}$

is a linear map . For any , and therefore holds ${\ displaystyle F \ to G}$${\ displaystyle x, x '\ in E}$${\ displaystyle y, y '\ in F}$${\ displaystyle \ alpha \ in K}$

{\ displaystyle {\ begin {aligned} f (x + x ', y) & = f (x, y) + f (x', y) \\ f (x \ cdot \ alpha, y) & = \ alpha \ cdot f (x, y) \\ f (x, y + y ') & = f (x, y) + f (x, y') \\ f (x, \ alpha \ cdot y) & = \ alpha \ cdot f (x, y). \\\ end {aligned}}}

One can say that the concept of bilinearity represents a generalization of the (left and right) distributive laws that apply to rings and especially to bodies . The bilinearity describes not only (like the distributive laws) the behavior of the mapping with regard to addition, but also with regard to scalar multiplication .

More precisely: If a (possibly non-commutative) ring is included , then the side of the modules must be taken into account, i.e. H. must be a right and a left module. The sidedness of remains freely selectable (in the equations it is on the left). ${\ displaystyle K}$${\ displaystyle 1}$${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle K}$${\ displaystyle G}$

## Continuity and differentiability

Bilinear mappings with a finite dimensional domain are always continuous .

If a bilinear mapping is continuous, it is also totally differentiable and it applies ${\ displaystyle B}$

${\ displaystyle DB (x_ {0}, y_ {0}) (x, y) \; = \; B (x_ {0}, y) \, + \, B (x, y_ {0})}$

Using the chain rule it follows that two differentiable functions, which are linked with a bilinear mapping , can be derived with the generalization of the product rule : If functions are totally differentiable, then it holds ${\ displaystyle f, g}$

{\ displaystyle {\ begin {aligned} DB (f (\ cdot), g (\ cdot \ cdot)) (x_ {0}, y_ {0}) (x, y) & = D (B \ circ (f , g)) (x_ {0}, y_ {0}) (x, y) \\ & = B (Df (x_ {0}) x, g (y_ {0}))) + B (f (x_ { 0}), Dg (y_ {0}) y) \ end {aligned}}}

## Examples

All common products are bilinear mappings: the multiplication in a field (real, complex , rational numbers ) or a ring ( whole numbers , matrices), but also the vector or cross product , and the scalar product on a real vector space.

The bilinear forms are a special case of bilinear mapping . With these, the range of values ​​is identical to the scalar field of the vector spaces and . ${\ displaystyle G}$${\ displaystyle K}$${\ displaystyle E}$${\ displaystyle F}$

${\ displaystyle f \ colon E \ times F \ to K}$

Bilinear forms are important for analytic geometry and duality theory.

In image processing, bilinear filtering is used for interpolation.

## Other properties

Symmetry and antisymmetry (for ) and other properties are defined as in the more general case of multilinear maps . ${\ displaystyle F = E}$

A bilinear map turns into an algebra . ${\ displaystyle E \ times E \ to E}$${\ displaystyle E}$

In the case of complex vector spaces, one also considers sesquilinear (“one and a half” -linear) mappings, which are antilinear in the second (or possibly in the first) argument, i.e.

${\ displaystyle f (x, \ alpha \ cdot y) = \ alpha ^ {*} \ cdot f (x, y)}$

(where denotes the complex conjugation ) while all other equations above remain. ${\ displaystyle *}$

## Relation to tensor products

Bilinear maps are classified by the tensor product in the following sense : Is

${\ displaystyle f \ colon E \ times F \ to G}$

a bilinear mapping, there is a uniquely determined linear mapping

${\ displaystyle E \ otimes F \ to G, x \ otimes y \ mapsto f (x, y);}$

conversely, defines any linear map

${\ displaystyle \ lambda \ colon E \ otimes F \ to G}$

a bilinear map

${\ displaystyle E \ times F \ to G, \ quad (x, y) \ mapsto \ lambda (x \ otimes y).}$

These two constructions define a bijection between the space of the bilinear map and the space of the linear map . ${\ displaystyle E \ times F \ to G}$${\ displaystyle E \ otimes F \ to G}$

## Bilinear mappings over finite-dimensional vector spaces

Are and finite -Vektorräume with randomly selected bases of , and by , then there is for any of the presentation ${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle K}$${\ displaystyle (b_ {i}) _ {i = 1, \ dotsc, n}}$${\ displaystyle E}$${\ displaystyle (c_ {j}) _ {j = 1, \ dotsc, m}}$${\ displaystyle F}$${\ displaystyle x}$${\ displaystyle E}$

${\ displaystyle x = \ sum _ {i} x_ {i} b_ {i}}$with coefficients from and analogously for any one from the representation${\ displaystyle x_ {i}}$${\ displaystyle K}$${\ displaystyle y}$${\ displaystyle F}$
${\ displaystyle y = \ sum _ {j} y_ {j} c_ {j}.}$

With the calculation rules of the bilinear mapping this then results

${\ displaystyle f (x, y) = \ sum _ {i} \ sum _ {j} x_ {i} y_ {j} f (b_ {i}, c_ {j}).}$

The bilinear mapping is thus determined by the images of all ordered pairs of the basis vectors of and . If there is also a K-vector space, the image spans a maximally dimensional sub-vector space of . In general, however, the image of a bilinear mapping between vector spaces is not a sub-vector space. ${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle G}$${\ displaystyle \ operatorname {Im} (f)}$${\ displaystyle n \ cdot m}$${\ displaystyle G}$

For bilinear forms the are off , so that they can be noted in a matrix in an obvious way. This matrix is ​​then the coordinate representation of the bilinear form with respect to the selected bases. ${\ displaystyle f (b_ {i}, c_ {j})}$${\ displaystyle K}$