Bilinear map

In the mathematical subfield of linear algebra and related fields, the bilinear mappings generalize the diverse terms of products ( in terms of a multiplication). The bilinearity corresponds to the distributive

In normal multiplication.

Definition

A bilinear map is a 2- multilinear mapping, that is, an image

So that from each (fixed selected )

A linear transformation, and for each of

Is a linear map. For arbitrary, and is therefore

One can say that the concept of bilinearity is a generalization applicable to rings and in particular body (left and right ) distributive. However, the bilinearity describes not only (such as the distributive ) the behavior of the Figure with respect to addition, but also in the scalar multiplication.

Continuity and differentiability

Bilinear mappings with a nite domain are always continuous.

If a bilinear mapping steadily, it is also totally differentiable and it holds

Using the chain rule, it follows that two differentiable functions that are associated with a bilinear mapping can be derived with the generalization of the product rule: Be totally differentiable functions, then applies

Examples

All commonly available products are bilinear mappings: the multiplication in a body ( real, complex, rational numbers ) or a ring ( integers, matrices ), but also the vector or cross product and the dot product on a real vector space.

A special case of bilinear mappings are the bilinear forms. In these, the range of values ​​with the Skalarkörper of vector spaces and the same.

Bilinear forms are important for analytic geometry and duality theory.

In image processing, a bi-linear filtering is used for the interpolation.

Other properties

Symmetry and anti-symmetry ( for ), and other properties are as defined in the more general case of multi- linear maps.

A bilinear map makes it an algebra.

In the case of complex vector spaces is also considered sesquilineare ( " half " - linear ) projections, which are anti- linear in the second argument (or possibly the first ), which means that

(the complex conjugation hereinafter), while all other above equations remain.

Respect to tensor products

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

A bilinear map, so there exists a unique linear map

Conversely, any linear map defined

A bilinear map

These two constructions define a bijection between the space of bilinear mappings and the space of linear maps.

Bilinear mappings on finite dimensional vector spaces

Are and finite vector spaces with arbitrarily chosen bases of and, then there is for any of the presentation

Using the calculation rules of the bilinear mapping is then as follows

The bilinear map is therefore through the images of all ordered pairs of basis vectors of and determined. Is also a K-vector space, spanned the image to a maximum dimensional subspace of. In general, the image of a bilinear mapping between vector spaces is but is not a vector space.

For bilinear forms are from the, so that they can be listed in an obvious way in a matrix. This matrix is then the coordinate representation of the bilinear form with respect to the chosen bases.

Swell

• Gerd Fischer: Linear Algebra. 17th edition. Vieweg Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4.

• Linear Algebra