Let be there is a vector space above the field (most often considered fields or ).
A bilinear form is a function. linear for each of the arguments :
- {\ displaystyle F (x + z, \, y) = F (x, \, y) + F (z, \, y)} ,
- ,
- ,
- ,
here and
The bilinear form is a special case of the notion of a tensor (a tensor of rank (0.2)).
Content
Alternative Definition
In the case of finite-dimensional spaces (for example, ) a different definition is often used.
Let be there are many view vectors Where .
Bilinear forms are called functions. kind of
Where but - some constants from the field
In other words, the bilinear form is a function of two groups of variables, which is a homogeneous polynomial of the first degree with respect to variables from each group.
Related definitions
- Bilinear form called symmetric if for any vectors .
- Bilinear form called antisymmetric (antisymmetric) if for any vectors .
- Vector is called orthogonal (more precisely, left orthogonal ) subspace regarding , if a for all . The set of vectors orthogonal to a subspace relative to this bilinear form is called the orthogonal complement of the subspace regarding and is denoted by .
- Radical bilinear form called the orthogonal complement of the space itself regarding that is the totality vectors for which for all .
Properties
- The set of all bilinear forms defined on an arbitrary fixed space is a linear space.
- Any bilinear form can be represented as a sum of symmetric and skew-symmetric forms.
- With selected basis at any bilinear form uniquely determined by the matrix
so for any vectors and
- {\ displaystyle F (x, \, y) = {\ begin {pmatrix} x ^ {1} & x ^ {2} & \ ldots & x ^ {n} \ end {pmatrix}} {\ begin {pmatrix} F ( e_ {1}, \, e_ {1}) & F (e_ {1}, \, e_ {2}) & \ ldots & F (e_ {1}, \, e_ {n}) \\ F (e_ {2 }, \, e_ {1}) & F (e_ {2}, \, e_ {2}) & \ ldots & F (e_ {2}, \, e_ {n}) \\\ vdots & \ vdots & \ ddots & \ vdots \\ F (e_ {n}, \, e_ {1}) & F (e_ {n}, \, e_ {2}) & \ ldots & F (e_ {n}, \, e_ {n}) \ end {pmatrix}} {\ begin {pmatrix} y ^ {1} \\ y ^ {2} \\\ vdots \\ y ^ {n} \ end {pmatrix}},}
i.e
- It also means that the bilinear form is completely determined by its values on the basis vectors.
- Space dimension there is .
- Although the bilinear matrix depends on the choice of basis, the rank of the matrix of the bilinear form in any basis is the same, it is called the rank of the bilinear form . A bilinear form is called nondegenerate if its rank is equal to .
- For any subspace orthogonal complement is a subspace .
- where - rank of bilinear form .
Convert the bilinear matrix by changing the base
The matrix representing the bilinear form in the new basis is connected with the matrix representing it in the old basis through the matrix, the inverse of the transition matrix to the new basis (Jacobi matrix), through which the coordinates of the vectors are transformed.
In other words, if the coordinates of the vector in the old basis expressed through coordinates in new through the matrix or matrix recording then bilinear form on any vectors and will be written as
- ,
that is, the components of the matrix representing the bilinear form in the new basis will be:
- ,
or, in matrix notation:
- ,
- where - direct coordinate transformation matrix .
Relationship with tensor products and the Hom functor
From the universal property of the tensor product, it follows that the bilinear forms on V are in one-to-one correspondence with the set where k is the main field.
Since the tensor product functor and the Hom functor are conjugate , , that is, the bilinear form corresponds to the linear mapping of in dual space . This correspondence can be carried out in two ways (since there are two functors of the tensor product — with a fixed left argument and with a fixed right); they are often designated as
.
See also
- Quadratic form
- Bilinear operation
- Bilinear transform
Literature
- Maltsev A. I. Basics of linear algebra. - M .: Science, 1975.
- Gelfand I.M. Lectures on linear algebra. - M .: Science, 1971.
- Faddeev DK Lectures on algebra. M .: Science, 1984.
- Kostrikin A.I. Introduction to Algebra, Moscow: Nauka, 1977.
- Beklemishev D.V. Analytic geometry and linear algebra. - M .: Higher. Shk., 1998. - 320 p.
- Gelfand IM , Linear Algebra . Lecture course.
- Shafarevich I.R. , Remizov A.O. Linear Algebra and Geometry, - Fizmatlit, Moscow, 2009.