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Encyclopedia > Bilinear form

In mathematics, a bilinear form on a vector space V over a field F is a mapping V × VF which is linear in both arguments. That is, B : V × VF is bilinear if the maps Euclid, detail from The School of Athens by Raphael. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

v mapsto B(v, w)
v mapsto B(w, v)

are linear for each w in V. This definition applies equally well to modules over a commutative ring with linear maps being module homomorphisms. In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a module is a generalization of a vector space. ...


Note that a bilinear form is a special case of a bilinear operator. In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...


When F is the field of complex numbers C one is often more interested in sesquilinear forms. These are similar to bilinear forms but are conjugate linear in one argument instead of linear. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and conjugate-linear in the other. ... In mathematics, a real linear transformation f from a complex vector space V to another is said to be antilinear (or conjugate-linear or semilinear) if for all c, d in C and all x, y in V. See also: complex conjugate, sesquilinear form ...

Contents


Coordinate representation

Let C={e_{1},ldots,e_{n}} be a basis for V, assuming the latter is of finite dimension. Define the ntimes n - matrix A by (Aij) = B(ei,ej). A is symmetric exactly due to symmetry of the bilinear form. Then if the ntimes 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, B(v,w) is given by :


xTAy


Suppose C' is another basis for V, with : begin{bmatrix}e'_{1} & cdots & e'_{n}end{bmatrix} = begin{bmatrix}e_{1} & cdots & e_{n}end{bmatrix}S with S an invertible ntimes n - matrix. Now the new matrix representation for the symmetric bilinear form is given by :


A' = STAS


Maps to the dual space

Every bilinear form B on V defines a pair of linear maps from V to its dual space V*. Define B_1,B_2colon V to V^* by In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

B1(v)(w) = B(v,w)
B2(v)(w) = B(w,v)

This is often denoted as

B1(v) = B(v, − )
B2(v) = B( − ,v)

where the (–) indicates the slot into which the argument is to be placed.


If either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...


If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

B * (v,w) = B(w,v).

If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to the dimension of V then B1 and B2 are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem, this is equivalent to the condition that the kernel of B1 be trivial. In fact, for finite dimensional spaces, this is often taken as the definition of nondegeracy. Thus B is nongenerate iff In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...

B(v,w)=0mbox{ for all w}Rightarrow v=0.

Given any linear map A : VV* one can obtain a bilinear form B on V via

B(v,w) = A(v)(w)

This form will be nondegenerate iff A is an isomorphism.


Reflexivity and orthogonality

A bilinear form B : V × VF is said to be reflexive if forall v,win V : B(v,w)=0Longleftrightarrow B(w,v)=0.



Reflexivity allows us to define orthogonality : two vectors v and w are said to be orthogonal with respect to the reflexive bilinear form if and only if : B(v,w)=0Longleftrightarrow B(w,v)=0


The radical of a bilinear form is the subset of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical if and only if : A x= 0 Longleftrightarrow x^{T} A=0 The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.


Suppose W is a subspace. Define : W^{perp}={v| B(v,w)=0 forall win W}


When the bilinear form is nondegenerate, the map Wleftarrow W^{perp} is bijective, and the dimension of W^{perp} is dim(V)-dim(W).


One can prove that B is reflexive if and only if it is :

  • symmetric : B(v,w) = B(w,v) for all v,win V
  • alternating if B(v,v) = 0 for all vin V

Every alternating form is skew-symmetric ( B(v,w) = − B(w,v)). This may be seen by expanding B(v+w,v+w). In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...


If the characteristic of F is not 2 then the converse is also true (every skew-symmetric form is alternating). If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...


A bilinear form is symmetric (resp. skew-symmetric) iff its coordinate matrix (relative to any basis) is symmetric (resp. skew-symmetric). A bilinear form is alternating iff its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(F) ≠ 2). ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...


A bilinear form is symmetric iff the maps B_1,B_2colon V to V^* are equal, and skew-symmetric iff they are negatives of one another. If char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and a skew-symmetric part as follows

B^{pm} = {1over 2}(B pm B^*)

where B* is the transpose of B (defined above).


Relation to tensor products

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps VVF. If B is a bilinear form on V the corresponding linear map is given by In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

votimes wmapsto B(v,w).

The set of all linear maps VVF is the dual space of VV, so bilinear forms may be thought of as elements of In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

(Votimes V)^{*} cong V^{*}otimes V^{*}.

Likewise, symmetric bilinear forms may be thought of as elements of S2V* (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*). In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...


On normed vector spaces

A bilinear form on a normed vector space is bounded, if there is a constant C such that for all u, vin V In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

B(u,v) le C |u| |v|.

A bilinear form on a normed vector space is elliptic, if there is a constant c such that for all uin V

B(u,u) ge c |u|^2.

See also

In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In multilinear algebra, a multilinear form is a map of the type , where V is a vector space over the field K, that is separately linear in each its N variables. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and conjugate-linear in the other. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product or dot product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

External links


  Results from FactBites:
 
Bilinear operator - Wikipedia, the free encyclopedia (644 words)
In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law.
The case where X is F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear operator V × V → F.
  More results at FactBites »

 
 

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