In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Definition

For a formal definition, given three vector spaces V, W and X over the same base field F, a bilinear operator is a function 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 function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

B : V × W → X

such that for any w in W the map

is a linear operator from V to X, and for any v in V the map In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

is a linear operator from W to X. In other words, if we hold the first entry of the bilinear operator fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.

If V = W and we have B(v,w)=B(w,v) for all v,w in V, then we say that B is symmetric. In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ...

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 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. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

The definition works without any changes if instead of vector spaces we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear. In abstract algebra, a module is a generalization of a vector space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

For the case of a non-commutative base ring R and a right module M_R and a left module _RN, we can define a bilinear operator B : M × N → T, where T is a commutative group, such that for any n in N, m |-> B(m, n) is a group homomorphism, and for any m in M, n |-> B(m, n) is a group homomorphism, and which also satisfies In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

B(mr, n) = B(m, rn)

for all m in M, n in N and r in R.

Properties

A first immediate consequence of the definition is that B(x,y) = o whenever x=o or y=o. (This is seen by writing the null vector o as 0·o and moving the scalar 0 "outside", in front of B, by linearity.) The term null vector can have two different meanings: null vector (vector space) null vector (Minkowski space) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

The set L(V,W;X)of all bilinear maps is a linear subspace of the space (viz vector space, module) of all maps from V×W into X. The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... Viz is a somewhat archaic abbreviation of the Latin word videlicet (which means that is to say) which is often used to introduce a list or series, in much the same way as a colon; it should be noted that is used differently from e. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a module is a generalization of a vector space. ...

If V,W,X are finite-dimensional, then so is L(V,W;X). For X=K, i.e. bilinear forms, the dimension of this space is dimV×dimW (while the space L(V×W;K) of linear forms is of dimension dimV+dimW). To see this, chose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(e_i,f_j), and vice versa. Now, if X is a space of higher dimension, we obviously have dimL(V,W;X)=dimV×dimW×dimX. In mathematics, the dimension of a vector space V is the cardinality (i. ...

Examples

• Matrix multiplication is a bilinear map M(m,n) × M(n,p) → M(m,p).
• If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear operator V × V → R.
• 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.
• If V is a vector space with dual space V*, then the application operator, b(f, v) = f(v) is a bilinear operator from V* × V to the base field.
• Let V and W be vector spaces over the same base field F. If f is a member of V* and g a member of W*, then b(v, w) = f(v)g(w) defines a bilinear operator V × W → F.
• The cross product in R³ is a bilinear operator R³ × R³ → R³.
• Let B : V × W → X be a bilinear operator, and L : U → W be a linear operator, then (v, u) → B(v, Lu) is a bilinear operator on V × U
• The null map, defined by B(v,w) = o for all (v,w) in V×W is the only map from V×W to X which is bilinear and linear at the same time. Indeed, if (v,w)∈V×W, then if B is linear, B(v,w) = B(v,o) + B(o,w) = o + o if B is bilinear.

For the square matrix section, see square matrix. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... 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. ... 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). ... In mathematics, the cross product is a binary operation on vectors in vector space. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Zero redirects here. ...

See also

• Multilinear map
• Sesquilinear form