Bilinear map - Citizendia

In mathematics, a bilinear map is a function which is linear in both of its arguments. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that An example of such a map is multiplication of integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

1 Definition
2 Properties
3 Examples
4 See also

Definition

Let V, W and X be three vector spaces over the same base field F. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division A bilinear map is a function

B : V × W → X

such that for any w in W the map

v ↦ B(v, w )

is a linear map from V to X, and for any v in V the map

w ↦ B(v, w )

is a linear map from W to X. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

In other words, if we hold the first entry of the bilinear map 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 term "symmetric function" can mean two different things

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 is a Bilinear mapping V × V → F, where In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property It also can be easily generalized to n-ary functions, where the proper term is multilinear.

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 map B : M × N → T, where T is an abelian 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

B(mt, n ) = B(m, tn )

for all m in M, n in N and t in R. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

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. In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero )

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

If V,W,X are finite-dimensional, then so is L(V,W;X). In Mathematics, the dimension of a Vector space V is the cardinality (i For X=F, 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, choose 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. Now, if X is a space of higher dimension, we obviously have dimL(V,W;X)=dimV×dimW×dimX.

Examples

Matrix multiplication is a bilinear map M(m,n) × M(n,p) → M(m,p). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally
If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear map V × V → R. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where
If V is a vector space with dual space V*, then the application operator, b(f, v) = f(v) is a bilinear map from V* × V to the base field. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals
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 map V × W → F.
The cross product in R³ is a bilinear map R³ × R³ → R³. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which
Let B : V × W → X be a bilinear map, and L : U → W be a linear operator, then (v, u) → B(v, Lu) is a bilinear map 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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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.

Contents

Definition

Properties

Examples

See also