Bilinear operator

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In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law.

1 Definition
2 Properties
3 Examples
4 See also

[edit] 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

B : V × W → X

such that for any w in W the map

$v \mapsto B(v, w)$

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

$w \mapsto B(v, w)$

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.

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).

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.

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 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.

[edit] 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 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.

If V,W,X are finite-dimensional, then so is L(V,W;X). 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. Now, if X is a space of higher dimension, we obviously have dimL(V,W;X)=dimV×dimW×dimX.

[edit] 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.