“Bilinear map”的意思、由来-开放百科全书

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Bilinear map

释义

Definition
Vector spaces Modules
Properties
Examples
See also
External links

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.

Definition

Vector spaces

Let V, W and X be three vector spaces over the same base field F. 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.

In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.

If {{nowrap|1=V = W}} and we have {{nowrap|1=B(v, w) = B(w, v)}} for all v, w in V, then we say that B is symmetric.

The case where X is the base field F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).

Modules

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It generalizes to n-ary functions, where the proper term is multilinear.

For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map {{nowrap|B : M × N → T}} with T an {{nowrap|(R, S)}}-bimodule, and for which any n in N, {{nowrap|m ↦ B(m, n)}} is an R-module homomorphism, and for any m in M, {{nowrap|n ↦ B(m, n)}} is an S-module homomorphism. This satisfies

B(r ⋅ m, n) = r ⋅ B(m, n)

B(m, n ⋅ s) = B(m, n) ⋅ s

for all m in M, n in N, r in R and s in S, as well as B being additive in each argument.

Properties

A first immediate consequence of the definition is that {{nowrap|1=B(v, w) = 0_X}} whenever {{nowrap|1=v = 0_V}} or {{nowrap|1=w = 0_W}}. This may be seen by writing the zero vector 0_X as {{nowrap|0 ⋅ 0_X}} (and similarly for 0_W) and moving the scalar 0 "outside", in front of B, by linearity.

The set {{nowrap|L(V, W; X)}} of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from {{nowrap|V × W}} into X.

If V, W, X are finite-dimensional, then so is {{nowrap|L(V, W; X)}}. For {{nowrap|1=X = F}}, i.e. bilinear forms, the dimension of this space is {{nowrap|dim V × dim W}} (while the space {{nowrap|L(V × W; F)}} of linear forms is of dimension {{nowrap|dim V + dim W}}). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix {{nowrap|B(e_i, f_j)}}, and vice versa.

Now, if X is a space of higher dimension, we obviously have {{nowrap|1=dim L(V, W; X) = dim V × dim W × dim X}}.

Examples

Matrix multiplication is a bilinear map {{nowrap|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 map {{nowrap|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 map {{nowrap|V × V → F}}.
If V is a vector space with dual space V^∗, then the application operator, {{nowrap|1=b(f, v) = f(v)}} is a bilinear map from {{nowrap|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 {{nowrap|1=b(v, w) = f(v)g(w)}} defines a bilinear map {{nowrap|V × W → F}}.
The cross product in R³ is a bilinear map {{nowrap|R³ × R³ → R³}}.
Let {{nowrap|B : V × W → X}} be a bilinear map, and {{nowrap|L : U → W}} be a linear map, then {{nowrap|(v, u) ↦ B(v, Lu)}} is a bilinear map on {{nowrap|V × U}}.

External links

{{springer|title=Bilinear mapping|id=p/b016280}}

2 : Bilinear operators|Multilinear algebra

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