雙線性形

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在F域中,向量空間V的雙線性形是一個映射:

V × V → F (線性變換)

B : V × V → F 是雙線性如果

$w \mapsto B(v, w)$

$w \mapsto B(w, v)$

是線性 $\forall ''v'' \in ''V''$ 。這個定義適用於交換環的模,其環擁有模同態的線性映射。

注意一個雙線性形是特別的雙線性運算元。

[编辑] 坐标表達方法

如果V是n維向量空間,在V上任意的雙線性形B能以矩陣B( $B i j = B (e i, e j)$ ) 表達坐标。向量u及v的雙線性形:

$B(u,v) = \mathbf{u}^T \mathbf{Bv} = \sum_{i,j=1}^{n}B_{ij}u^i v^j$

這裡的 uⁱ 和v^j 都是u 和v 的組成部分.

[编辑] 雙對空間映射

V的每一個雙線性形B定義為一對線性映射,由V射到它的雙對空間V*。 $B_1,B_2\colon V \to V^*$ 的定義

B 1 (v)(w) = B (v, w)

B 2 (v)(w) = B (w, v)

常常記作：

B 1 (v) = B (v, - )

B 2 (v) = B ( - , v)

這裡的(–)可以被任意的變數來取代。

如果 V的維度是有限，我們可以找到V的雙雙對空間V**。 B₂是B₁ 的線性映射的轉置。定義B的轉置為雙線性形：

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

如果 V的維度是有限，B₁ 及B₂ 擁有相同的等級。如果跟V的維度相同，B₁ and B₂ 是線性同構，由V映射到V*。如此，B就是非降解的。

例如一個線性映射A : V → V*，而B是V上的雙線性形。

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

假如A 是同構，這便是一個非降解的形。

[编辑] 對稱

雙線性形 B : V × V → F :

對稱條件: $B (v, w) = B (w, v)$ $\forall v,w\in V$
旋鈕對稱 條件: $B (v, w) = - B (w, v)$ $\forall v,w\in V$
alternating 條件: $B (v, v) = 0$ $\forall v\in V$

Every alternating form is skew-symmetric; this may be seen by expanding

B(v+w,v+w).

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.

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).

A bilinear form is symmetric iff the maps $B_1,B_2\colon V \to V^*$ 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 an skew-symmetric part as follows

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

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

[编辑] 張量乘積關係

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps V ⊗ V → F. If B is a bilinear form on V the corresponding linear map is given by

$v\otimes w\mapsto B(v,w).$

The set of all linear maps V ⊗ V → F is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of

$(V\otimes V)^{*} \cong V^{*}\otimes V^{*}.$

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*).