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Vector Space Filter Basis Proof Generating Set Of A Group Mapping Group Permutation Implication Field Determinant Relation Between Objects Propositional Logic Characteristic Signature Of A Permutation Transposition Cycle Notation Linearization Bilinear Map

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Anti Symmetric

Let and be a vector spaces over a field . A bilinear mapping $B:U\times U\rightarrow V$ is said to be antisymmetric if

$\displaystyle B(u,u)=0$

(1)

for all $u\in U$ .

If is antisymmetric, then the polarization of the anti-symmetry relation gives the condition:

$\displaystyle B(u,v) + B(v,u) = 0$

(2)

for all $u,v \in U$ . If the characteristic of

is not 2, then the two conditions are equivalent.

A multlinear mapping $M:U^k\rightarrow V$ is said to be totally antisymmetric, or simply antisymmetric, if for every $u_1,\ldots,u_k\in U$ such that

$\displaystyle u_{i+1} = u_i$

for some $i=1,\ldots,k-1$ we have

$\displaystyle M(u_1,\ldots,u_k)=0.$

Proposition 1 Let $M:U^k\rightarrow V$ be a totally antisymmetric, multlinear mapping, and let $\pi$ be a permutation of $\{1,\ldots,k\}$ . Then, for every $u_1,\ldots,u_k\in U$ we have

$\displaystyle M(u_{\pi_1},\ldots,u_{\pi_k}) = \mathrm{sgn}(\pi) M(u_1,\ldots,u_k),$

where $\mathrm{sgn}(\pi)=\pm1$ according to the parity of $\pi$ .

Proof. Let $u_1,\ldots,u_k\in U$ be given. multlinearity and anti-symmetry imply that

0	$\displaystyle = M(u_1+u_2,u_1+u_2,u_3,\ldots,u_k)$
	$\displaystyle = M(u_1,u_2,u_3,\ldots,u_k) + M(u_2,u_1,u_3,\ldots,u_k)$

Hence, the proposition is valid for $\pi=(12)$ (see cycle notation). Similarly, one can show that the proposition holds for all transpositions

$\displaystyle \pi=(i,i+1),\quad i=1,\ldots,k-1.$

However, such transpositions generate the group of permutations, and hence the proposition holds in full generality.

Note.

The determinant is an excellent example of a totally antisymmetric, multlinear mapping.

1	Vector Space
1	Filter Basis
1	Field
1	Proof
1	Mapping
1	Implication
1	Generating Set Of A Group
1	Permutation
1	Group
2	Relation Between Objects
2	Determinant
2	Propositional Logic
3	Characteristic
4	Signature Of A Permutation
5	Transposition
5	Bilinear Map
8	Linearization
9	Cycle Notation