antisymmetric mapping (original) (raw)

A multlinear mapping M:Uk→Vis said to be totally antisymmetric, or simply antisymmetric, if for every u1,…,uk∈U such that

for some i=1,…,k-1 we have

Proposition 1

Let M:Uk→V be a totally antisymmetric, multlinear mapping, and let π be a permutation of {1,…,k}. Then, for every u1,…,uk∈U we have

where sgn⁢(π)=±1 according to the parity of π.

Proof.Let u1,…,uk∈U be given. multlinearity and anti-symmetry imply that

Hence, the proposition is valid for π=(12) (see cycle notation). Similarly, one can show that the proposition holds for alltranspositions

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.