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%%% Primary Title: antisymmetric
%%% Primary Category Code: 15A63
%%% Filename: AntiSymmetric.tex
%%% Version: 5
%%% Owner: rmilson
%%% Author(s): Mathprof, rmilson
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\newtheorem{proposition}{Proposition}

\begin{document}

 Let $U$ and $V$ be a \htmladdnormallink{vector spaces}{http://planetmath.org/encyclopedia/LinearSpace.html} over a \htmladdnormallink{field}{http://planetmath.org/encyclopedia/Field.html} $K$. A \htmladdnormallink{bilinear mapping}{http://planetmath.org/encyclopedia/BilinearMapping.html}
$B:U\times U\rightarrow V$
is said to be \emph{antisymmetric} if
\begin{equation}
B(u,u)=0
\end{equation}
for all $u\in U$.

If $B$ is antisymmetric, then the \htmladdnormallink{polarization}{http://planetmath.org/encyclopedia/Linearization.html} of the anti-symmetry
\htmladdnormallink{relation}{http://planetmath.org/encyclopedia/Corelation.html} gives the condition:
\begin{equation}
B(u,v) + B(v,u) = 0
\end{equation}
for all $u,v \in U$. If the \htmladdnormallink{characteristic}{http://planetmath.org/encyclopedia/Characteristic.html} of $K$ is not 2, then
the two conditions are \htmladdnormallink{equivalent}{http://planetmath.org/encyclopedia/Equivalent4.html}.

A multlinear \htmladdnormallink{mapping}{http://planetmath.org/encyclopedia/Map2.html} $M:U^k\rightarrow V$
is said to be \emph{totally antisymmetric}, or simply antisymmetric, if
for every $u_1,\ldots,u_k\in U$ such that
$$u_{i+1} = u_i$$
for some $i=1,\ldots,k-1$ we have
$$M(u_1,\ldots,u_k)=0.$$
\begin{proposition}
Let $M:U^k\rightarrow V$ be a totally antisymmetric, multlinear
mapping, and let $\pi$ be a \htmladdnormallink{permutation}{http://planetmath.org/encyclopedia/Permutation.html} of $\{1,\ldots,k\}$. Then,
for every $u_1,\ldots,u_k\in U$ we have
$$M(u_{\pi_1},\ldots,u_{\pi_k}) = \mathrm{sgn}(\pi)
M(u_1,\ldots,u_k),$$
where $\mathrm{sgn}(\pi)=\pm1$ according to the \htmladdnormallink{parity}{http://planetmath.org/encyclopedia/SignatureOfAPermutation.html} of $\pi$.
\end{proposition}
{\em \htmladdnormallink{Proof}{http://planetmath.org/encyclopedia/Proof.html}.}
Let $u_1,\ldots,u_k\in U$ be given. multlinearity and anti-symmetry
\htmladdnormallink{imply}{http://planetmath.org/encyclopedia/VacuouslyTrue.html} that
\begin{align*}
0 &= M(u_1+u_2,u_1+u_2,u_3,\ldots,u_k) \\
&= M(u_1,u_2,u_3,\ldots,u_k) + M(u_2,u_1,u_3,\ldots,u_k)
\end{align*}
Hence, the \htmladdnormallink{proposition}{http://planetmath.org/encyclopedia/PropositionalLogic.html} is valid for $\pi=(12)$ (see \htmladdnormallink{cycle notation}{http://planetmath.org/encyclopedia/CycleNotation.html}).
Similarly, one can show that the proposition holds for all
\htmladdnormallink{transpositions}{http://planetmath.org/encyclopedia/Transposition.html}
$$\pi=(i,i+1),\quad i=1,\ldots,k-1.$$
However, such transpositions
\htmladdnormallink{generate}{http://planetmath.org/encyclopedia/SubgroupGeneratedBy.html} the \htmladdnormallink{group}{http://planetmath.org/encyclopedia/NontrivialElement2.html} of permutations, and hence the proposition holds in
full generality.

\paragraph{Note.} The \htmladdnormallink{determinant}{http://planetmath.org/encyclopedia/Determinant2.html} is an excellent example of a totally
antisymmetric, multlinear mapping.

\end{document}