Vector space/Wedge product/Construction/Section

Let ${}K$ a field, let ${}V$ be a ${}K$-vector space, and ${}n\in \mathbb {N}$. We construct the so-called ${}n$-th wedge product of ${}V$ with itself, written as ${}\bigwedge ^{n}V$. For this, we consider the set ${}S$ of all symbols of the form

${(v_{1},\ldots ,v_{n})}{\text{ with }}v_{i}\in V,$

and the corresponding set of the ${}e_{(v_{1},\ldots ,v_{n})}$. We consider the vector space

${}H=K^{(S)}\,,$

this is the set of all (finite) sums

$a_{1}e_{s_{1}}+\cdots +a_{k}e_{s_{k}}{\text{ with }}a_{i}\in K{\text{ and }}s_{i}\in S;$

the ${}e_{s}$ form a basis of this space. This is, with the natural addition and the natural scalar multiplication, a vector space; it is a linear subspace of the mapping space ${}\operatorname {Map} \,{\left(S,K\right)}$ (${}H$ is the set of those vectors where almost all elements ${}s\in S$ have the value ${}0$). In ${}H$, we consider the linear subspace ${}U$ that is generated by the following elements (they are called the standard relations of the wedge product).

$e_{(v_{1},\ldots ,v_{i-1},v+w,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},w,v_{i+1},\ldots ,v_{n})}$

for arbitrary ${}v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n},v,w\in V$.

$e_{(v_{1},\ldots ,v_{i-1},av,v_{i+1},\ldots ,v_{n})}-ae_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{n})}$

for arbitrary ${}v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n},v\in V$ and ${}a\in K$.

$e_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{j-1},v,v_{j+1},\ldots ,v_{n})}$

for ${}i<j$ and arbitrary ${}v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,,v_{j-1},v_{j+1},\ldots ,v_{n},v\in V$.

Here, the main idea is to enforce the rules that hold for an alternating multilinear mapping by making these relations to ${}0$. The first type represents the additivity in every argument, the second represents the compatibility with the scalar multiplication, the third represents the alternating property.

We define now

${}\bigwedge ^{n}V:=H/U\,,$

that is, we form the residue class space of ${}H$ modulo the linear subspace ${}U$.