[[Grassmannian]]
# Plücker embedding

The **Plücker embedding** $\iota : \mathrm{Gr}_{k}(V) \hookrightarrow \mathrm{P}({\textstyle\bigwedge}^k V)$ embeds the [[Grassmannian]] $\mathrm{Gr}_{k}(V)$ of $k$-dimensional subspaces of $V$ into the [[projectivization]] $\mathrm{P}({\textstyle\bigwedge}^k V)$ of the $k$-th [[Exterior algebra|exterior power]] of $V$. #m/def/geo 
$$
\begin{align*}
\iota : \Span \{w_{i}\}_{i=1}^k \mapsto \left[\bigwedge_{i=1}^kw_{i}\right]
\end{align*}
$$
Since the wedge product of linearly independent vectors are proportional iff they span the same linear subspace,
each $k$-dimensional subspace of $V$ is uniquely defined by a $k$-blade up to scaling,
This motivates the embedding as given above.

In the simplest non-projective case of $\mathrm{Gr}_{2}(\mathbb{K}^4)$ we get the [[Klein correspondence]].

## Properties

- The $k$-blades of $V$ satisfy a set of homogenous quadratic relations known as the [[Plücker relations]].
  Hence the Grassmannian is embedded as a [[Projective variety]].

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