The Plücker embedding maps the Grassmannian of 
-dimensional
subspaces of an 
-dimensional
vector space 
over a field 
into the projective space
of the 
th
exterior power of 
by
![Gr(k,V)∋W=span(v_1,...,v_k)|->[v_1 ^ ... ^ v_k] in P( ^ ^kV).](/images/equations/PlueckerEmbedding/NumberedEquation1.svg) |
Changing the basis 
multiplies the wedge product by the nonzero determinant
of the change-of-basis matrix, so the projective point
is independent of the chosen basis.
With respect to a basis of 
, the homogeneous coordinates
of the image are the 
minors of a matrix whose rows span 
. They are called Plücker
coordinates, also known as Grassmann coordinates.
The image is a closed projective variety of dimension 
characterized by quadratic Plücker relations. For
example, the image of 
in 
is the quadric
 |
