Editing Plücker coordinates (section)

=== Bijectivity ===
If <math>(q_{01}:q_{02}:q_{03}:q_{23}:q_{31}:q_{12})</math> are the homogeneous coordinates of a point in {{tmath|\mathbb P^5}}, without loss of generality assume that {{math|''q''<sub>01</sub>}} is nonzero. Then the matrix

: <math> M = \begin{bmatrix} q_{01} & 0 \\ 0 & q_{01} \\ -q_{12} & q_{02} \\ q_{31} & q_{03} \end{bmatrix} </math>

has rank 2, and so its columns are distinct points defining a line {{mvar|L}}. When the {{tmath|\mathbb P^5}} coordinates, {{mvar|q<sub>ij</sub>}}, satisfy the quadratic Plücker relation, they are the Plücker coordinates of {{mvar|L}}. To see this, first normalize {{math|''q''<sub>01</sub>}} to 1. Then we immediately have that for the Plücker coordinates computed from {{mvar|M}}, {{math|1=''p<sub>ij</sub>'' = ''q<sub>ij</sub>''}}, except for

: <math> p_{23} = - q_{03} q_{12} - q_{02} q_{31} . </math>

But if the {{mvar|q<sub>ij</sub>}} satisfy the Plücker relation 
:<math>q_{23} + q_{02}q_{31} + q_{03}q_{12} = 0,</math> 
then {{math|1=''p''<sub>23</sub> = ''q''<sub>23</sub>}}, 
completing the set of identities.

Consequently, {{math|α}} is a [[surjection]] onto the [[algebraic variety]] consisting of the set of zeros of the quadratic polynomial

: <math> p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12} . </math>

And since {{math|α}} is also an injection, the lines in {{tmath|\mathbb P^3}} are thus in [[bijection|bijective]] correspondence with the points of this [[quadric]] in {{tmath|\mathbb P^5}}, called the Plücker quadric or [[Klein quadric]].