Editing Homogeneous space (section)

== Example ==
For example, in the line geometry case, we can identify ''H'' as a 12-dimensional subgroup of the 16-dimensional [[general linear group]], GL(4), defined by conditions on the matrix entries
: ''h''<sub>13</sub> = ''h''<sub>14</sub> = ''h''<sub>23</sub> = ''h''<sub>24</sub> = 0,
by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that ''X'' has dimension 4.

Since the [[homogeneous coordinates]] given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers.

This example was the first known example of a [[Grassmannian]], other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.