every projection operator of rank defines a subspace as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold with the set of rank orthogonal projection operators :
::

In particular, taking or gives completely explicit equations for embedding the Grassmannians , in the space of real or complex matrices , , respectively.

Since this defines the Grassmannian as a closed subset of the sphere the Grassmannian

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