The Grassmannian 
is the set of 
-dimensional
subspaces in an 
-dimensional
vector space. For example, the set of lines 
is projective
space. The real Grassmannian (as well as the complex Grassmannian) are examples
of manifolds. For example, the subspace 
has a neighborhood
. A subspace 
is in 
if 
and 
and 
.
Then for any 
,
the vectors 
and 
are uniquely determined by requiring 
and 
. The other six entries provide coordinates for
.
In general, the Grassmannian can be given coordinates in a similar way at a point 
. Let 
be the open set of 
-dimensional subspaces which project
onto 
.
First one picks an orthonormal basis 
for 
such that 
span 
. Using this basis, it is possible to take any 
vectors and make a 
matrix. Doing this for the basis of 
, another 
-dimensional subspace in 
, gives a 
-matrix, which is well-defined up to linear combinations
of the rows. The final step is to row-reduce so that the first 
block is the identity matrix. Then the last 
block is uniquely determined by 
. The entries in this block give coordinates for the open set
.
If 
is the standard basis of 
, a basis of 
is given by the 
vectors 
, 
. If 
is a basis of a subspace 
of dimension 
of 
, 
corresponds to a point 
of 
, whose coordinates are
the components of 
with respect to the basis of 
given above. These coordinates are the so-called Grassmann coordinates of 
.
A different choice of the basis of 
yields a different 
-tuple of coordinates, which differs from the original 
-tuple by a nonzero multiplicative constant,
hence it corresponds to the same point.
The Grassmannian is also a homogeneous space. A subspace is determined by its basis vectors. The group that permutes basis vectors
is 
. The matrix that fixes 
is a diagonal block
matrix, with a 
nonsingular matrix in the top left, and a 
invertible matrix in the
lower right. 
acts transitively on the Grassmannian 
. Let 
be the stabilizer (or isotropy) of 
. Then 
is the subgroup of 
consisting of matrices ![A=[a_(i,j)]](/images/equations/Grassmannian/Inline65.svg)
such that 
for all 
, 
such that 
and 
.
is isomorphic to 
.
The tangent space to the Grassmannian is given by 
matrices, i.e., linear maps
from 
to the quotient vector space 
.
The elements 
are the 
-minors
of the 
matrix whose 
th
row contains the components of 
with respect to the basis 
. It corresponds to a linear transformation 
whose range is 
. In general, the range of such a linear transformation has
dimension 
iff the corresponding 
matrix has rank 
.
Let 
be the subset of 
defined by the condition that all the 
-minors of the matrix 
(which can be viewed as a sequence
of 
coordinates) be equal to zero, and
one 
-minor
be nonzero. The Grassmannian 
can be viewed as the image of the map 
which maps each matrix of 
to the sequence of its 
-minors.
It as an algebraic projective algebraic variety defined by equations called Plücker's
equations. It is a nonsingular variety of
dimension 
.
