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 vector basis,
it is possible to take any 
vectors and make a 
matrix. Doing this for the
vector 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 perform row reduction 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 
.
Let 
for a field 
, with standard basis 
. A vector
basis of 
is given by the 
vectors 
, 
. If 
is a vector basis
of a subspace 
of dimension 
of 
, the Plücker embedding
sends 
to a point 
of 
, whose coordinates
are the components of 
with respect to the vector basis of 
given above. These coordinates
are the so-called Grassmann coordinates
of 
.
A different choice of the vector basis of 
yields a different 
-tuple of coordinates,
which differs from the original 
-tuple by the nonzero determinant
of the change of basis matrix,
hence it corresponds to the same point.
The Grassmannian is also a homogeneous space. A subspace is determined by its vector
basis vectors. The general
linear group 
acts on bases. A matrix
fixing the subspace 
is upper block triangular, with nonsingular
matrices of sizes 
and 
on the diagonal and an arbitrary upper-right block.
acts transitively
on the Grassmannian 
.
Let 
be the stabilizer (or isotropy
group) of 
.
Then 
is the subgroup of 
consisting of matrices ![A=[a_(i,j)]](/images/equations/Grassmannian/Inline66.svg)
such
that 
for all 
, 
such that 
and 
. 
is isomorphic to 
.
The tangent space to the Grassmannian is given by 
matrices,
i.e., linear transformations 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 vector 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 
represented by 
matrices 
of matrix
rank 
.
Equivalently, at least one 
-minor is nonzero.
The Grassmannian 
can be viewed as the image of the map 
which maps each
matrix of 
to the sequence of its 
-minors.
It is a projective algebraic variety defined by equations called Plücker
relations. It is a smooth variety of dimension 
.
