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 such that for all , such that and . is isomorphic to .
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.