  We show how to compactly represent any $n$-dimensional subspace of $R^m$ as a
banded product of Householder reflections using $n(m - n)$ floating point
numbers. This is optimal since these subspaces form a Grassmannian space
$Gr_n(m)$ of dimension $n(m - n)$. The representation is stable and easy to
compute: any matrix can be factored into the product of a banded Householder
matrix and a square matrix using two to three QR decompositions.
