The space of linear maps into a Grassmann manifold

We show that the space of all holomorphic maps of degree one from the Rie-mann sphere into a Grassmann manifold is a sphere bundle over a flag manifold. Using the notions of "kernel" and "span" of a map, we completely identify the space of unparame-terized maps as well. The illustrative case of maps into the quadric Grassmann manifold is discussed in detail and the homology of the corresponding spaces is computed.