Grassmanian manifold

Given a vector space $V$ of dimension $n$ and an integer $r < n$ , $G (r, V)$ is the smooth manifold of $r$ -planes of $V$ . It is a generalization of the projective space.
The dimension is

r \times (n - r) .

The key geometric idea is: deforming an $r$ -plane means specifying how each of its $r$ directions tilts into the complementary $(n - r)$ -dimensional directions.