Geometric intuition of the dimension of Grassmannians and flag manfolds

Question

I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{n}k_{i}(k_{i-1}-k_{i})+Nk_{n}$ (in fact with understanding the Grassmanian case it would be enough because the flag are just "nested" Grassmanians).

I am thinking in a spatial way in the well known $G(2,5)$ but I am unable to see geometrically how the space of all $2$-planes in $\mathbb{P}^{5}$ can be $6$-dimensional.

Answer 1

The idea is to use the standard affine charts of $G(k,n)$. Start with the $k$-plane $P \subset \mathbb{R}^n$ (say, the one spanned by $e_1, \dots, e_k$) and a $(n-k)$-plane $P^\perp$ transverse to $P$ (say, the one spanned by $e_{k+1}, \dots, e_n$). The set of all $k$-planes transverse to $P^\perp$ is an open neighborhood of $P$. Each such $k$-plane $Q$ is the graph of a linear map $A: P \rightarrow P^\perp$ and vice versa. Therefore, the dimension of $G(k,n)$ is the dimension of the space of all linear maps from a fixed $k$-dimensional subspace to a fixed $(n-k)$-dimensional subspace, i.e., the space of all $(n-k)$-by-$k$ matrices, which has dimension $k(n-k)$.

When $k = 1, 2$ and $n = 2, 3$, this can be seen visually and therefore viewed as geometric intuition.

Similar coordinates can be defined for a flag manifold, where the matrices are block triangular.

Answer 2

Very simple examples of $k$-planes in $\mathbb{C}^n$ are the graphs linear operators $\mathbb{C}^k\to\mathbb{C}^{n-k}$. These linear operators form a complex vector space of dimension $k(n-k)$, that is exactly the dimension of $G(k,n)$. As an aside, this is how you construct coordinate charts on $G(k, n)$. For $k=1$ this construction corresponds to the classical way of producing coordinate charts on projective spaces.