How to construct a flag manifold / flag variety? I'm learning about flag manifolds and I'm trying to understand it better with an example.  My goal is to construct a flag manifold that will be a collection of 2 flags- xy, yx.
From here I understand that the definition of a flag is that it's an increasing sequence of subspaces of a finite-dimensional vector space V. So in the vector xy, y needs to contain x, and in the vector yx, x needs to contain y.
How can I construct it with simple vectors? Say, x = [1,2], y = [5,6]. That is, what will xy or yx be equal to in a vector form?
 A: $\DeclareMathOperator{\Span}{Span}\newcommand{\Reals}{\mathbf{R}}$This may not be an answer so much as a glossary based on clarifications in the comments.

If $V$ is a $k$-dimensional real vector space for some $k \geq 1$, and if $(x_{1}, x_{2}, \dots, x_{k}) = (x_{i})_{i=1}^{k}$ is an ordered basis of $V$, there is an induced flag $(V_{1}, \dots, V_{k}) = (V_{j})_{j=1}^{k}$ defined by $V_{j} = \Span(x_{i})_{i=1}^{j}$, the set of linear combinations of the first $j$ vectors in the ordered basis.
If we take $x = (1, 2)$ and $y = (5, 6)$, then $xy$ is the flag in the plane with $V_{1} = \Span(x)$—the set of scalar multiples of $(1, 2)$—and $V_{2} = \Reals^{2}$. Similarly, $yx$ is the flag in the plane with $V_{1} = \Span(y)$ and $V_{2} = \Reals^{2}$.

From a coding perspective, there's not much to say generally except the code presumably needs to supply an ordered list of ordered lists, e g., [[1,1,1], [1,2,3], [-1,0,0]] (caution: I don't speak Python), and needs to have some way of constructing, storing, and working with subspaces (sets of linear combinations).
