I'm struggling to understand the transversality of flags:
The statement "any general pair of distinct flags $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ can be mapped by a suitable element of GL(n) to a specific pair consisting of the standard flag $\mathbb{E}$, with $E_i = \operatorname{Span} \{e_1, \ldots, e_i\}$ and the opposite flag $\mathbb{E}'$ with $E_i' = \operatorname{Span} \{e_{n−i+1}, \ldots , e_{n}\}$" doesn't seem true to me.
For example if I take the flags
$$\mathbb{F}_{1}=\operatorname{Span}\{e_{1}\}\subsetneq \operatorname{Span}\{e_{1}, e_{2}\}\subsetneq \operatorname{Span}\{e_{1}, e_{2},e_{3}\}$$
and
$$\mathbb{F}_{2}=\operatorname{Span}\{e_{2}\}\subsetneq\operatorname{Span}\{e_{2}, e_{1}\}\subsetneq\operatorname{Span}\{e_{2}, e_{1},e_{3}\},$$
what is the right $g \in GL(n)$ to choose, to make the flags meet transversally?
It seems false to me as $\operatorname{Span} \{e_2\} \subset \operatorname{Span}\{e_1, e_2\}$, $ \ $and this will still be the case under the action of $g \in GL(n)$.