For each flag choose an adapted <basis>: its first $n_i$ vectors span $V_i$, obtained by extending a <basis> successively through the nested subspaces and finally through $V$. For two flags, the invertible <linear transformation> taking one adapted <basis> to the other takes each member of the first flag to the corresponding member of the second. Hence \b[the action is transitive].

Solution

Past exam of the options II course of the University of Oxford / 2008 printed variant / A / A2 / b / i / Solution

 Solution (source code)