\b[Yes.] Let $g$ send the first <flag> to the second and write $d=\det g\ne0$. In an adapted <basis> of the target <flag>, scale its first vector by $d^{-1}$ and fix the others. This defines a <flag>-preserving map $h$ with determinant $d^{-1}$. Then $hg$ still sends the first <flag> to the second and has determinant one. Thus $\operatorname{SL}(n,K)$ is <transitive group action>[transitive] as well.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / A / 2 / b / ii / Solution

 Solution (source code)