Let $(A,B)$ and $(C,D)$ be two pairs of $k$-subspaces of a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$, with $k\leq n$ and $\mathrm{dim}(A\cap B)=\mathrm{dim}(C\cap D)$. Does there exist an element $g\in \mathrm{SL}_{n}(q)$ which sends $A$ to $C$ and $B$ to $D$?
I know there is a similar question at: Prove two pairs of subspaces are in the same orbit using dimension which asks the same but for $GL_{n}(q)$. From there, I know that there exists $g\in GL_{n}(q)$ which sends $A$ to $C$ and $B$ to $D$, but how can I adjust this element to have determinant 1?
Many thanks.
Choose such a $g\in GL_n(q)$. Let $\{e_1,\dots,e_n\}$ be a basis for the vector space such that $\{e_1,\dots,e_k\}$ is a basis for $A$ and $\{e_\ell,\dots,e_{\ell+k-1}\}$ is a basis for $B$ for some $\ell$. Now let $h\in GL_n(q)$ be the map that sends $e_1$ to $\det(g)^{-1}e_1$ and fixes $e_m$ for all $m>1$. Note that $h$ sends $A$ to $A$ and $B$ to $B$, and $\det(h)=\det(g)^{-1}$. Thus $gh$ sends $A$ to $C$ and $B$ to $D$, and $\det(gh)=1$.