Suppose that $F$ is a subfield of a field $G$ and, for

$ntimes n$ matrices $A_1,dots,A_m, B_1,dots,B_m$

over $F$, there exists a matrix $Tin{rm GL}_n(G)$

such that $T^{-1}A_iT=B_i$ for all $i$.

Does this imply that such a matrix $T$ can be chosen from ${rm GL}_n(F)$?

Surely,

**yes**if $m=1$;- and
**yes**if the field $F$ is infinite.