Suppose that $F$ is a subfield of a field $G$ and, for  
$n\times n$ matrices $A_1,\dots,A_m, B_1,\dots,B_m$ 
over $F$, there exists a matrix $T\in{\rm GL}_n(G)$ 
such that $T^{-1}A_iT=B_i$ for all $i$. 

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

It is easy to see tha the answer is  

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