If $F$ is a field and $d \ge 1$.
Let $K$ be a subfield of $F$ with finite index $k = [F : K]$. Then $F$ is a $k$-dimensional vector space over $K$. Thus every $F$-vector space is also a $K$-vector space and any $F$-linear transformation is also $K$-linear. Specifically this means that $F^d$ is isomorphic to $K^{kd}$ as a $K$-vector space and that $GL_d(F)$ is isomorphic to a subgroup of $GL_{kd}(K)$.
This is taken from page 55 of Dixon & Mortimer Permutation Groups. I ask myself why $GL_d(F)$ is just isomorphic to a subgroup of $GL_{kd}(K)$, can you give an example of an element from $GL_{kd}(K)$ not contained in $GL_d(F)$ under the identification $F^d$ with $K^{kd}$?