Hellow. I'm sure that the following is truth, but I can't prove it.

Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = \{\theta\in K: \mathrm{ord}\theta = q^{mn}-1\}$ be the set of primitive elements. I want to prove the following statement. If $_RW$ be a subspace of $_RK$, $\mathrm{dim}_RW = n$ such that **for every** $\theta \in A$
$$
W\oplus W\theta\oplus\ldots\oplus W\theta^{m-1} = K,
$$
then $W = aS$ for some $a\in K$.

I will be grateful for the help.