how map a 1 dimensional subspace of $GF(q^r)^N$ to a r dimensional subspace of $GF(q)^{rN}$

Question

Let $K=GF(q)$ be a finite field and $L=GF(q^r)$. Let $n=Nr$. Is it true that to every $1$ dimensional subspace of $L^N$ over $L$, corresponds a $r$ dimensional subspace of $K^n$ over $K$? also I need the corresponding map. Many thanks