Let $F\subset E$ be a finite Galois extension. Prove that $\mathcal L_F(E,E)$, the vector space of all $F$-linear maps $E\to E$ over $E$ has dimension $[E:F]$, where the scalar multiplication is defined by multiplication in $E$ pointwisely.
My attempt is to take the Galois automorphisms in $\operatorname{Gal}(E/F)$ and it sufficies to show they generate $\mathcal L_F(E,E)$ since $|\operatorname{Gal}(E/F)|=[E:F]$ and by linear independence of characters the Galois automorphisms are linearly independent over $E$.
I am stuck here. How can I prove the Galois automorphisms generate $\mathcal L_F(E,E)$?