Existence of a supplementary subspace

Basis extension. Let $(e_1,\ldots,e_k)$ be a basis of $F$. By the incomplete basis theorem, it extends to a basis $(e_1,\ldots,e_k,e_{k+1},\ldots,e_n)$ of $E$.

Define $G$. Let $G=\operatorname{span}(e_{k+1},\ldots,e_n)$.

Direct sum. Any $v\in E$ is a combination of the full basis, hence $v=v_F+v_G$ with $v_F\in F$ and $v_G\in G$. If $v\in F\cap G$, then $v$ is a combination of the $e_i$ for $i\le k$ and of the $e_j$ for $j>k$; by uniqueness of coordinates in the basis, all coefficients are zero, so $v=0$.

Conclusion. $E=F\oplus G$. Any subspace has at least one supplementary subspace.