Given $n$ vectors $y_1,...,y_n$ of vector space $E$ of dimmension $n$, show that we can find bases $(b_1,...,b_n)$ and $(b_1',...,b_n')$ of $E$ such that for all $k \in \{1,2,..,n\}$, we have $y_k=b_k+b_k'$.
The reason why I know this is true is because it is equivalent to a reformulation of the problem which is magically much easier to solve, but posed as above I think it becomes harder.
Let $e_k$, $k=1,...,n$ a basis of $E$, and $f$ the linear application from $E$ to $E$ such that $f(e_k)=y_k$ for all $k$. One can write $f=g+h$, with $g, h$ isomorphisms of $E$ (For example, $g=\lambda id$ and $h=f-\lambda id$ with $\lambda$ not $0$ and not an eigenvalue of $f$). Then $b_k=g(e_k)$ and $b_k^{\prime}=h(e_k)$ do the job.