Letting $F$ be a field and $V$ a finite-dimensional vector space over $F$, we have $U_i,\ldots,U_k$ distinct proper subspaces of $V$. I'm asked to provide a condition on the $U_i$ such that there exists a subspace $W$ such that $U_i\oplus W = V$ for all $i$, where $\oplus$ denotes the internal direct sum. My claim is that $\dim(U_i)=\dim(U_j)$ for all $i$ and $j$ is the necessary condition. Moreover I want to know that this necessary condition is also sufficient, namely, there does exist such a unique $W$ in the event that $F$ is infinite.
Showing the necessary-ness seems reasonable in that each $U_i$ should be isomorphic to $V/W$ if such a $U$ exists, but actually constructing such a $W$ for the case that $F$ is infinite is tripping me up. I'm assuming there's a clever way of extending the bases of $U_i$, but I haven't really been able to make that precise.
It may be that it's closely related to a previous question I posed here
As always, any and all help is very much appreciated.