Let $F$ be a field and $V$ be a $n$-dimensional vector space over $F$.
For any subspace $W\subseteq V$ we say that $W'$ is a complement of $W$ if $$ W + W' = V.$$
I have the following question. Consider $W_1,\ldots, W_l$ finitely many subspaces of the same dimension $k$, then prove that there exists a subspace $W'$ of dimension $n-k$ such that it is a complement for all $W_i$.
I have tried by considering basis $B_i$ of $W_i$ and extending to basis of $V$; but I do not really know how to "simultaneously" extend the basis so I can define the complement $W'$.