Proof direct sum from isomorphism

Question

Let $ V $ be a vector space over $ \Bbb F $, and suppose U and W are vector subspaces of V. Suppose $ V/U \cong W $and $V/W \cong U$ . Show that $ V = U \oplus W $.

Answer 1

It is false in general: take a vector space of even dimension $2n$. Then for any two subspaces $U, W$ of dimension $n$, there will be isomorphisms $V/U\simeq W $, $V/W\simeq U$, yet we can choose $U,W$ such that $\;U\cap W\neq\{0\}$