Complements of a subspace

Question

Show that a non-trivial subspace $U$ of $V$ has two virtually disjoint complements iff $dim(U)\geq \frac{dim(V)}{2}$.

Definition 1:$S$ and $T$ are said to be virtually disjoint if $S\cap T=\{0\}$.

Definition 2: A subspace $T$ of $V$ is said to be complement of $S$ (a subspace of $V$) if $S\oplus T=V$ (i.e. $S+T$ is direct sum).

The if part is done, I am stuck with the only if part of the question. If possible try to find a suitable answer using Modular Law.

Any help will be appreciated.

Answer 1

for the only if, use the formula for the sum of subspaces, reported for example in here.

In fact, if $S,T$ are the disjoint complements, then $$ dim(S) + dim(T) = dim(S + T) \le dim(V) $$ but $dim(S) = dim(T) = dim(V) - dim(U)$ so $$ dim(U) = dim(V) - dim(S) \ge dim(V) - \frac{\dim(V)}2= \frac{\dim(V)}2. $$