TITLE: Unique complement of a subspace
QUESTION [8 upvotes]: Can we have a unique complement to a subspace of a finite-dimensional vector space $V$ (exclude the trivial cases $0$ and $V$)?

REPLY [18 votes]: No. Let $U$ be your subspace, with $\{ 0 \} \subset U \subset V$. (I mean, proper inclusions.) Let $W$ be a complement of $U$ in $V$, and pick an element $0 \ne w \in W$. Let $B$ be a basis of $W$ including $w$. Pick $0 \ne u \in U$, and modify $B$ by replacing $w$ with $w + u$, to obtain a new set $B'$. Now $W' = \langle B' \rangle$ is another complement of $U$, different from  $W$.