Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement.
If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = U \oplus V^0$ .
What does supplementaty subspace mean in above proposition? This is s a proposition in Serre's A Course in Arithmetic, and it only says that it's clear to prove, but never defines what a supplementary subspace is.
Since there is no answer, here is what I think the correct interpretation is:
If $U,U'$ are subspaces of a vector space $V$, then $U'$ is supplementary to $U$ iff $U\oplus U' = V$, in the sense that $U\cap U' = 0$ and $U + U' = V$.
With this definition, contrary to Marc van Leeuwen's comment, I think the statement of Serre is not totally trivial: all you need to show is that you have orthogonality of the subspaces.