Prove or disprove:
- If is orthogonal to , then
then is orthogonal to .
- Proof. Let and . Then we have
But, by hypothesis, so . Therefore,
since for all real
- Proof. Suppose otherwise, that . Then, we know that for all we have
for all . So, in particular is must hold for both
Both of these are nonzero since by assumption. But then,
But since we cannot have both and . Hence, this is a contradiction, so we must have .