If you have vector $v$ and vector $u$ that are linearly independent to each other. And vector $u$ is also linearly independent to vector $w$ and $w$ is not equal to vector $v$, then does that imply vector $v$ and vector $w$ are also linearly independent?
As a general principle, reflexivity is necessary for transitivity, as you can always just compose $u \sim v$ with $v \sim u$, and if the relation was transitive, this would yield $u \sim u$. So if the relation is not reflexive, it can't possible be transitive.
Edit: as pointed out in the comments, it is possible to have relations for which there is never any $ v \sim u$, even if $u \sim v$, such as with the relation $<$. In this case though, that's obvious so we're good.