Say we are considering some vector space $V$ and some non-zero unit vector $x$. Considering the one-dimensional vector space $\langle x \rangle = \text{span}(x)$, we know it admits a complementary subspace $H$ (per Zorn's lemma). I was wondering what is a sufficient argument then to establish that $H$ is a hyperplane then, as we can't just look at the sums/differences of dimension.
Would it be enough to claim that that $x + H$ provides a basis for the quotient space $V/H$ (and so it's one dimensional)? It's clear we can readily demonstrate this by using the definition of the internal direct sum, but I just wanted to check that my reasoning is correct as I am unfamiliar with working in this setting. Thanks for the help!
