I have a textbook question that asks me to prove that there is only one zero-element of a vector space. There are many other questions that have asked this, but I was unsatisfied with the answers.
As far as I know, there are two axioms involved:
$\mathbf v+\mathbf 0=\mathbf v$
$\mathbf v+(-\mathbf v)=\mathbf0$
Is there a way to prove that the zero-vector in (1) is the same as the zero-vector in (2)? That is, I want to prove:
if $\mathbf v+\mathbf z=\mathbf v$ and $\mathbf v+(-\mathbf v)=\mathbf0$, then $\mathbf z=\mathbf0$