Prove that the additive inverse, defined in Axiom 4 of a vector space is unique.

Solution: Suppose both $\bf w$ and $\bf w'$ are different additive inverses of $\bf v$, then we have

$$

{\bf 0}={\bf v}+{\bf w},\qquad {\bf 0}={\bf v}+{\bf w'}.

$$Therefore

\begin{align}

{\bf w}=&\ {\bf w}+{\bf 0}={\bf w}+({\bf v}+{\bf w'})\\

=&\ ({\bf w}+{\bf v})+{\bf w'}={\bf 0}+{\bf w'}\\ =&\ {\bf w'}+{\bf 0}={\bf w'}.

\end{align}Hence we obtain a contradiction. This implies that the additive inverse, defined in Axiom 4 of a vector space is unique.