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Solution to Linear Algebra Done Wrong Exercise 1.1.6


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.



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