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Solution to Linear Algebra Done Wrong Exercise 1.1.4

Prove that a zero vector ${\bf 0}$ of a vector space $V$ is unique.

Solution: Let ${\bf 0}$ and ${\bf 0}'$ be two different zero vectors in $V$, then by Property 3 we have
{\bf 0}={\bf 0}+{\bf 0}',\quad {\bf 0}'={\bf 0}'+{\bf 0}.
$$ Since by Property 1, we have ${\bf 0}+{\bf 0}'={\bf 0}'+{\bf 0}$. Therefore, we obtain ${\bf 0}={\bf 0}'$. But we assume that ${\bf 0}$ and ${\bf 0}'$ are different. Hence we get a contradiction. Therefore, the zero vector $\bf 0$ is unique.

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