Exercise 1.1.4

Question

Prove that a zero vector of a vector space $V$ is unique.

Jing Huang · Accepted Answer

\begin{proof}
Suppose there exist 2 different zero vectors $\mathbf{0}_1$ and $\mathbf{0}_2$. So for any $\mathbf{v} \in V$, we have
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 \begin{equation*}
    \begin{split}
    \mathbf{v} + \mathbf{0}_1 &= \mathbf{v}  \
   \mathbf{v} + \mathbf{0}_2 &= \mathbf{v}.
   \end{split}
\end{equation*}
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Find the difference of the equations above, we get
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\begin{equation*}
    \mathbf{0}_2 - \mathbf{0}_1 = \mathbf{v} - \mathbf{v} = \mathbf{0}_1 \,/(	ext{or}) \, \mathbf{0}_2,
\end{equation*}
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then
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\begin{equation*}
    \mathbf{0}_2 = \mathbf{0}_1 + \mathbf{0}_1 / \mathbf{0}_2 = \mathbf{0}_1.
\end{equation*}
%%%%%%%%%%%%%%%%
So the zero vector is unique. 
\end{proof}

Exercise 1.1.4

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Exercise 1.1.4

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