Exercise 1.1.6

Question

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

Jing Huang · Accepted Answer

\begin{proof}
Assume there exist two different additive inverses $\mathbf{w}_1$ and $\mathbf{w}_2$ of vector $\mathbf{v} \in V$. Then
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\begin{equation*}
\begin{split}
    \mathbf{v} + \mathbf{w}_1 &= \mathbf{0}  \
    \mathbf{v} + \mathbf{w}_2 &= \mathbf{0}.
\end{split}
\end{equation*}
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Obtain the difference of the two equations, we get
%%%%%%%%%%%%%%%%%
\begin{equation*}
    \mathbf{w}_1 - \mathbf{w}_2 = \mathbf{0},
\end{equation*}
%%%%%%%%%%%%%%%%
then
%%%%%%%%%%%%%%%%
\begin{equation*}
     \mathbf{w}_1 = \mathbf{w}_2.
\end{equation*}
%%%%%%%%%%%%%%%
So the additive inverse is unique.  
\end{proof}

Exercise 1.1.6

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Exercise 1.1.6

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