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I have a textbook question that asks me to prove that there is only one zero-element of a vector space. There are many other questions that have asked this, but I was unsatisfied with the answers.

As far as I know, there are two axioms involved:

  1. $\mathbf v+\mathbf 0=\mathbf v$

  2. $\mathbf v+(-\mathbf v)=\mathbf0$

Is there a way to prove that the zero-vector in (1) is the same as the zero-vector in (2)? That is, I want to prove:

if $\mathbf v+\mathbf z=\mathbf v$ and $\mathbf v+(-\mathbf v)=\mathbf0$, then $\mathbf z=\mathbf0$

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1 Answer 1

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It's a tautology that $v=v$. Using the additive identity, we have $v=v+0$. Add the additive inverse $-v$ to both sides. Then $v+(-v)=0$.

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  • $\begingroup$ It is a bit sloppy to "subtract" from both sides—it might be better to invoke additive cancelation. However, this is otherwise elegantly phrased. (+1) $\endgroup$
    – Xander Henderson
    Commented Sep 27, 2017 at 3:10
  • $\begingroup$ Good point. We get old and start to take shortcuts :D $\endgroup$ Commented Sep 27, 2017 at 3:11
  • $\begingroup$ I believe all this shows is that it is possible that the two zeros are the same. Clearly it is possible. I don't think this proves that it must be the case that they are the same. $\endgroup$ Commented Sep 28, 2017 at 7:23

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