Theorem The zero vector in a vector space is unique.

Proof. Suppose $0$ and $\tilde{0}$ are zero vectors in a vector space $V$ . Then both $0$ and $\tilde{0}$ must satisfy axiom 3, i.e., for all $v\in V$ , \begin{eqnarray*} v + 0 &=& v,\\ v + \tilde{0} &=& v. \end{eqnarray*}Setting $v=\tilde{0}$ in the first equation, and $v=0$ in the second yields $\tilde{0} + 0 = \tilde{0}$ and $0 + \tilde{0} = 0$ . Thus, using axiom 2, \begin{eqnarray*} {\displaystyle0} &= \tilde{0} + 0 \\ &= 0 + \tilde{0} \\ &= \tilde{0}, \end{eqnarray*}and $0=\tilde{0}$ .