zero vector in a vector space is unique

Theorem The zero vector in a vector space is unique.

Proof. Suppose $0$ and $\tilde{0}$ are zero vectorsin a vector space $V$. Then both$0$ and $\tilde{0}$ must satisfyaxiom 3 (http://planetmath.org/VectorSpace),i.e., for all $v\in V$,

$v+0$$=$$v,$

$v+\tilde{0}$$=$$v.$

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, usingaxiom 2 (http://planetmath.org/VectorSpace),

$0$$=\tilde{0}+0$

$=0+\tilde{0}$

$=\tilde{0},$

and $0=\tilde{0}$. $\mathrm{\square }$