# 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 vectors in a vector space $V$. Then both $0$ and $\tilde{0}$ must satisfy axiom 3 (http://planetmath.org/VectorSpace), i.e., for all $v\in V$,

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

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, using axiom 2 (http://planetmath.org/VectorSpace),

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

and $0=\tilde{0}$. $\Box$

Title zero vector in a vector space is unique ZeroVectorInAVectorSpaceIsUnique 2013-03-22 13:37:16 2013-03-22 13:37:16 matte (1858) matte (1858) 7 matte (1858) Theorem msc 16-00 msc 13-00 msc 20-00 msc 15-00 IdentityElementIsUnique