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
Canonical name	ZeroVectorInAVectorSpaceIsUnique
Date of creation	2013-03-22 13:37:16
Last modified on	2013-03-22 13:37:16
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Theorem
Classification	msc 16-00
Classification	msc 13-00
Classification	msc 20-00
Classification	msc 15-00
Related topic	IdentityElementIsUnique