Zero Vector is Unique

Theorem

Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms.

Then the zero vector in $\mathbf V$ is unique:

$\exists! \mathbf 0 \in \mathbf V: \forall \mathbf x \in \mathbf V: \mathbf x + \mathbf 0 = \mathbf x$

Proof

Proof of Existence

Follows from the vector space axioms.

$\Box$

Proof of Uniqueness

Let $\mathbf 0$, $\mathbf 0'$ be zero vectors.

Utilizing the vector space axioms:

\(\ds \mathbf 0\)

\(=\)

\(\ds \mathbf 0 + \mathbf 0'\)

\(\ds \)

\(=\)

\(\ds \mathbf 0' + \mathbf 0\)

\(\ds \)

\(=\)

\(\ds \mathbf 0'\)

$\blacksquare$

Also see

• Additive Inverse in Vector Space is Unique