Zero Vector is Unique

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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