# Additive Inverse in Vector Space is Unique

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## Theorem

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

Then for every $\mathbf v \in \mathbf V$, the additive inverse of $\mathbf v$ is unique:

$\forall \mathbf v \in \mathbf V: \exists! \paren {-\mathbf v} \in \mathbf V: \mathbf v + \paren {-\mathbf v} = \mathbf 0$

## Proof

### Proof of Existence

Follows from the vector space axioms.

$\Box$

### Proof of Uniqueness

Let $\mathbf v$ have inverses $\mathbf x$ and $\mathbf y$.

Then:

 $\displaystyle \mathbf v + \mathbf x$ $=$ $\displaystyle \mathbf 0$ $\quad$ $\quad$ $\displaystyle \land \ \$ $\displaystyle \mathbf v + \mathbf y$ $=$ $\displaystyle \mathbf 0$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \mathbf v + \mathbf x$ $=$ $\displaystyle \mathbf v + \mathbf y$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \mathbf x$ $=$ $\displaystyle \mathbf y$ $\quad$ Vectors are Left Cancellable $\quad$

$\blacksquare$