Additive Inverse in Vector Space is Unique

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

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


 * $\forall \mathbf v \in \mathbf V: \exists! \left({-\mathbf v}\right) \in \mathbf V: \mathbf v + \left({-\mathbf v}\right) = \mathbf 0$

Proof of Existence
Follows from the vector space axioms.

Proof of Uniqueness
Let $\mathbf v$ have inverses $\mathbf x$ and $\mathbf y$.

Then:

Also see

 * Zero Vector Unique
 * Vector Inverse is Negative Vector