Additive Inverse in Vector Space is Unique

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 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 is Unique
 * Vector Inverse is Negative Vector