# Vector Inverse is Negative Vector

## Theorem

Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$.

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

Then:

$\forall \mathbf v \in \mathbf V: -\mathbf v = -1_F \circ \mathbf v$

## Proof

 $\ds \mathbf v + \paren {-1_F \circ \mathbf v}$ $=$ $\ds \paren {1_F \circ \mathbf v} + \paren {-1_F \circ \mathbf v}$ Field Axiom $\text M3$: Identity for Product $\ds$ $=$ $\ds \paren {1_F + \paren {- 1_F} } \circ \mathbf v$ Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition $\ds$ $=$ $\ds 0_F \circ \mathbf v$ Field Axiom $\text A4$: Inverses for Addition $\ds$ $=$ $\ds \mathbf 0$ Vector Scaled by Zero is Zero Vector

so $-1_F \circ \mathbf v$ is an additive inverse of $\mathbf v$.

$-1_F \circ \mathbf v = -\mathbf v$

$\blacksquare$