Vector Inverse is Negative Vector

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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

\(\displaystyle \mathbf v + \paren {-1_F \circ \mathbf v}\) \(=\) \(\displaystyle \paren {1_F \circ \mathbf v} + \paren {-1_F \circ \mathbf v}\) Field Axiom $M \, 3$: Identity Element for Product
\(\displaystyle \) \(=\) \(\displaystyle \paren {1_F + \paren {- 1_F} } \circ \mathbf v\) Vector Space Axiom $V \, 5$: Distributivity over Scalar Addition
\(\displaystyle \) \(=\) \(\displaystyle 0_F \circ \mathbf v\) Field Axiom $A \, 4$: Inverse Elements for Addition
\(\displaystyle \) \(=\) \(\displaystyle \mathbf 0\) Vector Scaled by Zero is Zero Vector

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

From Additive Inverse in Vector Space is Unique:

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

$\blacksquare$


Sources