Vector Inverse is Negative Vector

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Theorem

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

Then:

$\forall \mathbf v, -\mathbf v \in \mathbf V: -\mathbf v = -1_{\mathbb F} \circ \mathbf v$


Proof

Utilizing the vector space axioms:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \mathbf v + \left({-1_{\mathbb F} \circ \mathbf v }\right)\) \(=\) \(\displaystyle \) \(\displaystyle \left({1_{\mathbb F} \circ \mathbf v}\right) + \left({-1_{\mathbb F} \circ \mathbf v }\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \left({1_{\mathbb F} - 1_{\mathbb F} }\right) \circ \mathbf v\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle 0 \circ \mathbf v\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \mathbf 0\) \(\displaystyle \) \(\displaystyle \)          Vector Scaled by Zero is Zero Vector          

... so $-1_{\mathbb F} \circ \mathbf v$ is an additive inverse of $\mathbf v$.

From Vector Inverse Unique, $-1_{\mathbb F} \circ \mathbf v = -\mathbf v$.

$\blacksquare$