Sufficient Condition for Vector Equals Inverse iff Zero

Theorem

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

Let $\mathbb F$ be infinite.

Then:

$\forall \mathbf v, -\mathbf v \in \mathbf V: \mathbf v = - \mathbf v \iff \mathbf v = \mathbf 0$

Proof

Necessary Condition

\(\ds \mathbf v\)

\(=\)

\(\ds \mathbf 0\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds -1_{\mathbb F} \circ \mathbf v\)

\(=\)

\(\ds -1_{\mathbb F} \circ \mathbf 0\)

scaling both sides by the negative of the unity of $\mathbb F$

\(\ds \leadsto \ \ \)

\(\ds -\mathbf v\)

\(=\)

\(\ds \mathbf 0\)

Vector Inverse is Negative Vector, Zero Vector Scaled is Zero Vector

\(\ds \leadsto \ \ \)

\(\ds \mathbf v\)

\(=\)

\(\ds -\mathbf v\)

$\Box$

Sufficient Condition

Utilizing the vector space axioms:

\(\ds \mathbf v\)

\(=\)

\(\ds -\mathbf v\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds \mathbf v - \mathbf v\)

\(=\)

\(\ds -\mathbf v - \mathbf v\)

adding $-\mathbf v$ to both sides

\(\ds \leadsto \ \ \)

\(\ds \mathbf 0\)

\(=\)

\(\ds \left({-1_{\mathbb F} \circ \mathbf v}\right) + \left({-1_{\mathbb F} \circ \mathbf v}\right)\)

Vector Inverse is Negative Vector

\(\ds \leadsto \ \ \)

\(\ds \mathbf 0\)

\(=\)

\(\ds \left({-2 \cdot 1_{\mathbb F} }\right)\circ \mathbf v\)

$\blacksquare$

By hypothesis, $\mathbb F$ is infinite.

By Characteristic of Division Ring is Zero or Prime:

$\operatorname{Char} \left({\mathbb F}\right) = 0$

so $-2 \cdot 1_{\mathbb F} \ne 0$.

Thus from Vector Product is Zero only if Factor is Zero, $\mathbf v = \mathbf 0$.

$\blacksquare$