Additive Inverse in Vector Space is Unique

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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

Proof of Existence

Follows from the vector space axioms.

$\Box$


Proof of Uniqueness

Let $\mathbf v$ have inverses $\mathbf x$ and $\mathbf y$.

Then:

\(\displaystyle \mathbf v + \mathbf x\) \(=\) \(\displaystyle \mathbf 0\)
\(\, \displaystyle \land \, \) \(\displaystyle \mathbf v + \mathbf y\) \(=\) \(\displaystyle \mathbf 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \mathbf v + \mathbf x\) \(=\) \(\displaystyle \mathbf v + \mathbf y\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \mathbf x\) \(=\) \(\displaystyle \mathbf y\) Vectors are Left Cancellable

$\blacksquare$


Also see


Sources