# Vector Space has Unique Additive Inverses

## Theorem

A vector space has unique additive inverses for every element.

## Proof 1

Suppose $y$ and $z$ are both additive identities for some vector space $V$ of which $x$ is an element. Then $y=y+0$ because $0$ is an additive identity. Now $y+0=y+(x+z)=(y+x)+z=0+z=z$ follows from the definition of an inverse and the associativity of vector spaces. Hence $y=z$. Thus each element $x \in V$ has a unique additive inverse. $\blacksquare$

## Sources

- 2015: Sheldon Axler:
*Linear Algebra Done Right*(3rd ed.): p. 15