Scalar product with -1 is additive inverse in vector spaces

Last updated Nov 1, 2022

Suppose $v \in V$. Then $-1 v = -v$.

$$\begin{align*} v + (-1) v &= 1 v + (-1) v\\ &=(1 - 1)v\\ &= 0 v\\ & \mathbf{0}, \end{align*}$$

where the last line follows from the property above. Because $V$ is an Abelian Group, we know Left and Right Inverses in an Abelian Group are the Unique Inverse, so $-1 v$ is the unique Additive Inverse of $v$. $\blacksquare$