I started a proof by contradiction and would like to know if I'm on the right track, and get some direction please. I know there are solutions out there for this but I would like to try finishing my proof if possible.
Exercise: Prove that the zero vector is unique. This means that if is any other vector in with the property that for any , then .
By contradiction, assume that the zero vector equals some other vector .
Then by the definition of the zero vector, this means that
Since , then for some vector , we can write
(2) , therefore
(3) . By substituting equation (3) into equation (1) we get,
(4) , and thus
We have reached a contradiction and therefore proven that the zero vector is unique. QED