Suppose

is a vector space over a field

. Prove that the additive inverse of a vector

is unique.

A hint provided for the question is as follows:

Suppose

are additive inverses of

. Prove that

using the axioms of a vector space.

My work so far (uses the hint):

Suppose

such that

are inverses of

.

Then

.

I'm pretty sure that my current answer is lacking substance, so I could use a hand in making it more solid.

The main issue I'm having is some of the wording. I feel as though there's something I could be missing in the question. The part on "axioms of a vector space" is one I don't entirely understand.