Question

Let u be an arbitrary vector from vector space V

Prove that -u = (-1)u

Answer 1

List of axioms:
Let u, v be vectors
k,m be scalars

1. If u and v are in V, then u+v is in V
2. u + v = v + u
3. u + (v + w) = (u+v) + w
4. 0 + u = u + 0 = u
5. u + (-u) = (-u) +u = 0
6. If k is any scalar and u is in V, the ku is in V
7. k(u+v) = ku+kv
8.(k+m)u = ku + mu
9. k(mu) = (km)u
10. 1*u = u

Answer 2

Axiom 5 is the only one that has a (-u) in it, but I'm not sure how to derive (-1)u=-u from it.

Answer 3

What do you think about 6?

Answer 4

I could use 6 to show that (-1)u is in V.
But I can't say that (-1)u=-u
(like seriously what the heck math makes no sense at this point)