Let v ∈ V and c ∈ ℂ, with c ≠ 0. Prove that if cv = 0, then v = 0.
Vector space axioms.
The Attempt at a Solution
Simple proof overall, but I have one major clarification question.
v = 1v
= c^(-1) (cv)
= c^(-1) 0
v = 0
My question is, in a complex vector space, is it safe to assume that c^(-1) exists in this proof?
If it does, I feel very confident about this proof.
If it doesn't then I need to do something else.
I don't see anywhere in the vector space axioms that states the multiplicative inverse exists.
But, it makes sense to me that an inverse does exist for any possible scalar choice here.