The vector 0 is the only null vector?

Question

To prove the some set is vector space, I must prove 8 axioms. Two of then is the existence of the null vetor and the identity. May I have some vector diferent of 0 for the null vector, and some vector diferent of 1 for the identity?

Answer 1

Let's say you have two vectors $0_1$ and $0_2$, both fulfilling the axiom given in the definition of vector spaces (i.e. $0_1 + x = x + 0_1 = x$ and $0_2 + x = x + 0_2 = x$ for all vectors $x$ in the vector space). We would then have $$ 0_1 = 0_1+0_2 = 0_2 $$ so they are equal. In other words, no, you cannot have more than one null vector, becuase the very axiom that defines them also forces all of them to be the same.