When is the additive identity not the zero vector? My teacher cryptically mentioned today that the zero vector is not always the additive identity. When asked for clarification I was told "we'll get there".
He did confirm it is always 0 in matrices filled with real numbers, but I can't think of or find any matrix, whether complex or variable or whatever where anything else would work, or where the zero vector wouldn't work.
It might be half a joke to keep me interested, but I'll be a minkeys uncle if it didn't work! 
I don't know, any ideas?

A vector space forms a group under addition:
1) If $a,b$ are vectors, $a+b$ is a vector.
2) $(a+b)+c = a+(b+c)$ for vectors $a,b,c$.
3) $a+0 = 0+a = a$ by definition of the zero vector $0$ (i.e. the zero vector is defined to be an additive identity as in the vector space axioms). 
4) $a+(-a) = (-a) + a = 0$ , i.e. additive inverses exist.
In a group, one can easily show an additive identity is unique (hence why we say "the" additive identity).
Thus, the zero vector is the only additive identity. 
