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: That depends on what you mean by the zero vector. 
If you want, you can consider $\mathbb{R}_+$ (the set of positive real numbers) as a vector space, where you define $x\oplus y = xy$ and $\lambda x = x^\lambda$. Then the "additive identity" is actually $1$ (but should probably be called the zero vector in this strange context).
A: 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. 
A: The difficulty is not to confuse the abstract notion of a vector space with concrete vector spaces. What do I mean  by this?
The notion of an abstract vector space
Well, if $0_\mathbb{V} \in \mathbb{V}$ is a distinguished element of a non-empty set $\mathbb{V}$, $\mathbb{K}$ a field, and $\oplus : \mathbb{V} \times \mathbb{V} \to \mathbb{V}$ and $\otimes : \mathbb{K} \times \mathbb{V} \to \mathbb{V}$ two operation such that $0_\mathbb{V}$, $\oplus$, and $\otimes$ satify the vector space axioms; then we can call the algebraic structure
$$
(\mathbb{V}, \oplus, \odot, 0_\mathbb{V})
$$
an abstract $\mathbb{K}$-vector space. It is abstract in the sense that $\mathbb{V}$, $\mathbb{K}$, $\oplus$, $\odot$, and $0_\mathbb{V}$ are placeholders for concrete mathematical objects. Nonetheless, we want to be able to talk to others about the different placeholders, which is why we agree upon calling the symbol $\oplus$ vector addition, the symbol $\odot$ scalar multiplication, and the symbol $0_\mathbb{V}$ additive identity.
Supplement:
To show how important it is to keep the different notations apart, consider the linear combination
$$
v = k_1 \odot v_1 \oplus k_2 \odot v_2 \oplus \ldots \oplus k_n \odot v_n = \bigoplus\limits_{i=1}^{n} k_i \odot v_i.
$$
Then we call the symbol $\bigoplus$ vector summation.
A concrete vector space 1
A standard example of a concrete $\mathbb{R}$-vector space is $(\mathbb{R}, +, \cdot, 0)$. In this example, vector addition, scalar multiplication, and additive identity actually coincide with usual addition, usual multiplication, and the number zero, respectively. A linear combination of vectors $v_1,v_2,\ldots,v_n \in \mathbb{R}$ is
$$
v = \sum\limits_{i=1}^{n} k_i \cdot v_i,
$$
i.e., the vector summation coincides with the usual sum.
A concrete vector space 2
Another example was given by @mrf: If you consider the concrete $\mathbb{R}$-vector space $(\mathbb{R}_{>0}, \cdot, \hat{},1)$, where the symbol $\hat{}$ shall denote usual exponentiation, then vector addition, scalar multiplication, and additive identity correspond to usual multiplication, usual exponentiation, and the number one, respectively. A linear combination of vectors $v_1,v_2,\ldots,v_n \in \mathbb{R}_{>0}$ is
$$
v = \prod\limits_{i=1}^{n} (v_i)^{k_i},
$$
i.e., the vector summation coincides with the usual product.
