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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?
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 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.
The difficulty is not to confuse the abstract notion of a vector space with concrete vector spaces. What do I mean by this?
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 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.
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.
