What are some alternative definitions of vector addition and scalar multiplication?

My favorite example is the set of subsets of a set under the operation of symmetric difference (otherwise known as bitwise XOR). This forms a vector space over the finite field $\mathbb{F}_2$. This example is important in computer science, coding theory, combinatorics, ...

This example is certainly not "widely used", but I think it's worth thinking about anyway. This answer comes from an MO post by John Goodrick: https://mathoverflow.net/questions/9402/pedagogical-question-about-linear-algebra

I'll quote the entirety of his post here in case you don't feel like clicking on the link (and since I've done no work on my own, I'll make this post CW)

You could try giving the following example: the set of all positive real numbers, considered as a vector space over the field R, with vector addition given by multiplication and scalar multiplication given by taking exponents.

As a first step, you could verify that this satisfies a few of the vector-space axioms, and then let students check the rest of them (say, as homework). Then, you could ask questions like, "what is the dimension of this vector space?" or, "give an example of a (nontrivial) linear transformation from this space into R^3."

What do you "normally" call addition and multiplication?

Just those operations with real numbers, or all kind of addition and multiplication "derived" from the well-known operations with real numbers, or that "look like" these operations?

Because, in the first case, you have plenty of elementary and widely used vector spaces with operations which are not those of real numbers:

1. $\mathbb{R}^2$, the set of ordered pairs of real numbers $(x,y)$, is a real vector space, with addition and multiplication defined as $(x,y) + (u,v) = (x+u, y+v)$ and $\lambda (x,y) = (\lambda x , \lambda y)$. These operations are defined using the "normal" addition and multiplication of real numbers, but are not the "normal" addition and multiplication of real numbers just because $(x,y)$ is not a real number.
2. ${\cal C}^0 (\mathbb{R}, \mathbb{R})$, the set of continuous functions $f:\mathbb{R} \longrightarrow \mathbb{R}$, is a real vector space, with addition and multiplication defined point-wise; that is $(f+g)(x) = f(x) + g(x)$ and $(\lambda f)(x) = \lambda f(x)$. Again, these addition and multiplication are defined using the "normal" addition and multiplication of real numbers, but are not the "normal" addition and multiplication of real numbers for the same reason.
3. $\mathbb{Z}/2\mathbb{Z}$, the set of integers mod 2, is a $\mathbb{Z}/2\mathbb{Z}$-vector space, with addition and multiplication $\widetilde{m} + \widetilde{n} = \widetilde{n+m}$ and $\widetilde{\lambda}\widetilde{m} = \widetilde{\lambda m}$, where $\widetilde{m}$ denotes the class of $m$ mod 2. Ditto.
4. $\mathbb{R}(x)$, the field of rational functions $\frac{p(x)}{q(x)}$, where $p(x), q(x) \in \mathbb{R}[x]$ are polynomials, $q(x) \neq 0$, is a $\mathbb{R}(x)$-vector space, with addition and multiplication $\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{p(x) s(x) + r(x) q(x)}{q(x)s(x)} $ and $\frac{p(x)}{q(x)} \frac{r(x)}{s(x)} = \frac{p(x)r(x)}{q(x)s(x)}$. Ditto.
5. $\mathbb{C}$, the set of complex numbers, is a $\mathbb{C}$-vector space, whit the addition and multiplication of complex numbers. Ditto.
6. $\mathbb{K}^n$, the set of ordered families $(x_1, \dots , x_n)$ of elements of any field $\mathbb{K}$, is a $\mathbb{K}$-vector space, with addition and multiplication defined as in example 1. Examples 3, 4 and 5 are particular cases of this one with $n=1$ and $\mathbb{K} =$ $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{R}(x)$ and $\mathbb{C}$, respectively. Example 1 is also a particular case, with $n=2$ and $\mathbb{K} = \mathbb{R}$. Addition and multiplication in $\mathbb{K}$ may have nothing in common with the operations with real numbers.