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While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commutativity of addition, etc.). Is there any widely used vector space in which alternative functions are used as addition/multiplication?

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Feel free to ask/edit my question since my wording is a bit vague. – F M Sep 15 '10 at 2:21
up vote 5 down vote accepted

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.
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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, ...

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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:

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."

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Many of the operations commonly encountered in vector spaces, such as modular arithmetic, functional convolution, or even p-adic arithmetic, ultimately are related to the addition and multiplication with which you are familiar. One interesting exception is Conway's field of nimbers (which is effectively a one-dimensional vector space). I wouldn't say it's "widely used," but it's well enough known to qualify as a genuine example.

References: (nimbers form a subfield of the surreal numbers).

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Nimbers and Surreals are related off course.But I don't think that nimbers form a subfield of the surreal numbers. For every nimber x, x+x=0, while this is not true for any surreal other than 0 itself. – ypercubeᵀᴹ Feb 23 '11 at 1:47
@ypercube Are you claiming, then, that nimbers do not form a field? – whuber Feb 23 '11 at 14:55
No, i'm saying that both nimbers and surreals are Fields (and not fields as they are proper classes and not sets). But the Field of nimbers (called No2 by Conway) has characteristic 2, while the Field of surreals (No) does not. – ypercubeᵀᴹ Feb 24 '11 at 12:18
There is however a way to to see Nimbers (No2) as corresponding 1-1 to Ordinals (On). But then the operations (addition and multiplication) that make it a Field are not the usual addition and multiplication of ordinals. – ypercubeᵀᴹ Feb 24 '11 at 12:23
Nimbers emerge as a class of games, as you said the impartial ones (those where every left member is a nimber and also a right member and vice-versa). So do surreals, as the class of games where every left member is not >= than every right member and all (left and right) members are surreals. But the only surreal that is also a nimber is zero. – ypercubeᵀᴹ Feb 25 '11 at 16:37

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