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What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties.

Perhaps this question could be put more generally as: what's the difference between a vector space and a field?

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3 Answers 3

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In a field you can multiply any two elements, as well as add them, while in a vector space you can add vectors, but you can only multiply by scalars; you can't multiply two vectors. (Multiplication in a field has to satisfy some axioms, but I think this is the essential point for your question.)

A field containing $\mathbb R$, or more generally a system of hypercomplex numbers containing $\mathbb R$, will in particular be a vector space (since you can add, and you can multiply by elements of $\mathbb R$, i.e. by scalars), but it will have extra structure. (E.g. you can multiply quaternions, and this is extra structure which is not at all obvious if you just know about $4$-vectors.)

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They are separate concepts, it's just that hypercomplex numbers can be used to model the behavior of vectors in $\mathbb{R}^n$.

A vector is an element of a vector space. Vector spaces are (up to isomorphism) direct sums of the field $k$, in that a vector is an ordered $n$-tuple $(a_1,\ldots,a_n)$ with each $a_i \in k$ which can be multiplied by an element $b \in k$, which is called a scalar, as follows: $k*(a_1,\ldots,a_n) = (k*a_1,\ldots,k*a_n)$. In the case of real vectors, this field is $\mathbb{R}$.

Hypercomplex numbers, such as the quaternions, are different in that they are elements of certain algebras over the real numbers. Algebras are defined similarly to vector spaces, only they also allow you to multiply elements of the algebra by other elements of the algebra.

Essentially, vectors and hypercomplex numbers are different concepts and cannot be used interchangeably. However, they are similar in that often the same problem can be solved using either.

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You may mean vector space where you say vector field . –  Myself Mar 16 '11 at 10:54
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I cannot believe I did that. Late night. –  Alex Becker Mar 16 '11 at 14:12

A hypercomplex number is an element of a certain kind of distributive unital algebra over $\mathbb{R}$. An algebra over $\mathbb{R}$ is, by definition, a vector space over $\mathbb{R}$ with additional structure so that you can multiply elements together. It's not very hard to turn a given finite-dimensional vector space into an algebra of hypercomplex numbers, but the question is whether you can do so in a natural, useful way.

Of course, there are vector spaces over any field at all, so it's also easy to come up with a vector space which cannot be turned into an algebra of hypercomplex numbers. For instance, a finite dimensional vector space over a finite field has only finitely many elements, so obviously cannot be turned into an algebra over $\mathbb{R}$.

The interesting thing is that there are objects which are simultaneously all of these things, and in a recursive way: for example, $\mathbb{C}$ is both a field and an algebra over $\mathbb{R}$, and a fortiori a vector space over $\mathbb{R}$. But $\mathbb{R}$ is in turn both a field and an algebra over $\mathbb{Q}$... The study of fields which are algebras over other fields is called Galois theory, which also finds applications in algebraic geometry (though probably not in the way you expect!).

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Great answer – thanks for the extra insights! –  msutherl Mar 16 '11 at 15:54

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