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It turns out that, if you restrict to spaces of finite dimension, then Euclidean space is the only vector space out there, up to your choice of labelling the ba
by j2kun 13y ago
It turns out that, if you restrict to spaces of finite dimension, then Euclidean space is the only vector space out there, up to your choice of labelling the basis elements and up to a certain linear transformation.
In this sense, vectors can always be written as lists of numbers, but no, they don't have to be this in general. For example, you could have a vector space of smooth functions. It's infinite dimensional, and its elements (its vectors) are functions, not tuples of numbers.
- hacman 13y agoYou must also add the condition that your scalars are real numbers. There are many finite dimensional vector spaces that are nothing like Euclidean space: any finite extension of a finite field, for example. This comes from Galois theory, but is not just abstract nonsense. One application to computing is if your scalars are only 0 or 1 and all operations are done mod 2, then you have a framework for doing error correcting codes, among other things. We call this set of scalars either the finite field of size 2 or the Galois field of size 2, aka GF(2).
- j2kun 13y agoYes, what I should have said is: if k is an algebraically closed field then n-dimensional vector spaces over k are isomorphic to k^n. Unless I've had too much to drink this evening.