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In other words, can a vector space exist, and not have a coordinate system? I'm asking because in the definitions that I've seen of a vector space, there's no mention of a coordinate system.

set5
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  • Coordinates are not inherent to vector spaces. That's why we can change coordinates whenever we feel like -- they're just something extra we add to the vector space to make them easier to work with and to get actual numbers out of the manipulations we do on vectors. –  Oct 18 '15 at 15:18
  • The set of continuous functions $\Bbb R\to\Bbb R$ form a vector space. Don't even think you can find coordinates – Hagen von Eitzen Oct 18 '15 at 15:22
  • the components of a vector are its coordinates – janmarqz Oct 18 '15 at 15:22
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    @janmarqz How do you define vector components without also specifying a basis (which determines a coordinate system in the finite-dimensional case)? – amd Oct 18 '15 at 17:13
  • @amd. To have a vector space you need a basis. Any other vector is a linear combination of basis vectors. The scalars used in the linear combination are unique, then you can take them as the coordinates of the vector. – janmarqz Oct 18 '15 at 17:19
  • @janmarqz One doesn’t need a basis to have a vector space. That a basis can be found is an important theorem, but bases aren’t a part of the definition of a vector space. – amd Oct 18 '15 at 17:31
  • Ok,@amd... Having a vector space (finite dimensional at least) you got a basis. Any other vector is a linear combination of basis vectors. The scalars used in the linear combination are unique, then you can take them as the coordinates of the vector – janmarqz Oct 18 '15 at 17:37

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For any finite dimensional vector space $V$, there exists a coordinate system (it is not unique). Indeed, let $ n := \dim V$. Then you can find a linearly independent set $\mathcal{B} = \{u_1, \dots, u_n\}$ of $n$ vectors that will generate $V$, and $\mathcal{B}$ can be taken as a coordinate system if your order its elements.

However, the vector space structure doesn't include one particular specified coordinate system. In other words, to define a finite dimensional vector space, you don't need to specify a coordinate system, but you can always find one if you want.

In the general case of (maybe infinite) vector space, the existence of a coordinate system is not a requirement, and there are examples of vector spaces with no coordinate system, as @Hagen von Eitzen pointed out in a comment.

Olivier
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