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I know that a vector space over field F satisfies the 8 axioms, but does a vector space satisfy this also?

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- Thread starter kman12
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- #1

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I know that a vector space over field F satisfies the 8 axioms, but does a vector space satisfy this also?

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- #3

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This doesn't answer my question, I want to know the difference between a vector space and a vector space over field F.

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Fredrik

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Right this would make sense. So that means the 8 axioms for a "Vector Space over a field F" also hold for a "Vector space".

Because i know that the basic

1) It contains a non empty set V whose elements are vectors

2) A field F whose elements are scalars

3) A binary operation + on V Under which V is closed

4) A multiplication . of a vector by a scalar.

So on top of this the 8 axioms (That hold for a vector space over a field F) also hold for a vector space (i cant be asked to write all axioms)?

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Fredrik

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(i) [itex](x+y)+z=x+(y+z)[/itex] for all [itex]x,y,z\in V[/itex]

...and so on. (You seem to know the rest).

Note that V is just a set. It's convenient to call V a vector space, but you should be aware that this is actually a bit sloppy. It's certainly OK to do it when it's clear from the context what field [itex]\mathbb F[/itex] and what addition and scalar multiplication functions we have in mind. For example, it's common to refer to "the vector space [itex]\mathbb R^2[/itex] " because everyone is familiar with the standard vector space structure on that set.

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I find it useful to think of vector spaces as special cases of modules over rings.

A module is just an abelian group together with a distributive multiplication by elements of a ring. If the ring is a field then the module is a vector space.

The distinguishing feature of a field is that is has multiplicative inverses.

Much of the theory of vector spaces actually comes from considering modules where the ring is a principal ideal domain. This is because the ring of polynomials over a field is a principal ideal domain.

A module is just an abelian group together with a distributive multiplication by elements of a ring. If the ring is a field then the module is a vector space.

The distinguishing feature of a field is that is has multiplicative inverses.

Much of the theory of vector spaces actually comes from considering modules where the ring is a principal ideal domain. This is because the ring of polynomials over a field is a principal ideal domain.

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right thanks fredrik

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