is the natural numbers can be a valid subspace over some field?

sharon

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if i choose the real numbers as vector space over SOME field, and i look on the subset of the natural numbers group including zero. so according to subspaces: 1) 0 included 2) closed under addition of vectors. so, is there a Field that can make the natural numbers as a valid subspaces? (field that multipication from natural numbers with scalar is closed) or it doesnt exist
 
if i choose the real numbers as vector space over SOME field, and i look on the subset of the natural numbers group including zero. so according to subspaces: 1) 0 included 2) closed under addition of vectors. so, is there a Field that can make the natural numbers as a valid subspaces? (field that multipication from natural numbers with scalar is closed) or it doesnt exist
You are incorrect about what must be true in order for a subset of a vector space to be a subset.
A subspace must be (1) closed under addition of vectors and (2) closed under scalar multiplication. Multiplying an odd natural number by 1/2 does NOT give a natural number so the set of natural numbers is not closed under scalar multiplication.

(And you do not need "0 included", only "is not empty". If v is any vector in the subset, then 0v= 0 so if a subset is non-empty and is "closed under scalar multiplication, 0 is necessarily included. Of course, if 0 is in the subset then it is non-empty so "not empty" and "includes 0" are equivalent.)
 
You are incorrect about what must be true in order for a subset of a vector space to be a subset.
A subspace must be (1) closed under addition of vectors and (2) closed under scalar multiplication. Multiplying an odd natural number by 1/2 does NOT give a natural number so the set of natural numbers is not closed under scalar multiplication.

(And you do not need "0 included", only "is not empty". If v is any vector in the subset, then 0v= 0 so if a subset is non-empty and is "closed under scalar multiplication, 0 is necessarily included. Of course, if 0 is in the subset then it is non-empty so "not empty" and "includes 0" are equivalent.)

i know that i'm wrong i just dont understand why:
my question is if there can be a field that has only positive scalars
like modulo3 (0 1 2) for example and then all multipication from natural numbers with that field will be closed
thank you for your time
 
i know that i'm wrong i just dont understand why:
my question is if there can be a field that has only positive scalars
like modulo3 (0 1 2) for example and then all multipication from natural numbers with that field will be closed
thank you for your time

In its normal sense, "positive" is a meaningless term in modular arithmetic and in non-real fields. is [4] positive modulo 3? then so must be [-2]. Is 4-2i positive in C?

You may want to look up modules over rings. It is a generalization of vector spaces over fields that meet some of your requirements. Note that N is not a ring, however
 
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