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March 19th, 2011, 07:56 PM   #1
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About Vector spaces, subspace :).

In the book "Lectures in algebra abstract" by Jacobson I came across the following:

Other examples of vector spaces can be obtained as subspaces
of the spaces defined thus far. Let be any vector space over
and let be a subset of that is a subgroup and that is closed
under multiplication by elements of . By this we mean that if
and a is arbitrary in then in. Then it is clear that
the trio consisting of , and the multiplication is a vector
space; it is obvious that they hold
also in the subset .

i donīt know because , since nothing tells me that the sum of these elements is in :S.

Thanks in andvace
eraldcoil is offline  
March 19th, 2011, 10:20 PM   #2
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Re: About Vector spaces, subspace :).

Well, we know S is a subgroup. What else could you want? Subgroups are closed under the group operation, and here we have that it is closed under multiplication over K.
DLowry is offline  
March 20th, 2011, 09:34 PM   #3
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Re: About Vector spaces, subspace :).

In case it is not clear, the group operation of a vector space IS its addition. So for S to be a subgroup is exactly for it to be closed under addition.
Turgul is offline  

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