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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
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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.
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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.
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