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March 19th, 2011, 07:56 PM  #1 
Newbie Joined: Feb 2011 From: Chile Posts: 14 Thanks: 0  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 
March 19th, 2011, 10:20 PM  #2 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  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.

March 20th, 2011, 09:34 PM  #3 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 5  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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