My Math Forum About Vector spaces, subspace :).

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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 $V$ be any vector space over $K$ and let $S$ be a subset of $V$ that is a subgroup and that is closed under multiplication by elements of $A$. By this we mean that if $S$ and a is arbitrary in $K$ then $ax$ in$S$. Then it is clear that the trio consisting of $S$, $K$ and the multiplication $ax$ is a vector space; it is obvious that they hold also in the subset $S$. i don´t know because $ax+by$$\in$ $S$ , since nothing tells me that the sum of these elements is in $S$ :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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