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Math Help - Subfields and Vector Spaces

  1. #1
    Senior Member I-Think's Avatar
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    Subfields and Vector Spaces

    Consider a vector space V over a field F. Let K be a subfield of F.
    Prove that V over K is a vector space if we retain the same addition operation and restrict scalar multiplication to scalars from K

    Work so far
    I figure if we just prove that there exists the 0 element, and that addition and scalar multiplication are closed, the statement.

    I know there exists a zero element, as 0x=0 for x\in{V} and 0\in{F}, and this 0 is also in K

    Under scalar multiplication, all scalars from K are also in F, so if it was closed under F it will also be closed under K

    But how do I resolve addition if the same operation is retained.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    Consider a vector space V over a field F. Let K be a subfield of F.
    Prove that V over K is a vector space if we retain the same addition operation and restrict scalar multiplication to scalars from K

    Work so far
    I figure if we just prove that there exists the 0 element, and that addition and scalar multiplication are closed, the statement.

    I know there exists a zero element, as 0x=0 for x\in{V} and 0\in{F}, and this 0 is also in K

    Under scalar multiplication, all scalars from K are also in F, so if it was closed under F it will also be closed under K

    But how do I resolve addition if the same operation is retained.
    What do you mean? Since it's a field the sum of two field elements is in the field?
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  3. #3
    Senior Member I-Think's Avatar
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    Yes. If the same operation is retained as V over F, then the sums of elements of V over K should also be closed, shouldn't it?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    Yes. If the same operation is retained as V over F, then the sums of elements of V over K should also be closed, shouldn't it?
    Yes, I think I understand where the confusion is coming in. We have in general for a vector space \left(V,+,\mu,F\right) where \mu is scalar multiplication that \left(V,+\right) is an abelian group, this is independent of the other parts of the vector space. Does that help?
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