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Math Help - Annihilators and subspaces

  1. #1
    Member Mollier's Avatar
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    Annihilators and subspaces

    Hi,

    Problem:
    If M and N are subspaces of a finite-dimensional vector space, then (M \cap N)^0=M^0+N^0 and (M+N)^0=M^0 \cap N^0.
    ( M^0 and N^0 are the annihilators of M and N, respectively)

    Attempt:
    Not much I can do here, but I know that if M \subset N, then  N^0 \subset M^0.
    I also know that M^{00}=(=(M^0)^0)=M. I think the author wants me to use these two facts to prove the given statement.

    Any hints are greatly appreciated.

    Thanks.
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  2. #2
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    dont think that you need to use those facts that you mentioned, basicly what you have is two sets, so you need to show that one set is a subset of other and opposite.
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  3. #3
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    Quote Originally Posted by Mollier View Post
    Hi,

    Problem:
    If M and N are subspaces of a finite-dimensional vector space, then (M \cap N)^0=M^0+N^0 and (M+N)^0=M^0 \cap N^0.
    ( M^0 and N^0 are the annihilators of M and N, respectively)

    Attempt:
    Not much I can do here, but I know that if M \subset N, then  N^0 \subset M^0.
    I also know that M^{00}=(=(M^0)^0)=M. I think the author wants me to use these two facts to prove the given statement.

    Any hints are greatly appreciated.

    Thanks.

    Hints (easy): (a) U\subset W\Longrightarrow W^0\subset U^0 , (b) A\subset A^{00}\,,\,\,\forall subset A\subset V , (c) U=U^{00} if U is a subspace of  V , and thus we get:

    (1) M^0+N^0\subset (M\cap N)^0 (why?);

    (2) (M+N)^0\subset M^0\cap N^0 (why?)

    Take now the annihilator in both sides in (1) and use (a) and (c) above:

    (3) M\cap N=(M\cap N)^{00}\subset (M^0+N^0)^0 , and now use (2) above (with M^0\,,\,N^0 instead of M\,,\,N), and you get

    (4)  (M^0+N^0)^0\subset M^{00}\cap N^{00}= M\cap N , so from (3)-(4) you get

    (5) M\cap N\subset (M^0+N^0)^0\subset M\cap N , and this means the inclusions (3)-(4) must be equalities ; but this means that the annihilators of both sides in (1) are equal , and thus we get the wanted equality because of the

    Hint (easy, too) (d) If U\,,\,W subspaces of V\,,\,\,s.t.\,\,\,U\neq W , then U^0\neq W^0

    Try now to complete the proof of the second equality by yourself.

    Tonio

    Pd. I don't know any proof of these equalities that is less messy than the above, which is not hard but requires concentration in details and lots of careful "connecting the dots"
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  4. #4
    Member Mollier's Avatar
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    Hi, apologies for the late reply. The last few days have been a mess so no time for math

    Quote Originally Posted by tonio View Post
    (1) M^0+N^0\subset (M\cap N)^0 (why?);

    (2) (M+N)^0\subset M^0\cap N^0 (why?)
    The only thing I can come up with is that since M \cap N \subset M+N , then (M+N)^0 \subset (M \cap N)^0, but that doesn't really prove either (1) nor (2)..

    Quote Originally Posted by tonio View Post
    Try now to complete the proof of the second equality by yourself.
    I figure that I need to show that M^0 \cap N^0 \subset (M^0+N^0). If I assume that (2) is true, I should be able to take the annihilator of both sides of (2) and get to the wanted result.
     ((M+N)^0)^0 \subset (M^0 \cap N^0)^0
    Then by (a) I have that
    (M^0 \cap N^0) \subset (M+N)^0

    Ah, I find this to be really hard!

    Thanks!
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    Quote Originally Posted by Mollier View Post
    Hi, apologies for the late reply. The last few days have been a mess so no time for math


    The only thing I can come up with is that since M \cap N \subset M+N , then (M+N)^0 \subset (M \cap N)^0, but that doesn't really prove either (1) nor (2)..


    Indeed, that doesn't...but M\cap N\subset M\Longrightarrow M^0\subset (M\cap N)^0\,,\,\,and\,\,\,also\,\,\,M\cap N\subset N\Longrightarrow N^0\subset (M\cap N)^0 and now you get what you wanted...


    I figure that I need to show that M^0 \cap N^0 \subset (M^0+N^0). If I assume that (2) is true, I should be able to take the annihilator of both sides of (2) and get to the wanted result.
     ((M+N)^0)^0 \subset (M^0 \cap N^0)^0
    Then by (a) I have that
    (M^0 \cap N^0) \subset (M+N)^0

    Ah, I find this to be really hard!


    Yes, it really requires concentration. Try again, since the last days you did no much math (or at all).

    Tonio

    Thanks!
    .
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