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Math Help - can you help me?

  1. #1
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    subset R^w is a subspace?

    I have trouble to prove that

    Is the following subset R^w(the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for R^n) a subspace?
    D={a R^w | the entries a i are limited to finitely many values}

    I appreciate to help...thank you...
    Last edited by iloveyou7; October 23rd 2009 at 03:47 PM.
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  2. #2
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    Quote Originally Posted by iloveyou7 View Post
    I have trouble to prove that

    Is the following subset R^w(the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for R^n) a subspace?
    D={a R^w | the entries a i are limited to finitely many values}

    I appreciate to help...thank you...

    How many possible scalars can multiply an element in D?

    Tonio
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  3. #3
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    Quote Originally Posted by iloveyou7 View Post
    I have trouble to prove that

    Is the following subset R^w(the vector space of real-valued sequences, with vector addition and scalar multiplication defined as for R^n) a subspace?
    D={a R^w | the entries a i are limited to finitely many values}

    I appreciate to help...thank you...
    I would interpret "the entries a_i are limited to finitely many values" to mean "there are only a finite number of real numbers that you can have as entries" and I think that is how Tonio interpreted it.

    But I suspect you mean "only a finite number of the entries a_i are non-zero" which is a very different thing. The former is NOT a subspace (for the reason Tonio suggested) and the latter is.

    Which do you mean?
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  4. #4
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    A sequence a will belong to GAMMA iff there is a set of values M=
    {r1...rk}, such that ai must come from M
    Thus <1,2,2,1,2,2,1,...> would belong to GAMMA since all of its
    entries come from the finite set {1,2}. NOTE: This does not mean that
    members of GAMMA need to have a repeating pattern. It simply means
    that the set of numbers which occur in the sequence is finite.
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  5. #5
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    Quote Originally Posted by iloveyou7 View Post
    A sequence a will belong to GAMMA iff there is a set of values M=
    {r1...rk}, such that ai must come from M
    Thus <1,2,2,1,2,2,1,...> would belong to GAMMA since all of its
    entries come from the finite set {1,2}. NOTE: This does not mean that
    members of GAMMA need to have a repeating pattern. It simply means
    that the set of numbers which occur in the sequence is finite.

    Ok, and this is what I understood from your question and thus my hint answers it.

    Tonio
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  6. #6
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    But i don't know how to prove it...
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  7. #7
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    Quote Originally Posted by iloveyou7 View Post
    But i don't know how to prove it...

    Take your example with all the sequences whose entries can be only 1 or 2. If this is a subspace, then it is closed under multiplication by scalar. Take, for example, the sequence  \left\{1,2,1,2,1,2....\right\} and multiply it by the scalar -1: what happens?

    Tonio
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