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Math Help - Vector Subspaces

  1. #16
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    Re: Vector Subspaces

    Quote Originally Posted by Plato View Post
    That is exactly what I am saying.
    Zero is a finite number.
    You are confusing the difference between I have two books and I have 2 books.

    Two books: "War and Peace," "Uncle Vanya."
    2 books: 2, 2, 2, 2, 2,
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  2. #17
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    Re: Vector Subspaces

    A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.
    Last edited by Hartlw; October 15th 2011 at 11:03 AM.
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  3. #18
    MHF Contributor FernandoRevilla's Avatar
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    Re: Vector Subspaces

    Quote Originally Posted by Hartlw View Post
    A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.
    Imprecisions of usual language. What about the following?:

    Definition We say that the sequence (x_n)_{n\geq 0} has finitely many non zero terms iff there exists p\in\mathbb{N} such that x_n=0 for all n\geq p .

    This is just what all mathematicians interpret with the expression the sequence (x_n)_{n\geq 0} has finitely many non zero terms.

    Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.
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  4. #19
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    Re: Vector Subspaces

    Quote Originally Posted by FernandoRevilla View Post
    Imprecisions of usual language. What about the following?:

    Definition We say that the sequence (x_n)_{n\geq 0} has finitely many non zero terms iff there exists p\in\mathbb{N} such that x_n=0 for all n\geq p .

    This is just what all mathematicians interpret with the expression the sequence (x_n)_{n\geq 0} has finitely many non zero terms.

    Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.

    I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

    I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

    I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

    So W is a subspace because it contains only 0.

    * a zero number of non-zero terms. zero is a finite number. Reference Plato
    Last edited by Hartlw; October 15th 2011 at 12:56 PM.
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  5. #20
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    Re: Vector Subspaces

    Quote Originally Posted by Hartlw View Post
    I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

    I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

    I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

    So W is a subspace because it contains only 0.

    * a zero number of non-zero terms. zero is a finite number. Reference Plato
    Actually, if finite means nore than a zero number of terms, the answer is W is not a subspace because it doesn't contain 0.
    If "finite number" is restricted to "zero number," W is a subspace because it only contains 0
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  6. #21
    Grand Panjandrum
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    Re: Vector Subspaces

    Quote Originally Posted by Hartlw View Post
    This is what I am saying: A sequence of zeros does not have any non-zero entries, so it cannot have finiteley many of them. 0 does not belong to W.

    If I have no books do I have a finite numbr of them?

    What are you saying?
    You won't have a chance to reply to this as I have closed the thread but consider your definition of a finite set, then check how a selection of text or reference books define a finite set.

    CB
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