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Math Help - basis of a vector space

  1. #1
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    basis of a vector space

    Is the basis of a vector space unique? Justify your answer.

    I know that the answer is 'No'. I can justify the answer by providing two different basis of the same vector space. Is there any other way to defend the answer by giving logic and arguments.
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    Re: basis of a vector space

    Quote Originally Posted by Suvadip View Post
    Is the basis of a vector space unique? Justify your answer.

    I know that the answer is 'No'.
    The correct answer depends on the vector space.
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    Re: basis of a vector space

    I think if we change the question to

    Is it possible to have more than one basis of a vector space ? Justify answer

    Then the correct answer will be unique.
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    Re: basis of a vector space

    Why don't you formulate your statement precisely, using quantifiers "for all" and "there exists"? I would say that the answer to "Is it possible to have more than one basis of a vector space?" also depends on the vector space.
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    Re: basis of a vector space

    Quote Originally Posted by Suvadip View Post
    Is the basis of a vector space unique? Justify your answer.

    I know that the answer is 'No'. I can justify the answer by providing two different basis of the same vector space. Is there any other way to defend the answer by giving logic and arguments.
    without specifying that your basis vectors are unit (or some other) length you can always choose scalar multiple of a given basis and clearly this will also be a basis.
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    Re: basis of a vector space

    Is there exist any vector space whose basis is not unique? Justify
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    Re: basis of a vector space

    Quote Originally Posted by romsek View Post
    without specifying that your basis vectors are unit (or some other) length you can always choose scalar multiple of a given basis and clearly this will also be a basis.
    And yet any scalar multiple of a certain vector is equal to that original vector. That is, if there is a vector in the basis at all.

    Quote Originally Posted by Suvadip View Post
    Is Does there exist any vector space whose basis is not unique? Justify
    Yes, the basis of any nontrivial vector space is not unique, as romsek said.
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