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Math Help - Finite dimensional vector space

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    Finite dimensional vector space

    Def: A finite dimensional vector space has a finite number of base vectors.

    What is a vector space without any base vectors?
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    Re: Finite dimensional vector space

    Quote Originally Posted by Hartlw View Post
    Def: A finite dimensional vector space has a finite number of base vectors.

    What is a vector space without any base vectors?
    Depending on the definitions of your book. The Empty Set spans the space \mathbf{0}. The vectorspace with only the zero vector in it.

    PlanetMath: span
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    Re: Finite dimensional vector space

    Quote Originally Posted by TheEmptySet View Post
    Depending on the definitions of your book. The Empty Set spans the space \mathbf{0}. The vectorspace with only the zero vector in it.

    PlanetMath: span
    The vector space with only the 0 vector has one base vector.

    EDIT: If you have a set which contains only the empty set and you define a scalar times an empty set as an empty set, then the empty set is a base vector of the set and the set has ONE base vector.

    On the other hand, if you claim the empty space is a vector space, you have the problem that it doesn't have any elements, which actually raises an interesting question:

    Let A be all sets such that a proposition P(A) is true for all members of A. Is the proposition true for the empty set (every member of the set satisfies the proposition because it has no members).
    Last edited by Hartlw; October 17th 2011 at 03:28 PM.
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    Re: Finite dimensional vector space

    Quote Originally Posted by Hartlw View Post
    The vector space with only the 0 vector has one base vector.

    EDIT: If you have a set which contains only the empty set and you define a scalar times an empty set as an empty set, then the empty set is a base vector of the set and the set has ONE base vector.

    On the other hand, if you claim the empty space is a vector space, you have the problem that it doesn't have any elements, which actually raises an interesting question:

    Let A be all sets such that a proposition P(A) is true for all members of A. Is the proposition true for the empty set (every member of the set satisfies the proposition because it has no members).
    OK. An empty set is a vector space because each one of its members (it has no members) satisfies the definition of a vector space, and it has no base vectors. But, please.
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    Re: Finite dimensional vector space

    Frrom Wolfram Mathworld: "Finite set: A set whose elements can be numbered through from 1 to , for some positive integer ." Which is same as def from Birkhoff-McLane.

    HOWEVER Kelley (Topology) gives a definition of a finite set which is unintelligible unless you have the time and interest to learn the language, which you can do in one of two ways depending on how your brain works:

    1) Memorize words and symbols, their parts of speech, and the rules of the "grammar." In this way you will always put together a logical sentence, which others will recognize as logical. You could accidentally put together a sentence like "the dog is a cat," or the most profound thought in the history of the world.

    2) Same as above but you also try for some meaning to the words.

    I don't have the brain for 1) and some trouble with 2), which is why I look for the minimum number of meaningful concepts with the greatest relevance.

    A link to a page written in a language I don't know doesn't help me much.

    EDIT If the elements are books, and a bag has a finite number of books, a book has to exist in order to create a one-one correspondence with it. Is a book with the characteristic that it doesn't exist a book?
    Last edited by Hartlw; October 18th 2011 at 11:31 AM.
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