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Span of an infinite set. Exam Question.

  1. Dec 31, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose V=<Z> where Z is an infinte set, So Z spans V.
    Suppose that every pair of distinct elements of Z is linearly independent.
    Is it possible that V is finite dimensional? Justify your answer.

    2. Relevant equations

    3. The attempt at a solution
    All the vectors must be linearly independent if any two are so...
    I dont think you can as a basis for the space would have infinte linaerly independent vectors.
  2. jcsd
  3. Dec 31, 2008 #2


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    All of the vectors don't have to be linearly independent just because every pair is. Let Z be the set of all unit vectors in the first quadrant in the two dimensional plane. Since they all point in different directions, any pair is linearly independent.
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