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Span and vector space

  1. Mar 4, 2017 #1
    1. The problem statement, all variables and given/known data

    The question is:
    if vectors v1, v2, v3 belong to a vector space V does it follow that:

    span (v1, v2, v3) = V

    span (v1, v2, v3) is a subset of V.

    2. The attempt at a solution:

    If I understand it correctly the answer to both questions is yes.
    The first: the linear combinations of these three vectors fill the space V.

    Am I correct? I would like to make sure if I have understood the definitions.
  2. jcsd
  3. Mar 4, 2017 #2


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    Why must the span of three vectors be the full vector space? If so, then why not the span of just two vectors?
  4. Mar 4, 2017 #3
    Yes, I know, they should be linearly independent to be a spanning set. Well, we don't know if they are independent. :( I guess there may be also other vectors needed for a spanning set for the vector space V: we don't know this either. In this case the answer to the first question would be negative. But the second would hold. Have I got it?

    I have read the definitions so many times that I am somewhat dizzy.
  5. Mar 4, 2017 #4


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    There are two reasons why the span of ##v_1, v_2, v_3## may not be all of ##V##. If ##V## is 3-dimensional, the vectors may be linearly dependent. And, ##V## may be more than 3-dimensional in the first place.
  6. Mar 4, 2017 #5
    Thank you very much. Everything is clear to me now. :)
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