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Span{} and the zero vector

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution

    It's not so much a homework problem as it is something I was wondering. Our book is horrible, and does not explicitly state that the zero vector is always in the span of two vectors. If I am understanding things right:

    if v and u are vectors
    [tex]span(v, u)[/tex]
    is the collection of all points that can be reached via a linear combination of v and u. My reasoning is that if v is equal to u, then span{v,u} = span{v} = span{u}, which is essentially a line. However, it seems to me that in any space, R^2, R^3,...,R^n, the span{} of any n vectors will always go through the origin and thus, the zero vector will always be in that collection. Is that accurate?
     
  2. jcsd
  3. Mar 30, 2012 #2

    jgens

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    Yes. Notice that if [itex]S[/itex] is a set and [itex]v \in S[/itex], then [itex]0 \cdot v = 0 \in \mathrm{Span}(S)[/itex].
     
  4. Mar 30, 2012 #3
    Yup, since the weights of the linear combination can be zero. Thanks, I just wanted clarification due to the books grey area. I know it seems obvious, but you would have to read this book to understand the confusion :rofl:
     
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