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Linear algebra: spanning

  1. Oct 25, 2012 #1
    1. The problem statement, all variables and given/known data

    The problem is : Let S = [ (1,-1,3) , (-1,3, -7) , (2,1,0) ]. Do the vectors u = (5,1,3) and v = (2,3,6) belong to span(S)


    2. Relevant equations



    3. The attempt at a solution

    So span means that I could take linear combinations of u and v and they should end up giving (1,-1,3) , (-1,3,-7) and (2,1,0). Right?

    I could take x [5 1 3 ] + y [ 2 3 6 ] = [1 -1 3] or [-1 3 -7 ] or [2 1 0 ] (btw i meant to write [ 5 1 3] as a column matrix but I am not sure of how to using Latex. So anyway, is what I am trying to do correct? Is that what it means for the vectors to span S?

    Thanks
     
  2. jcsd
  3. Oct 25, 2012 #2

    Mark44

    Staff: Mentor

    No, it's the other way around.

    Span(S) is the set of all linear combinations of the vectors in S. u is in Span(S) if there are constants a, b, and c for which a(1, -1, 3) + b(-1, 3, -7) + c(2, 1, 0) = u.

    Similarly for v.
     
  4. Oct 25, 2012 #3

    Zondrina

    User Avatar
    Homework Helper

    Put your vectors from S into matrix form, augmenting them with either u or v ( You'll have to do both at some point so pick one at a time ).

    Solve the corresponding system and check if the following system is linearly independent or dependent.

    If the system is dependent for your choice of u or v, then you can conclude that the vector is not in the span of your set. Otherwise if your system is independent, you can exhibit a unique solution for your system implying that your vector IS in the span of your set.
     
  5. Oct 25, 2012 #4
    Ahh I see thanks

    Yes, it works out...thanks
     
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