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Span Problem

  1. Mar 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose that u, v, w and x are vectors in a vector space V . If x !∈ span(v,w) and x ∈ span(u, v,w),
    does it follow that u ∈ span(x, v,w)? Justify your answer.

    2. Relevant equations



    3. The attempt at a solution
    Not really sure how to even start this problem. Any hints would be good thanks!
     
  2. jcsd
  3. Mar 30, 2009 #2

    lanedance

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    Homework Helper

    Hi misterau

    I would start with the defintion of a span, or how any vector in a given span can be written...
     
  4. Mar 30, 2009 #3
    I really not sure, probably not correct...
    x != c1*v + c2*w
    x = c1*v + c2*w + c3*u
    u = c1*v + c2*w + c4*x

    c1*v + c2*w != c1*v + c2*w + c3*u
    0 != c3*(c1*v + c2*w + c4*x)
    0 != c1*v + c2*w + c4*x
     
  5. Mar 30, 2009 #4
    First hint:
    Given:
    x != c1*v + c2*w
    x = c1*v + c2*w + c3*u

    you know that c3 != 0.

    Second hint:
    Starting from this, can you find a more explicit expression for u in terms of x,v,w and c1,c2 and c3?
     
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