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Linearly Dependence

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

    If u1 and u2, u2 and u3, u1 and u3 are Linearly Independent, does it follow that {u1,u2,u3} is Linearly Independent?
     
  2. jcsd
  3. Mar 1, 2009 #2

    Defennder

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    It helps if you write it out as follows:
    [tex]a_1 u_1 + a_2 u_2 + a_3 u_3 = \textbf{0}[/tex].

    Suppose one of the ai's is non-zero. Can you derive a contradiction with what you are given? Then, next suppose 2 of the coefficients are non-zero. Apply the same consideration.
     
  4. Mar 1, 2009 #3
    No. Try to find a counterexample (this is possible in R^2).
     
  5. Mar 1, 2009 #4

    lurflurf

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    Try this first
    if u1 and u2 are linearly dependent does it follow that
    v1 and v2 are linearly independent where
    v1=a*u1+b*u2
    v2=c*u1+d*u2

    or

    if span(V)=n
    does that mean any n vectors are linearly independent?
     
  6. Mar 1, 2009 #5

    Defennder

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    Oops, can't believe I missed such a simple counter-example.
     
  7. Mar 2, 2009 #6

    I could not find a counter example. I think it is LI.
     
  8. Mar 2, 2009 #7

    Mark44

    Staff: Mentor

    In R^2 there are zillions of counterexamples where v1, v2, and v3, are pairwise linearly independent. If you can't find any, you aren't looking very hard.
     
  9. Mar 2, 2009 #8

    Defennder

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    You can try thinking about the orthogonal standard basis vectors.
     
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