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Linear Dependence Definition

  1. Nov 8, 2016 #1
    1. The problem statement, all variables and given/known data

    True or False:

    If [itex]u[/itex], [itex]v[/itex], and [itex]w[/itex] are linearly dependent, then [itex]au+bv+cw=0[/itex] implies at least one of the coefficients [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] is not zero

    2. Relevant equations

    Definition of Linear Dependence:

    Vectors are linearly dependent if they are not linearly independent; that is there is an equation of the form [itex]c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}[/itex] with at least one coefficient not zero

    3. The attempt at a solution

    I said true, but the book says false. It gives the reason, "for any vectors [itex]u[/itex], [itex]v[/itex], [itex]w[/itex] - linearly dependent or not - [itex]0u+0v+0w = 0[/itex]" . But isn't the problem a direct restatement of the definition? Or am I missing something subtle here.
     
  2. jcsd
  3. Nov 8, 2016 #2

    PeroK

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    You're missing something subtle.
     
  4. Nov 8, 2016 #3

    haruspex

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    If u, v, w linearly independent, au+bv+cw=0 implies a=b=c=0.
    Inverting that, if u, v, w linearly dependent, au+bv+cw=0 does not imply a=b=c=0. But they still could be 0.
     
  5. Nov 8, 2016 #4
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