1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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
    Thanks
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted