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: 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


    User Avatar
    Homework Helper

    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


    User Avatar
    Homework Helper

    Try this first
    if u1 and u2 are linearly dependent does it follow that
    v1 and v2 are linearly independent where


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


    User Avatar
    Homework Helper

    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


    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


    User Avatar
    Homework Helper

    You can try thinking about the orthogonal standard basis vectors.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook