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Homework Help: Linear Algebra linear independence

  1. Jan 25, 2010 #1
    If u,v andw are three linearly independent vectors of some vectorial space V, show that u + v , u-v and u -2v + w are also linearly independent.


    Okay, first of all, i know that:
    [tex]\lambda_{1} \times u + \lambda_{2} \times v + \lambda_{3} \times w = (0,0,0)[/tex]

    admits only the solution that all lambdas = 0, but how can I proove that they are linearly independent, knowing so little?
     
  2. jcsd
  3. Jan 25, 2010 #2
    I'd say to set it up in matrix form and check to see if the determinant is non-zero or row-reduce and if there is no row of all zeros at the end, it's linearly independent.

    Edit: If matrices aren't allowed, show that for a system with constants multiplied by your u, v and w coefficients, each constant must be zero.

    e.g.

    Solve

    1*a+1*b=0
    1*a-1*b=0
    1*a-2*b+c=0
     
    Last edited: Jan 25, 2010
  4. Jan 25, 2010 #3

    Mark44

    Staff: Mentor

    Show that the equation c1(u + v) + c2(u - v) + c3(u - 2v + w) = 0 has only a single solution for the constants ci, using the fact that u, v, and w are linearly independent.
     
  5. Jan 25, 2010 #4
    that's the process you would normally use when dealing with coordinatse.
    But since we are dealing with whole vectors (instead of each vector's coordinates), would that really work?

    For example, if I wanted to proove that vectors a b and c were linear independent:
    Given a = (1,0,0), b = (0,1,0) and c = (0,0,1), we'd just do that same process, dealing with coordinates (c1(1,0,0) + c2(0,1,0)... = (0, 0, 0)

    The confusion rises because we are multiplying constants with vectors, not coordinates.
     
  6. Jan 25, 2010 #5

    Mark44

    Staff: Mentor

    This definition applies whether you know the coordinates or not.
    Again, you are making a false distinction. Try what I suggested.
     
  7. Jan 25, 2010 #6
    didn't know it applied to vectors too. Thanks.
    Is there anywhere i can read on about that to get a better feel for the theory behind it?

    And could I use the same principle to prove linear dependece on a problem, again with three vectors (u, v and w), but not necessairly linear independent, such that : w = 2u + v
    ?
     
  8. Jan 25, 2010 #7

    Mark44

    Staff: Mentor

    Presumably you have a textbook. Look up the definitions of linear independence and linear dependence.
     
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