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How to show functions are linearly dependent?

  1. Oct 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that the set of functions:

    x^2
    3x+2
    x-1
    2x+5

    are linearly dependent.

    2. Relevant equations

    -

    3. The attempt at a solution

    I know that you have to show that you can put constants in front of each equation (that aren't all zero) such that:

    c1y1 + c2y2 + c3y3 + c4y4 = 0

    i.e. c1(x^2) + c2(3x+2) + c3(x-1) + c4(2x-5) =0

    But I have no idea how to do this?
     
  2. jcsd
  3. Oct 25, 2008 #2
    work out c1(x^2) + c2(3x+2) + c3(x-1) + c4(2x-5) =0

    by grouping all the same factors of x, ie:

    c1x^2 = 0
    (3 c2 + c3 + 2 c4)x = 0
    ... = 0

    and figure out what set of constants makes all of these equations true, can you do that?
     
  4. Oct 25, 2008 #3
    i'm struggling i must admit :S

    this would give me:

    (c1)x^2 = 0
    (3c2 + c3 + 2c4)x = 0
    (2c2 + c3 + 5c4) =0

    yes?
     
  5. Oct 25, 2008 #4

    Mark44

    Staff: Mentor

    Yes. From above, it's clear that c1 = 0, but since you have two more equations in three unknowns (c2, c3, c4) it's pretty likely you're going to get a whole lot of nonzero solutions for these constants.
     
  6. Oct 25, 2008 #5
    but do you think i have to find the values of the constants?

    surely it's impossible? because I have three unknowns - I need three equations?
     
  7. Oct 25, 2008 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Therefore what? If the coefficients are nopt all 0, then the functions are dependent!
     
    Last edited: Oct 25, 2008
  8. Oct 28, 2008 #7
    I used a wronskian in the end!
     
  9. Oct 28, 2008 #8

    Mark44

    Staff: Mentor

    That's like going after a housefly with a bulldozer! It would have been much simpler to just solve this system algebraically:
    (c1)x^2 = 0
    (3c2 + c3 + 2c4)x = 0
    (2c2 + c3 + 5c4) =0
     
  10. Oct 28, 2008 #9
    but i still cant see how that proves it?

    and i needed the marks, so overkill it was :P
     
  11. Oct 28, 2008 #10

    Mark44

    Staff: Mentor

    The functions x^2, 3x+2, x-1, and 2x+5 are linearly dependent iff the equation c1x^2 + c2 (3x+2) + c3 (x-1) + c4(2x+5) = 0 has a nontrivial solution. I.e., at least one of the constants ci is nonzero.

    You were well on your way to establishing this with this set of equations:
    (c1)x^2 = 0
    (3c2 + c3 + 2c4)x = 0
    (2c2 + c3 + 5c4) =0

    I guarantee you, if you can't find a solution to this system (and there are lots of them), you will have a difficult time of it, and knowledge of how to apply the Wronskian will be of little help to you. Guaranteed.
    Mark
     
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