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Linear Independence

  1. Oct 8, 2007 #1
    Using the wronskian (determinant basically), determine if e^x, sin(x), cos(x) are linearly independent

    2. Relevant equations
    I used this:
    | e^{x} sin(x) \:cos(x)|[/tex]
    |e^{x} cos(x) -sin(x)|[/tex]
    [tex]|e^{x} -sin(x) -cos(x)|[/tex]

    But pretend that's just a 3x3 matrix and you take the determinant of it

    3. The attempt at a solution

    After finding the determinant I get -2e^x which is never 0 so they're linearly independent. Am I right in this?
  2. jcsd
  3. Oct 8, 2007 #2


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    Yes, but you know the wronskian only has to be nonvanishing someplace for the functions to be linearly independent, right?
  4. Oct 8, 2007 #3
    Oh wow, I thought it was at all points. If possible, would you mind telling me why that is?
  5. Oct 8, 2007 #4


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    Sure. If f1(x), f2(x) and f3(x) are linearly dependent, then there are nonzero constants such that c1*f1(x)+c2*f2(x)+c3*f3(x) is identically zero over some interval. So a linear combination of columns in your matrix is zero. This tells you det=0 over the interval. So if det is non-zero anywhere, you know they aren't linearly dependent.
    Last edited: Oct 8, 2007

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