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!

Checking Linear Independence. Using Wronskian vs. Using Definition

  1. Feb 10, 2013 #1
    1. The problem statement, all variables and given/known data
    Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?


    2. Relevant equations

    Definition: Linear Independence
    A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero
    such that a linear combination of the functions in the set are equal to zero.

    Definition: Wronskian
    http://en.wikipedia.org/wiki/Wronskian

    Theorem
    (see wiki link as well)
    If the Wronskian of a set of n functions defined on the interval a ≤ x ≤ b is nonzero for at least one point then the set of functions is linearly independent there.

    3. The attempt at a solution

    Let's say I'm using the interval [-∞, ∞]. First, I'll use the definition.

    Consider
    $$ a*cos(x) + b*cos(2x) $$
    Now, pick x = 0, a = 1, b = 1
    $$ 1*cos(0) - 1*cos(0) = 0 $$
    Since a ≠ 0 and b≠ 0, I conclude from the definition that the functions are linearly dependent.

    Now, I'll use the Wronskian.

    $$ W(cos(x), cos(2x)) = \left| \begin{array}{cc}
    cos(x) & cos(2x) \\
    -sin(x) & -2sin(2x) \end{array} \right| =
    -2sin(2x)cos(x) + sin(x)cos(2x) $$

    Pick x = ∏/4. Then,

    $$ W = -2sin(\frac{\pi}{2})cos(\frac{\pi}{4}) + sin(\frac{\pi}{4})cos(\frac{\pi}{2}) =
    \frac{-2}{\sqrt{2}} ≠ 0$$

    So, by the Theorem above, since the Wronskian is nonzero, I conclude that the functions are linearly independent.

    A contradiction. What in flying flip went wrong?
     
    Last edited: Feb 10, 2013
  2. jcsd
  3. Feb 10, 2013 #2
    Hey got it!

    I misinterpreted the definition of linear dependence.

    The constants need to be non-zero for all x on the interval. I just chose one x.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook