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Homework Help: 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

    (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.

    $$ a*cos(x) + b*cos(2x) $$
    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.
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