Checking Linear Independence. Using Wronskian vs. Using Definition

[mef51]

Homework Statement

Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?

Homework 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.

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?

 

[mef51]

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.