Linear Independence: Determining w/ Wronskian Matrix

[dashkin111]

Homework Statement

Using the wronskian (determinant basically), determine if e^x, sin(x), cos(x) are linearly independent

Homework Equations

I used this:
$$| e^{x} sin(x) \:cos(x)|$$
$$|e^{x} cos(x) -sin(x)|$$
$$|e^{x} -sin(x) -cos(x)|$$

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

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?

 

[Dick]

Yes, but you know the wronskian only has to be nonvanishing someplace for the functions to be linearly independent, right?

 

[dashkin111]

Yes, but you know the wronskian only has to be nonvanishing someplace for the functions to be linearly independent, right?

Oh wow, I thought it was at all points. If possible, would you mind telling me why that is?

 

[Dick]

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.