# Homework Help: Linear Independence

1. Oct 8, 2007

### dashkin111

1. The problem statement, all variables and given/known data
Using the wronskian (determinant basically), determine if e^x, sin(x), cos(x) are linearly independent

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

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

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. Oct 8, 2007

### Dick

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

3. Oct 8, 2007

### dashkin111

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

4. Oct 8, 2007

### 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.

Last edited: Oct 8, 2007