Linear dependence

Pulsar06 · #1 $Old$ July 5th, 2009, 09:48 PM

Hey

Are sin(x)cos(x) and sin(2x) linearly dependent or independent on the interval (0,1)? I say dependent since sin(2x) = 2sin(x)cos(x) which is simply twice the first function, but according to my book it's independent. Who's wrong?

Thanks

Chris L T521 · #2 $Old$ July 5th, 2009, 10:18 PM

Quote:

Originally Posted by Pulsar06 $View Post$

Hey

Are sin(x)cos(x) and sin(2x) linearly dependent or independent on the interval (0,1)? I say dependent since sin(2x) = 2sin(x)cos(x) which is simply twice the first function, but according to my book it's independent. Who's wrong?

Thanks

It looks like you're right.

There are two ways to do this:

1) Form a linear combination and find $\alpha,\beta$ such that $\alpha\sin x\cos x+\beta \sin\left(2x\right)=0$ . If $\alpha,\beta$ are not both zero, then its dependent; otherwise, its independent. In our case, if $\alpha=-2$ and $\beta=1$ , then we have shown that the two functions are linearly dependent.

2) Use the Wronksian. If $\exists\,x\in\left(0,1\right):W=0$ , then it's dependent.

So our Wronskian is $W=\begin{vmatrix}\tfrac{1}{2}\sin\left(2x\right) & \sin\left(2x\right)\\ \cos\left(2x\right) & 2\cos \left(2x\right)\end{vmatrix}=\sin\left(2x\right)\cos\left(2x\right)-\cos\left(2x\right)\sin\left(2x\right)=0$ . Thus its dependent $\forall\,x\in\left(0,1\right)$

Pulsar06 · #3 $Old$ July 5th, 2009, 10:27 PM

Thanks. I had trouble believing it was an error, since this is the fifth edition of the book, but I suppose it could be a new problem.