1. Linear dependence

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

2. Originally Posted by Pulsar06
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 $W=\begin{vmatrix}\tfrac{1}{2}\sin\left(2x\right) & \sin\left(2x\right)\\ \cos\left(2x\right) & 2\cos \left(2x\right)\end{vmatrix}=\sin\left(2x\right)\c os\left(2x\right)-\cos\left(2x\right)\sin\left(2x\right)=0$. Thus its dependent $\forall\,x\in\left(0,1\right)$

3. 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.