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
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 $\displaystyle \alpha,\beta$ such that $\displaystyle \alpha\sin x\cos x+\beta \sin\left(2x\right)=0$. If $\displaystyle \alpha,\beta$ are not both zero, then its dependent; otherwise, its independent. In our case, if $\displaystyle \alpha=-2$ and $\displaystyle \beta=1$, then we have shown that the two functions are linearly dependent.
2) Use the Wronksian. If $\displaystyle \exists\,x\in\left(0,1\right):W=0$, then it's dependent.
So our Wronskian is $\displaystyle W=\begin{vmatrix}\tfrac{1}{2}\sin\left(2x\right) & \sin\left(2x\right)\\ \cos\left(2x\right) & 2\cos \left(2x\right)\end{vmatrix}=\sin\left(2x\right)\c os\left(2x\right)-\cos\left(2x\right)\sin\left(2x\right)=0$. Thus its dependent $\displaystyle \forall\,x\in\left(0,1\right)$