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 such that . If are not both zero, then its dependent; otherwise, its independent. In our case, if and , then we have shown that the two functions are linearly dependent.
2) Use the Wronksian. If , then it's dependent.
So our Wronskian is . Thus its dependent