Results 1 to 3 of 3

Math Help - Linear dependence

  1. #1
    Newbie
    Joined
    Apr 2007
    Posts
    16

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    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)\c  os\left(2x\right)-\cos\left(2x\right)\sin\left(2x\right)=0. Thus its dependent \forall\,x\in\left(0,1\right)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2007
    Posts
    16
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. linear dependence
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 5th 2009, 06:43 AM
  2. Linear Dependence
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: November 2nd 2009, 09:09 PM
  3. Linear Dependence
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 1st 2009, 02:15 AM
  4. Linear dependence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 10th 2007, 02:33 PM
  5. Linear Dependence
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 7th 2007, 11:16 AM

Search Tags


/mathhelpforum @mathhelpforum