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Math Help - find if the set is dependent or not?

  1. #1
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    find if the set is dependent or not?

    The set is:
    {1, cos(2x) 3*(sin(x))^2}

    I figure that:
    C1(1 +C2(cos(2x) + C3*(3*sin(x))^2 =0

    That gives us:
    C1(1) + C2(Cos(0) + C3(3(sin(0))^2 = 0

    Thus C1 + C2 + 0 = 0
    So:
    C1 + C2 = 0

    How do I find C3?

    If my process is correct...Linear dependence exists if either C1 , C2 or C3 are non zero...
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  2. #2
    MHF Contributor arbolis's Avatar
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    Wait, is the set \{ 1, cos(2x) 3*(sin(x))^2\} as you wrote, or \{ 1 , cos(2x) , 3*(sin(x))^2 \}?
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  3. #3
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    Hello, My apologies. Your comma is correct.
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  4. #4
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    Quote Originally Posted by orendacl View Post
    The set is:
    {1, cos(2x) 3*(sin(x))^2}
    Hint: 2\sin^2 x = 1 - \cos 2x.
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  5. #5
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    Hence one of the terms changes...

    1, cos 2x, 1-cos3x thus our eq'n should look like:

    c1(1) + c2(cos 2x) +c3(1-cos3x) = 0

    That gives us
    c1+c2=0 while c3=1?

    Thus the solution is that
    matrix looks like |1|
    |0|
    |0|

    Thus the solution is dependent because one of the values is non-zero...
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by orendacl View Post
    The set is:
    ...That gives us:
    C_1(1) + C_2(Cos(0) + C_3(3(sin(0))^2 = 0...

    You're trying to show that some linear combination of the three functions is zero, not that they pass through zero for a certain x. You're trying to show that there exist some C_1, C_2, C_3 \in \mathbb{Z} such that f(x) =  C_1 +C_2*cos(2x) + C_3*(3*sin(x))^2 and f(x)=0 for all x.

    HINT: Look at ThePerfectHacker's post...
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  7. #7
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    I still don't see it...
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  8. #8
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by orendacl View Post
    I still don't see it...
    Don't think of the sine as functions, think of them as unknowns - call them \alpha, \beta and \gamma: \alpha = 1, \beta = cos(2x), \gamma = 3(sin(x))^2. So, we want to find integers C_1, C_2, C_3 such that C_1* \alpha + C_2* \beta + C_3 * \gamma = 0, by the definition of linear dependence.

    Substitute in the \alpha, \beta, \gamma into TPH's equality and stick everything on one side. You should have something of the form C_1* \alpha + C_2* \beta + C_3 * \gamma = 0 and thus your set is not linearly independent.
    Last edited by Swlabr; May 28th 2009 at 12:18 AM. Reason: I say "so" to much...
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