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Math Help - T: R -> R^2 defined by T(x) = [(sinx) (x)].

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    Exclamation T: R -> R^2 defined by T(x) = [(sinx) (x)].

    I need to show that this is not linear. From what I understand, sinx makes this a nonlinear function because it is a periodic function and only defined if: -π/2 ≤x≤π/2. Is my conjecture correct? How can I prove it?

    Thanks for your help.
    Yvonne
    Last edited by yvonnehr; April 8th 2008 at 07:40 PM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by yvonnehr View Post
    I need to show that this is not linear. From what I understand, sinx makes this a nonlinear function because it is a periodic function and only defined if: -π/2 ≤x≤π/2. Is my conjecture correct? How can I prove it?

    Thanks for your help.
    Yvonne
    A function f(x) is linear iff
    f(x + y) = f(x) + f(y) for all x, y in the domain
    and
    f(ax) = a \cdot f(x) for any scalar a and x in the domain.

    Neither condition is satisfied for your function.

    -Dan
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    definition of linear transformation

    I assumed that this function did not satisfy the conditions stated by the definition. However, I don't know how to show that for the sin function.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by yvonnehr View Post
    I assumed that this function did not satisfy the conditions stated by the definition. However, I don't know how to show that for the sin function.
    Well,
    sin(x + y) = sin(x)~cos(y) + sin(y)~cos(x) \neq sin(x) + sin(y)

    And as a particular example:
    sin(2x) = 2~sin(x)~cos(x) \neq 2~sin(x)

    -Dan
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