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Math Help - Subspaces and functions

  1. #1
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    Subspaces and functions

    Hi

    The complex functions z(t), t
    ∈ R, can be differentiated a number of times and are a vector space V :
    Subspaces and functions-billede-4.png
    W = span(e-t, e-it, et, eit) and W is a subspace of V.

    Which of the following 3 functions belong to W ?

    z1(t) = 3et, z2(t) = cos(t), z3(t) = sinh(t)

    If the functions were vectors for example like this W= span((1,2),(2,1)) and I had to determine if the vector (-2,3) belonged to W, I would solve the inhomogeneous linear system but since it's not I have no idea how to determine this.

    I hope that somebody can give me a hint.
    Sorry for the bad description.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    Which of the following 3 functions belong to W ? z1(t) = 3et, z2(t) = cos(t), z3(t) = sinh(t)
    We have z_1(t)=3e^{t},\;z_2(t)=\dfrac{1}{2}e^{it}+\dfrac{1  }{2}e^{-it},\;z_3(t)=\dfrac{1}{2}e^{t}-\dfrac{1}{2}e^{-t}, so the three funtions belong to W.
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  3. #3
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    Re: Subspaces and functions

    Can you give an explanation ?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    Can you give an explanation ?
    Better ask your concrete doubts.
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  5. #5
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    Re: Subspaces and functions

    I don't understand how you get z2 to that form ?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    I don't understand how you get z2 and z3 to that form ?
    \sinh t=\frac{e^t-e^{-t}}{2} by definition of the function \sinh . On the other hand, e^{it}=\cos t+i\sin t (Euler's formula), and from here you can deduce de expression for z_2(t).
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  7. #7
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    Re: Subspaces and functions

    This might be stupid but I don't get your z2 expression when I isolate cost in e^it = cost + i*sint ?
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    This might be stupid but I don't get your z2 expression when I isolate cost in e^it = cost + i*sint ?
    Don't worry. We have \begin{Bmatrix} e^{it}=\cos t+i\sin t\quad (1)\\e^{-it}=\cos t-i\sin t\quad (2)\end{matrix}

    Now, sum (1) and (2).
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  9. #9
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    Re: Subspaces and functions

    I see ! That makes sense.
    And one last question the functions belong to W because they each can be written as a linear combination of span(e-t, e-it, et, eit) right ?
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  10. #10
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    I see ! That makes sense.
    And one last question the functions belong to W because they each can be written as a linear combination of span(e-t, e-it, et, eit) right ?
    Well, as linear combination of those four functions.
    Last edited by FernandoRevilla; November 10th 2012 at 11:59 AM.
    Thanks from Tala
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  11. #11
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    Re: Subspaces and functions

    Thank you for your help !
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  12. #12
    MHF Contributor FernandoRevilla's Avatar
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    Re: Subspaces and functions

    Quote Originally Posted by Tala View Post
    Thank you for your help !
    You are welcome!
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