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Math Help - Showing that a function is a linear map

  1. #1
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    Showing that a function is a linear map

    Show that the function T : \mathbb{P}_3(\mathbb{R})\rightarrow \mathbb{P}_3(\mathbb{R}) defined by

    T(p)=4p'+3p where p'(x)=\frac{dp}{dx}

    is a linear map

    Would I begin the question like the following:

    p(x)=a_0+a_1x+a_2x^2+a_3x^3\Rightarrow p'(x)=a_1+2a_2x+3a_3x^2

    T\begin{pmatrix}a_0\\a_1\\a_2\\a_3\end{pmatrix}=4\  begin{pmatrix}a_1\\2a_2\\3a_3\\0\end{pmatrix}+3\be  gin{pmatrix}a_0\\a_1\\a_2\\a_3\end{pmatrix}
    Last edited by acevipa; October 8th 2010 at 02:15 AM.
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  2. #2
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    That seems more tedious than necessary. Just use the definition of "linear map".

    Suppose p and q are two polynomials in  \mathbb{P}_3(\mathbb{R}) and a is a real number.

    What is T(p+ q)? Can you show that is the same as T(p)+ T(q)? What is T(ap)? Can you show that is the same as aT(p)?
    Use basic properties of the derivative.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    That seems more tedious than necessary. Just use the definition of "linear map".

    Suppose p and q are two polynomials in  \mathbb{P}_3(\mathbb{R}) and a is a real number.

    What is T(p+ q)? Can you show that is the same as T(p)+ T(q)? What is T(ap)? Can you show that is the same as aT(p)?
    Use basic properties of the derivative.
    By the way, was what I was doing before still correct?

    Is this proof good enough

    T(p+q)=4(p+q)'+3(p+q)

    Now using properties of derivatives and functions

    =4p'+q'+3p+3q

    =(4p'+3p)+(q'+3q)

    =T(p)+T(q)\Rightarrow \textrm{Addition condition satisfied}

    T(ap)=4(ap)'+3(ap)

    =4ap'+3ap

    =a(4p'+3p)

    =a(T(a))\Rightarrow \textrm{Scalar multiplication condition}

    Therefore T is a linear map
    Last edited by acevipa; October 8th 2010 at 05:15 PM.
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