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Math Help - Proving a function is a linear map

  1. #1
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    Proving a function is a linear map

    Is the function T : \mathbb{P}_3(\mathbb{R})\rightarrow \mathbb{P}_3(\mathbb{R}) defined by T(p)=q where

    q(x)=4xp'(x)-8p(x) for x\in\mathbb{R}

    a linear map? Prove your answer


    Would I set my proof as follows

    Let p, q\in \mathbb{P}_3(\mathbb{R}), \lambda\in\mathbb{R}

    q(x)=4xp'(x)-8p(x), p(x)=4xq'(x)-8q(x)

    T(p+q)=4x(p+q)'(x)-8(p+q)(x)

    =4x(p'(x)+q'(x))-8(p(x)+q(x))

    =4xp'(x)+4xq'(x)-8p(x)-8q(x)

    =(4xp'(x)-8p(x))+(4xq'(x)-8q(x))

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


    T(\lambda p)=4x(\lambda q)'(x)-8(\lambda q)(x)

    =4x\lambda q'(x)-8\lambda q(x)

    =\lambda (4xq'(x)-8q(x))

    =\lambda T(p)\Rightarrow \textrm{Scalar multiplication condition satisfied}

    Therefore, T is a linear map
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  2. #2
    Member Haven's Avatar
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    Seems right to me.
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