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Math Help - Kernel and Range

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    Kernel and Range

    Determine the kernel and range of the following linear transformation from P3 to R2:

    L(p) =
    ( p'(0)
    Integral from -1 to 1 of p(x) dx )
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by harvardgurl29 View Post
    Determine the kernel and range of the following linear transformation from P3 to R2:

    L(p) =
    ( p'(0)
    Integral from -1 to 1 of p(x) dx )
    What in the name of God....


    L:\mathcal{P}_3\to\mathbb{R}^2 given by p(x)\mapsto \left(p'(0),\int_{-1}^{1}p(x)dx\right)?

    I'll help with the kernel. Let's see some effort on the image.

    \ker L=\left\{p(x):L(p(x))=(0,0)\right\}

    So, let p(x)=a_0+a_1 x+a_2 x^2+a_3 x^3.

    We know that p'(0)=a_1+2a_2\cdot 0+3a_3\cdot0^2=a_1=0. Thus, p(x)=a_0+a_2x^2+a_3 x^3.

    Furthermore, 0=\int_{-1}^{1}p(x)dx=a_0+\frac{a_2}{2}+\frac{a_3}{4}+a_0+\  frac{a_2}{2}-\frac{a_3}{4}=2a_0+a_2=0

    It follows that \ker L=\left\{a_0+a_1 x+a_2 x^2+a_3 x^3:a_1=0,2a_0+a_2=0\right\}
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