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Thread: Linear Algebra of polynomials

  1. #1
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    Linear Algebra of polynomials

    Linear Algebra of polynomials-screenshot_8.png

    I need help with this specific problem.

    a) prove that T is linear
    b) give the kernel dimension of T et dertermine a base of the image of T
    c) The application T is it injective? Is T surjective?
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  2. #2
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    Re: Linear Algebra of polynomials

    Quote Originally Posted by steelmaste View Post
    Click image for larger version. 

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    I need help with this specific problem. :
    a) prove that T is linear
    b) give the kernel dimension of T et dertermine a base of the image of T
    c) The application T is it injective? Is T surjective?
    $\displaystyle T(a{x^2} + bx + c) = \left( {\begin{array}{*{20}{c}} {p(0)}&{p'(0)}\\ {p'(0)}&{p''(0)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} c&b\\ b&{2a} \end{array}} \right)$
    a) It should be easy to show $T(\alpha p+q)=\alpha T(p)+T(q)$

    b) Looking at the mapping what polynomial(s) map to $\displaystyle \left( {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right)~?$

    c) for injectivity If T(p)=T(q) does it follow that $p=q$ HINT: what if $\displaystyle \left( {\begin{array}{*{20}{c}} c&b\\ b&2a \end{array}} \right)=\left( {\begin{array}{*{20}{c}} f&e\\ e&2d \end{array}} \right)~?$

    For surjectivity what happens in the case of $\displaystyle \left( {\begin{array}{*{20}{c}} 0&2\\ 1&3 \end{array}} \right)~?$
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  3. #3
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    Re: Linear Algebra of polynomials

    $\displaystyle T(1),T(x),T\left(x^2\right)$ are linearly independent matrices ("vectors")

    so $dim Im T =3$

    therefore $dim Ker T = 0$

    and $T$ is injective
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