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Math Help - Linear Transformation in polynomial vector space

  1. #1
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    Linear Transformation in polynomial vector space

    Define T: P1 --> P1 by

    T(ax + b) = (2b - a)x + (b + a)

    Show that T is both one-to-one and onto, and find the inverse transformation to T.

    This problem showed up in my homework, and I'm not sure how to finish it. I was able to prove that T is both on-to-one and onto, but don't know how to find T^(-1). I hope someone can help. Thanks in advance!
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  2. #2
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    Quote Originally Posted by dan213 View Post
    Define T: P1 --> P1 by

    T(ax + b) = (2b - a)x + (b + a)

    Show that T is both one-to-one and onto, and find the inverse transformation to T.

    This problem showed up in my homework, and I'm not sure how to finish it. I was able to prove that T is both on-to-one and onto, but don't know how to find T^(-1). I hope someone can help. Thanks in advance!
    Let e_1=1 and e_2=x be the basis vectors for
    \mathbb{P}_1

    Then T(1)=T(e_1)=2x+1=e_1+2e_2
    and
    T(x)=T(e_2)=-x+1=e_1-e_2

    So the matrix representation of the linear transformation is

    \begin{bmatrix}1 & 1 \\ 2 & -1 \end{bmatrix}

    Can you finish finding the inverse from here?
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  3. #3
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    Yes I can - thank you. I appreciate your help.
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