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Math Help - Linear map

  1. #1
    Member kjchauhan's Avatar
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    Linear map

    Show that the linear map T:\wp_2 \to \wp_2 defined by T(\alpha_0+\alpha_1 x+\alpha_2x^2)=(\alpha_0+\alpha_1)+(\alpha_1+2\alp  ha_2)x+(\alpha_0+\alpha_1+\alpha_2)x^2 is non-singular and find its inverse.

    Thanks in advance.
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  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by kjchauhan View Post
    Show that the linear map T:\wp_2 \to \wp_2 defined by T(\alpha_0+\alpha_1 x+\alpha_2x^2)=(\alpha_0+\alpha_1)+(\alpha_1+2\alp  ha_2)x+(\alpha_0+\alpha_1+\alpha_2)x^2 is non-singular and find its inverse.

    Thanks in advance.
    It is hard to answer a question like that when we do not know your definition of singular linear transformation or the theorems you know.
    Try one of these:
    1)Show the determinant of T is non zero by computing the matrix of T (fixing a basis) and then invert the matrix and then finally get inverse of T.
    2)Show that T is one-one and onto. Let F be the inverse map, then T\circ F = \text{Id}. So compute F directly from that.

    The second method is messy but it only uses definition of inverse.
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