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Math Help - Bijective Linear Transformation Proof

  1. #1
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    Bijective Linear Transformation Proof

    Let
    T : V W be a linear transformation that is bijective, that is, it is

    injective1 and surjective2. For each w W let T1(w) be the unique v V such that
    T(v) = w. v exists because T is surjective and it is unique because T is injective. T1 is characterized by the two conditions3 that T T1 = idW and T1 T = idV .



    We obtain a map T1 : W V



    Show that T1 is again a linear transformation.
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  2. #2
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    Quote Originally Posted by amm345 View Post
    Let
    T : V W be a linear transformation that is bijective, that is, it is

    injective1 and surjective2. For each w W let T1(w) be the unique v V such that

    T(v) = w. v exists because T is surjective and it is unique because T is injective. T1 is characterized by the two conditions3 that T T1 = idW and T1 T = idV .





    We obtain a map T1 : W V




    Show that T1 is again a linear transformation.

    Let w_1\,,\,w_2\in W\Longrightarrow\,\exists\,\,v_1\,,\,v_2\,\in\,V\,  \,s.t.\,\,Tv_1=w_1\,,\,Tv_2=w_2\,\,(why?) , so:

    As T(v_1+v_2)=Tv_1+Tv_2=w_1+w_2\,,\,then\,\,T^{-1}(w_1+w_2)=v_1+v_2 ....take it from here, and very simmilar with multiplication by a scalar.

    Tonio
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