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Math Help - help with adjoints of linear operators!

  1. #1
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    help with adjoints of linear operators!

    I need help with the following proof:
    Let V be a finite=dimensional inner product space and let T be a linear operator on V. Prove that if T is invertible, then T* is invertible and (T*)^-1 =(T^-1)*.
    Thanks for any suggestions and help!
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    I need help with the following proof:
    Let V be a finite=dimensional inner product space and let T be a linear operator on V. Prove that if T is invertible, then T* is invertible and (T*)^-1 =(T^-1)*.
    Thanks for any suggestions and help!

    \forall x\in V\, <x,x>=<T^{-1}Tx,x>=<Tx,\left(T^{-1}\right)^{*}x>=<x,T^{*}\left(T^{-1}\right)^{*}x> \,\Longrightarrow\,<x,\left(T^{*}\left(T^{-1}\right)^{*}-I\right)(x)>=0\,\Longrightarrow\,T^{*}\left(T^{-1}\right)^{*}-I=0 ...

    Tonio
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  3. #3
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    Quote Originally Posted by dannyboycurtis View Post
    I need help with the following proof:
    Let V be a finite=dimensional inner product space and let T be a linear operator on V. Prove that if T is invertible, then T* is invertible and (T*)^-1 =(T^-1)*.
    Thanks for any suggestions and help!
    <x,y>=<TT^{-1}x,y>=<T^{-1}x,T^*y>=<x,(T^{-1})^*T^*y>, for all x,y \in V, and so (T^{-1})^*T^*=\text{id}_V.
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