help with adjoints of linear operators!

**dannyboycurtis** · December 3rd 2009, 06:42 PM

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!

**tonio** · December 3rd 2009, 07:29 PM

Originally Posted by dannyboycurtis

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

**NonCommAlg** · December 3rd 2009, 07:30 PM

Originally Posted by dannyboycurtis

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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