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