# help with adjoints of linear operators!

• Dec 3rd 2009, 06:42 PM
dannyboycurtis
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!
• Dec 3rd 2009, 07:29 PM
tonio
Quote:

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!

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

Tonio
• Dec 3rd 2009, 07:30 PM
NonCommAlg
Quote:

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!

$\displaystyle <x,y>=<TT^{-1}x,y>=<T^{-1}x,T^*y>=<x,(T^{-1})^*T^*y>,$ for all $\displaystyle x,y \in V,$ and so $\displaystyle (T^{-1})^*T^*=\text{id}_V.$