# help with adjoints of linear operators!

• Dec 3rd 2009, 07:42 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, 08:29 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!

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

Tonio
• Dec 3rd 2009, 08:30 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!

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