Question

Suppose T is invertible then show that (T^-1)^* = (T^*)^-1

Answer 1

$Solution: ~Let~u\in V. Since~ T ~ is ~ invertible~ and ~ hence ~surjective, \\ \therefore u=Tw~for ~some~ vector ~w.\\ If ~ v \in V,< (T^{-1})^ * v,Tw>~=~<v,T^{-1}Tw>~=~<v,w> \\and~ < (T^{-1})^ * v,Tw>~=~<T^*(T^*)^{-1}v,w>~=~<v,w> \\Since ~ for ~all ~ u,v \in V, <(T^{-1})^ * v,u>=<(T^*)^{-1}v,u> \\ \therefore ~ (T^{-1})^ *=(T^*)^{-1} \\Hence ~ proved.$