Suppose V is an inner product..

• March 26th 2013, 01:20 AM
dave52
Suppose V is an inner product..
Suppose V is an inner product sapce
and T:V-->V
Show that dim(Im(T)) = dim(Im(T*))
• March 26th 2013, 04:01 AM
HallsofIvy
Re: Suppose V is an inner product..
You mean "Suppose V is an inner product space". Now, what is the definition of T*?
• March 26th 2013, 04:16 AM
dave52
Re: Suppose V is an inner product..
T* satisfies the equation <T(v),w> = <v,T*(w)> for all v,w in V?
• March 26th 2013, 04:32 AM
dave52
Re: Suppose V is an inner product..
Thanks.
• March 26th 2013, 05:25 AM
HallsofIvy
Re: Suppose V is an inner product..
Quote:

Originally Posted by dave52
T* satisfies the equation <T(v),w> = <v,T*(w)> for all v,w in V?

Good! Now choose a basis for Im(T), $w_1$, ... $w_n$, and let $v_i= T^*(w_i)$. Can you show that $v_1$,... $v_n$ is a basis for Im(T*)?
• March 26th 2013, 05:43 AM
dave52
Re: Suppose V is an inner product..
linear independent property: for all a1,....,an ∈ R if a1v1+.......+anvn = 0
spanning property: for every x in Im(T*) such that x = a1v1+.....+anvn