# Suppose V is an inner product..

• Mar 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*))
• Mar 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*?
• Mar 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?
• Mar 26th 2013, 04:32 AM
dave52
Re: Suppose V is an inner product..
Thanks.
• Mar 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), \$\displaystyle w_1\$, ... \$\displaystyle w_n\$, and let \$\displaystyle v_i= T^*(w_i)\$. Can you show that \$\displaystyle v_1\$,... \$\displaystyle v_n\$ is a basis for Im(T*)?
• Mar 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