# Math Help - Suppose V is an inner product..

1. ## 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*))

2. ## Re: Suppose V is an inner product..

You mean "Suppose V is an inner product space". Now, what is the definition of T*?

3. ## Re: Suppose V is an inner product..

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

Thanks.

5. ## Re: Suppose V is an inner product..

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*)?

6. ## 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