Suppose V is an inner product sapce and T:V-->V Show that dim(Im(T)) = dim(Im(T*))
Last edited by dave52; Mar 26th 2013 at 04:14 AM.
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You mean "Suppose V is an inner product space". Now, what is the definition of T*?
T* satisfies the equation <T(v),w> = <v,T*(w)> for all v,w in V?
Thanks.
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), , ... , and let . Can you show that ,... is a basis for Im(T*)?
linear independent property: for all a_{1},....,a_{n} ∈ R if a_{1}v_{1}+.......+a_{n}v_{n }= 0 spanning property: for every x in Im(T*) such that x = a_{1}v_{1}+.....+a_{n}v_{n}
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