The question:
Let Y be a finite-dimensional inner product space and T a linear operator on Y. Show that the range of T* is the orthogonal complement of the null space of T
Think i got one way:
took v$\in$Im(T*)
and u$\in$Ker(T)
will show $(u|v)=0$
v$\in$Im(T*) => there is v'$\in$Y for it T*(v') = v
(u|T*(v')) = (T(u)|v') = (0|v') = 0
and this is true for any u$\in$Ker(T)
=> v is in the complement of Ker(T)
Best Answer
What do you know about $\dim(\ker(T))$ and $\dim({\rm Im}(T)$? What do you know about $\dim({\rm Im}(T))$ and $\dim({\rm Im}(T^*))$?