i know i need to prove that if
x null (T)
therefore, T(x)=0
but i'm unsure as to how i get towards the other side. I know the operator is normal so it is diagonalizable in some basis due to the spectral theorem.
......?
ok so i settled on this as my way to solve the first part.
Since, T is normal, there exists an ONB of V so that the matrix representation of T is diagonal (by the complex spectral theorem)
Since A is diagonal, the null(A)=0 (the zero vector).
can be represented as A but with its entries to the power of k.
Since can be represented as A' which is also diagonal, the null( )=0 the zero vector.
Therefore the null(T)=null( )=0 the zero vector.
****AM I ON THE RIGHT TRACK?!?!?!!*******
also, how do i prove the range side? thanks all