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## Homework Statement

Show that if T is a normal operator on a finite dimensional vector space than it has the same image as its adjoint.

## Homework Equations

N/A

## The Attempt at a Solution

I have been able to show that both T and [itex]T^{*}[/itex] have the same kernel. Thus, by using the finite dimension property and the rank nullity theoremit just suffices to show containment one way.

However, if you suppose that a vector v is in the Im(T) I haven't been able to find some representative w such that T*(w) = v.

Does anyone have any idea how to proceed?

Thanks!