Assume $V$ is a finite dimensional vector space over $\mathbb{C}$ that does not come with an inner product. For $T: V \to V$, suppose $A = A^*$ where $A = [T]_\mathcal{X}$. Is it possible to find another basis/matrix where $B \neq B^*$ for $B = [T]_\mathcal{Y}$?
In answer this question, I'm trying to show that the property of "self-adjoint"-ness, the operator $T^*$, can only be defined when $V$ is equipped with an inner product. But I'm failing to find a counter-example. How would one go about finding this?