Show that following three statements are equivalent

Question

Proposition: Let $V$ be finite-dimensional inner-product space and $A\in L(V)$. Show that following three statements are equivalent:

1) A is hermitian operator.
2) For every orthonormal basis $b$ in $V$ matrix $[A]_{b}^b$ is hermitian.
3) There exists orthonormal basis $e$ in $V$ such that matrix $[A]_{e}^e$ is hermitian.

I have to show that $1)$ implies $2)$ and $3)$ implies $1)$. $2)$ obviously implies $3)$.

Answer 1

Show that if $b$ is an orthonormal basis of $V$ then $[A^{*}]_{b}^{b} = \left( [A]_{b}^{b} \right)^{*}$. This shows the direction $(1) \implies (2)$. For $(3) \implies (1)$, use the fact that if $[A]_{b}^{b} = [B]_{b}^{b}$ for some basis $b$ then $A = B$.