My definition of unitary is given a form, $<Tv,Tw>$=$<v,w>$ so form is preserved.
I don’t get why the matrix representation should not be unitary with respect to any basis. And why the theorem is true. I know that the matrix of the form with respect to a given basis can be given by $a_{ij}=<v_i,v_j>$
Please prove the theorem and say what’s different if our basis is not orthonormal. Pls help
Edit: I found this proof but I do not understand it:
Let $\beta$ be an orthonormal basis for a vector space $\textsf V$ and let $\textsf T \in \mathcal L(\textsf V)$. Suppose that $\textsf T$ is unitary and let $A$ be the matrix representation of $\textsf T$ with respect to $\beta$. Since $\textsf T$ is unitary we have that $$\langle \textsf T^*\textsf Tv,w \rangle = \langle \textsf Tv,\textsf Tw \rangle = \langle v,w \rangle \quad \textrm{ for all } v,w\in \textsf V $$ so, $\textsf T^*\textsf T = \textsf I_\textsf{V}$ and also $\textsf T \textsf T^* = \textsf I_\textsf{V}$. Now, since $\beta$ is orthonormal, the matrix representation of $\textsf T^*$ with respect to that basis is $A^*$ (this is why we need that the basis is orthonormal) and $\textsf T \textsf T^* = \textsf I_\textsf{V}$ implies that $AA^* = I$.
I add a proof of the last result concerning the orthonormal basis and the matrix of the adjoint operator.
Check it in Linear Algebra by Hoffman & Kunze.