I'm given the problem of showing that the change of basis matrix between two orthonormal bases of a complex inner product space is a unitary matrix. A solution to this problem is given here: prove change of basis matrix is unitary
My book gives the definition that an $n \times n$ complex matrix is said to be unitary if the conjugate transpose of the matrix is equal to the inverse of the matrix. I also have a theorem that says $T$, a linear operator on an inner product space $V$, is an isometry if and only if the inner product of $T(v)$ against $T(w)$ is equal to the inner product of $v$ against $w$ for all $v,w \in V$. In the given proof he has the change of basis matrix given by M, and says we need to show that the inner product of $Mx$ against $My$ is equal to the inner product of $x$ against $y$ for all $x,y \in V$ in order to show that $M$ is a unitary matrix. But nowhere does it have that the conjugate transpose of the matrix is equal to its inverse. Is saying that a complex matrix is unitary basically just saying that the linear operator represented by the matrix is an isometry? Any insight would be much appreciated.