The first postulate of quantum mechanics that can be found in the M. Nielsen and I. Chuang textbook:
Postulate 1: Associated to any isolated physical system is a complex vector space with the inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.
So, If one has some state $|\psi\rangle = \alpha |0\rangle + \alpha |1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$, it can be described as a vector in the Hilbert space with $\alpha$ and $\beta$ complex coordinates. Because of normalization $|\alpha|^2 + |\beta|^2 = 1$, the length of that vector is $1$. When we apply a unitary operator the length doesn't change (still $|\alpha'|^2 + |\beta'|^2 = 1$). So unitary operations are just rotations of that vector in the Hilbert space (they don't change the length of it). If we change the basis with a unitary operator then it can be described as a rotation of basis vectors in the Hilbert space (the basis vectors will not change their lengths, they will just point to the other directions).