Density Matrix: Theorem & Normality Conditions

[Nusc]

I have a question regarding the slide:

http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture3.pdf

On page 18-21 it gives the proof of the theorem that | \psi_i^{~} \rangle and |\phi_{i}^{~}\rangle generate the same density matrix iff |\psi_{i}^{~}\rangle = \sum_{j} u_{ij} |\phi_{j}^{~}\rangle assuming that | \psi_i^{~}\rangle is not necessarily normalized.

What if | \psi_i^{~}\rangle is normalized and | \phi_i^{~}\rangle not independent?

Would the necessary condition for which p = | \psi_i \rangle \langle \psi_i |= q = | \phi_j \rangle \langle \phi_j | require that you have |\psi_{i}^{~}\rangle = \sum_{j} u_{ij} |\phi_{j}^{~}\rangle ?

 

[Nusc]

We know for normalized states psi and phi that p = | \psi_i \rangle \langle \psi_i |= q = | \phi_j \rangle \langle \phi_j | iff \sqrt{p_{i}} | \psi_i \rangle = \sum_j u_{ij} \sqrt{q_j} | \phi_j \rangle