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I was following the above pdf and got through most of it but am not quite understanding how (41), (42), and (43) are derived.

It appears that (31) and (41) are representing the same states and are still orthogonal, but how exactly is (41) derived? Aside from a scalar multiplication, the equation in (40) works perfectly well when applying (31) instead of (41).

For (42), I understand this is a unitary matrix, but what's the motivation and how exactly is it derived from the basis states? To clarify, how many basis states are there?

For (43), I tried doing the matrix multiplication, and I'm not sure if it's a typo, but how are there two possible solutions for the m = 3/2 and j = 3/2 states when multiplied by U?

These final equations have given me a bit of trouble in understanding the material, so any help with understanding their derivations would be greatly appreciated!