Deriving transformation matrix from clues

GooseMunch
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I have PTV=R where P and T are square matrices (4x4) and V and R are non-square (4x3).
P and V are known, T is unknown, and R is partially known (3 unknown elements).
Seems impossible, but T is a transformation matrix (ie upperleft 3x3 is a rotation matrix) which gives me additional clues.

I'm trying to find T, but I can't figure out how to approach this problem. It's been almost 10 years since I took linear algebra course and I feel very lost.
Seems like I can just rearrange it like so:
T = P'RV' where P' and V' are inverses of P and V respectively but V is non-square so I can only get left-inverse. I don't think pseudoinverse of V is the answer because I'm afraid I'll end up with overdetermined system and the whole thing becomes unsolvable.

Can anyone nudge me in the right direction? Thanks.
 
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I would write out the entire system of linear equations, and see what can be said about kernel and rank of its coefficient matrix.
 

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