so in general, what physical property besides linearity that motivate us to use matrices and vectors (and tensors) (or in general, multilinear objects)) in quantum mechanics.
I mean, (forgetting about pictures for the moment), vectors and tensors are characterised by how they transform under coordinate transformations. What is the physical meaning behind the geometric meaning of a state vector or density matrix (since they obey transformation rules like a vector does)?
(putting in pictures in again) It is easy to see what the norms of these state vectors and matrices mean, but what physic…