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i am a beginner in quantum mechanics and i have those basic questions on the bra-ket notation:

Which dimension has a ket [itex] | \phi >[/itex] describing a state normally? Maybe[itex] \quad C ^n[/itex]?

Which dimension has a bra-ket [itex] <\psi | \phi >[/itex]then? Maybe [itex] \quad C [/itex]?

How do you get the matrix representation of an operator in general? I have been reading something like this: [itex] <\phi_n |\hat{A}| \phi_m >[/itex]? I think i have to figure out then, how A works on phi_m but what to do next?

Thanks for helping