I think for a generic 2 by 2 matrix we would require 4 linearly independent matrices to span the whole space, 1 per element. Then I would assume that the two constraints for a legitimate density matrix (trace equals 1, Hermitian) would each subtract one from this number, leaving me with 2 linearly independent matrices.
This logic definitely doesn't constitute as a proof however (or is even particularly convincing), so, firstly am I correct in saying that there are only 2 linearly independent density matrices in 2D? And secondly, if I am correct, does anybody have a more convincing proof for it?
Also if it is only 2 linearly independent density matrices in 2D, is it $(d^2-2)$ linearly independent ones in $d$ dimensions?
P.S. I'm not a quantum information guy or a mathematician so please speak slowly.