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I was wondering what conditions must two linear maps obey in order to commute?

If they are described by two matrices A and B, then one condition would be:

AB-BA=0

but what if we don't know the matrices, so we cannot compute AB adn BA? How is one supposed to proceed, is there a more general condition?

And another question: Suppose A and B commute. Will they still commute if I change the basis, I mean is the commutation coordinate independent?

thanks a lot for the help,

marin