Lets say you have a inner product between two state vectors with an operator in between A|X|B. I can write this as a summation over I and j as A|i i|X|j j|B (sorry for notation). But I don't understand what j is. I get that i is the basis vectors and we are summing over these basis vectors, so does that mean that j is another basis? Going off of this, in Feynman's Quantum Book equation 8.13 he just changes an equation summing over i's to summing over j's, so I don't understand how those would be equivalent if they are different basis. Does this have to do with rows and columns (i.e. i is row number and j is column number)? Also, you can describe j|i as the dirac delta function which is only true for when i=j, but I can't really make sense of that either. Confusing stuff.
Edit: The major reason i'm confused is that I remember calling the basis vectors i,j,k so I don't understand how you can sum over i and j in these equations. I assumed that when you wrote i and summed over it you were summing i,j,k so I don't see how you can sum over i's and j's