We are given the vectors la> = (1,0) and lb> = (0,1) and then a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entires and zero elsewhere. I am asked to now represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H because of the degeneracy of the eigenvalues of H. I always get confused with these basis-change problems. First of all: What does it even mean that a matrix is represented in a particular basis? Why is H necessarily represented in the basis (1,0) and (0,1)? Secondly I know that the basis change equation is: H' = SHS^-1 (1) But my QM also tells me that the elements of H' can be found by taking the inner products <a'lHla'>, <a'lHlb'> etc. How is this equivalent to (1)?