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_{ij}of its matrix representation is just the ith component of A acting on the jth basis vector. We can also represent the action of A on a ket as the matrix product of A's matrix with the column matrix representing the ket.

We can also represent the action of A on a bra vector as matrix product of the row matrix of the bra with another matrix. If the basis was orthonormal, it would be the same matrix A

_{ij}as above. But if the basis isn't orthonormal, is it a different matrix?