If A is a linear operator, and we have some ordered basis (but not necessarily orthonormal), then the element Aij 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 Aij as above. But if the basis isn't orthonormal, is it a different matrix?