
(a) If f_{1}>, f_{2}> is an orthonormal
basis, what are the conditions on the coefficients {a_{ij} so that e_{2}> = a_{21} f_{1}> + a_{22} f_{2}> is
also an orthonormal bases?
(b) Construct the 2x2 matrices A_{ij} = <f_{i}e_{j}> and B_{ij} = <e_{i}f_{j}>.
(c) Prove that these are the matrices that transform the coordinates of an arbitrary vector in the fbasis to its representation in the ebasis and back.
(d) Calculate the matrix product (AB)_{ij} = Σ_{k} A_{ik} B_{kj} in two ways; first, by putting your matrices in explicitly from part (b) and multiplying them, second, by writing the A and B matrix elements as Dirac inner products and isolating the k summation.
Last revision 28. October, 2004.