Problem 8+ In this problem, we represent the spin eigenfunctions -- as vectors, while spin operators - as matrices. + a) The spin eigenfunctions are often represented as the column vectors:+ a = la) = B = 18) = Complex conjugated functions are represented as row vectors:+ a* = (a = (1 0): p*= (81=(0 1)+ Verify the orthogonality of spin eigenfunctions in this representation+ b) If the spin angular momentum operators are represented as matrices:+ h Sy = $2 = 1 0 Show that the commutation rule [s,, sy.] = ths, holds.+ c) Evaluate the matrix representation of s- operator:+ + 82+ and+ d) verify the value of the commutator [s=, s,].+