The following questions will take a different and more revealing look at spin, spin operators, and eigenfunctions. In this question we will limit our consideration to a one electron system. a. The spin eigenfunctions are represented as column vectors =(10) and =(01). Show that and are orthogonal using this representation. Hint: here you do either matrix multiplication or take the dot product of the vectors. b. If the spin angular momentum operators are represented by the matrices S^x=2(0110), sy=2(0ii0), and sz=2(1001), show that the commutator rule [sx,sy]=ihsz holds. c. Show that S^2=s^x2+S^y2+s^z2=42(3003). d. Show that and are eigenfunctions of S^2 and S^z. What are the eigenvalues? e. Show that and are not eigenfunctions of s^x and s^y