S = σ in a basis where σx = ( ), σy = ( ), and σz = ( ) (the Pauli matrices).

1. Show that any 2x2 Hermitian matrix H can be written in the form H = Aσx + Bσy + Cσz, where A, B, and C are real numbers.
2. Show that the components of σ obey the angular momentum commutation relations, [σi, σj] = iεijkσk.
3. Show that σ^2 = σx^2 + σy^2 + σz^2.
4. Find the eigenspinors and eigenvalues of σ, denoted as |+⟩ and |-⟩, and of σy, denoted as |y+⟩ and |y-⟩.
5. An electron is prepared in the state |x⟩ = c|+⟩ + d|-⟩, where c and d are complex numbers. Find ⟨σx⟩, ⟨σy⟩, ⟨σz⟩, ⟨σ⟩, ⟨σy⟩, ⟨σz⟩, and ⟨σ^2⟩.
6. What are the possible values returned by a measurement of σ on this state, and what are their probabilities?