Consider a spin 1 interferometer which prepares the state as $\vert 1\rangle$, then sends this state through an $S_x$ apparatus and then an $S_z$ apparatus. Measure the relative probabilities after the final Stern-Gerlach analyzer for the seven possible cases where one beam, a pair of beams, or all three beams from the $S_x$ Stern-Gernach analyzer were used. Compare the simulation results to theory. Make sure that, for the theory section, you explicitly discuss your choice of projection operators.
Show that the matrix $$A\doteq\pmatrix{a_{11} & a_{12}\cr a_{21}& a_{22}}$$ is a linear operator on the space of all vectors, i.e.:
Show that $$A\left(\vert v_1\rangle + \vert v_2\rangle\right) =A\vert v_1\rangle + A\vert v_2\rangle $$
Also show that $$A\left(\lambda\vert v_1\rangle\right) =\lambda A\vert v_1\rangle $$