Operators can be represented by matrices. From linear algebra one knows that a certain matrix A can be represented in another basis through the transformation A ′ = P A P − − 1 where P is the basis transformation matrix. If the vectors b respectively c are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle t between them. The spin operator S b in the first basis can then be transformed into the spin operator S c of the other basis through the following transformation: S c = D ⁡ ⁡ ( y , t ) S b D − − 1 ⁡ ⁡ ( y , t )