Let be a nonzero quaternion.

(a) Prove that the left multiplication given by is an isomorphism.

By def., the Quaternion Algebra is a vector space equipped with a ring. Therefore, exist.

Thus, is monic.

Therefore, f is an epimorphism and f is an isomorphism.

(b) Find the matrix of relative to the basis

Can someone walk step by step through this piece? I don't remember what to do.