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.