Let $\displaystyle q=a+bi+cj+dk$ be a nonzero quaternion.

(a) Prove that the left multiplication $\displaystyle L_q:\mathbb{H}\to\mathbb{H}$ given by $\displaystyle L_q(x)=qx$ is an isomorphism.

$\displaystyle L_q(x_1)=L_q(x_2)\Rightarrow qx_1=qx_2$

By def., the Quaternion Algebra is a vector space equipped with a ring. Therefore, $\displaystyle q^{-1}$ exist.

$\displaystyle q^{-1}qx_1=q^{-1}qx_2\Rightarrow x_1=x_2$

Thus, $\displaystyle L_q(x)$ is monic.

$\displaystyle y=L_q(x)=qx\Rightarrow q^{-1}y=x$

$\displaystyle L_q(q^{-1}y)=qq^{-1}y=y$

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

(b) Find the matrix of $\displaystyle L_q$ relative to the basis $\displaystyle 1,i,j,k\in\mathbb{H}$

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