Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by
$ f(a+bi) = \left[ \begin{align} a && {-b} \\ b && a \end{align} \right]$
Prove that $f$ is an injective homomorphism of rings. Is it an isomorphism?
So that is my problem. Okay, so I don't really know what $f(a)$ and $f(bi)$ are, so it is kinda hard to show it is homomorphic.
So I tried this(don't want to do matrices they take forever. Read element break as comma, and line break as semi colon like in matlab):
$f(a+bi)=f(c+di) \to [a,-b;b,a] = [c,-d;d,c]$ take det and get $a^2 + b^2 = c^2 + d^2$
so since these matrices are equal, then $a=c, d=b$, hence $f$ is an injective homomorphism.
Now for isomorphism we need surjection.