Ok, so I think I understand how to find a homomorphism and I get the concept that it is a mapping of the same structure but I’m confused by the $a - (c/2)i$ part of this question?
I’m confused because if $f(xy)$ needs to equal $f(x) + f(y)$ then how do we do that with $a - (c/2)i$ ?
*also, does the x and y used above represent the G matrix and a - (c/2)i respectively? Or do they represent something by else?
In which case, how would I multiply the G matrix by a - (c/2)i and also add them?
Here's what I have done
Given $x,y\in G$, we know these are matrices of the form $$ x = \begin{pmatrix} 1&a&b\\&1&c\\&&1\end{pmatrix},\ y = \begin{pmatrix} 1&a'&b'\\&1&c'\\&&1\end{pmatrix} $$ for some $a,b,c,a',b',c'\in\mathbb R$.
Computing $\phi(xy)$, we get
\begin{align*} \phi(xy) &= \phi\left( \begin{pmatrix} 1&a+a'&b+b'+ac'\\&1&c+c'\\&&1\end{pmatrix}\right) \\ &= (a+a')-\frac{c+c'}{2}\mathrm i \\ &= \left( a-\frac c 2 \mathrm i\right) + \left(a'-\frac {c'} 2 \mathrm i\right) \\ &= \phi(x) + \phi(y). \end{align*}