I was trying to understand Example 7.15 from Rotman's Introduction to the Theory of Groups. The problem I'm having is maybe a bit silly and not strictly about semidirect products, but I don't understand how I apply $\theta_{x}$ to an element $a \in K$. My understanding is that, for each $x^i$, we have a map defined like this
$$\theta_{x^i} = \begin{bmatrix} 1&0\\i&1 \end{bmatrix}$$
but then what is $\theta_{x^i}(a)$? How is this map even an automorphism of $K$? $K$ isn't necessarily going to be a group of matrices, right?
Thanks!