I have to prove that the set $A$ of affine transformations $T(u)=au+b$ where $a, b \in \mathbb{F}$ and $a \ne 0$ forms a group under function composition and then I need to find a generating set for $A$. I am able to prove that $A$ is group. But, for the generators part, I am thinking that each transformation is obtained by a translation (by $b$) and a dilation by $a$, but but when I use these two transformations I couldn't write $T$ by these two elements. I don't know if I did a mistake but I believe the argument and the basic idea is correct.
I am also thinking to visualize each transformation as a matrix and follow the idea of Mobius transformation, but not sure if this also works.
The group $T$ is isomorphic to the group of matrices $$ G= \left(% \begin{array}{cc} a & b \\ 0 & 1 \\ \end{array}% \right),\ a,b\in F,\ a\neq0. $$ If $F$ is a finite field, then this group is generated by two elements: $$ g_1=\left(% \begin{array}{cc} \alpha & 0 \\ 0 & 1 \\ \end{array}% \right),\ g_2=\left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array}% \right), $$ where $\alpha$ a primitive element of $F$.