# How to show that a group $G$ is isomorphic to a subgroup of $GL_2(\mathbb{R})$

Let $$a\in\mathbb{R}^∗$$ and $$b\in \mathbb{R}$$. Consider the function $$f_{a,b} \in \operatorname{Fun}(\mathbb{R},\mathbb{R})$$ given by $$f_{a,b}(x)= ax + b$$.

a) Show that $$f_{a,b}$$ is a bijection, and find its inverse function.

b) Let $$G$$ be the set of functions $$\{f_{a,b}\mid a \in \mathbb{R}^∗ , b \in \mathbb{R}\}$$. Show that $$G$$ is a group, where the group operation is composition of functions. (Thus $$G$$ is a subgroup of $$\operatorname{Bij}(\mathbb{R}, \mathbb{R})$$.)

c) Show that the group $$G$$ is isomorphic to a subgroup of $$GL_2(\mathbb{R})$$

I managed to solve parts a) and b) but part c) is a bonus question (which we didn't cover yet in the lecture) and I don't know how to solve it. Please help?

• What is your definition of $\operatorname{GL}_2(\Bbb{R})$? And what are your thoughts on the problem? – Servaes Sep 5 '19 at 10:33
• Here's a thought. Can you represent the function composition in terms of a linear map? – Cameron Williams Sep 5 '19 at 10:41
• I don't understand the close votes here. Posting a three part question and stating that you are only struggling with the final part gives a decent amount of context. We have as much knowledge as can be expected of what level of complexity an answer would require (for example, it is clearly not a research or graduate problem!). – user1729 Sep 5 '19 at 11:06

$$\phi:f_{a,b} \mapsto \left(\begin{matrix} a& b\\0& 1 \end{matrix}\right)$$
On one hand, we have $$f_{a,b}\circ f_{c,d}=f_{ac, ad+b}$$.
On the other hand, $$\left(\begin{matrix} a& b\\0& 1 \end{matrix}\right)\left(\begin{matrix} c& d\\0& 1 \end{matrix}\right)=\left(\begin{matrix} ac& ad+b\\0& 1 \end{matrix}\right)$$
Therefore, $$\phi$$ is an (obviously injective) group morphism. Therefore $$G$$ is isomorphic to $$\phi(G)\subset GL_2(\mathbb{R})$$.