$GL_2(\mathbb R)$ acting on $\hat{\mathbb R}=\mathbb R\cup \{\infty\}$.

Question

The question and its hint is given below:

But I could not understand what the question is trying to teach me, could anyone explain this for me please?

Also I could not understand how the hint could be used in the solution, could anyone explain this for me please?

Thank you!

Answer 1

But I could not understand what the question is trying to teach me

The problem is testing your understanding of the definition of a group acting on a set by having you work on a specific example: the group $GL_2(\mathbb R)$ acting on the set $\hat{\mathbb{R}}$. If you can see what the connection is to the suggested transformation is and the group, then all you're doing is verifying group action axioms.

Also I could not understand how the hint could be used in the solution

The hint is in case you fail to see the connection between the proposed action and $GL_2(\mathbb R)$. (Yes, there are four numbers $a,b,c,d$ in the original proposition, but what do they have to do with matrices? Well, here's a hint...)

It explicitly outlines a correspondence between elements in $\hat{\mathbb R}$ and one dimensional subspaces of $\mathbb R^2$ plus the vertical line through $(0,0)$, and notes that the proposed action is simply matrix multiplication plus the extra rule that assigns meaning to the edge cases when $\infty$ appears for $x$ and $0$ appears in the denominator.