Let $M$ be the collection of all affine lines on the plane $\mathbb{R}^2$. Introduce an atlas of two charts on $M$. The chart $U_1$ consists of all non-vertical lines, and the line $L : y = a_1x +b_1$ has coordinates $(a_1, b_1)$. The chart $U_2$ consists of all non-horizontal lines and the line $L : x = a_2y + b_2$ has coordinates $(a_2, b_2)$. Find the transition functions and show that $M$ is connected and non-orientable.
I understand what a transition function is, but I'm not sure how to find them. Any help would be much appreciated.
Let $L:U_1\rightarrow R^2$ defined by $L(y=a_1x+b_1)=(a_1,b_1)$.
$M:U_2\rightarrow R^2$ defined by $M(x=a_2y+b_2) =(a_2,b_2)$. You have to compute $L\circ M^{-1}$ and $M\circ L^{-1}$.
You have $M^{-1}(a,b)=(x=ay+b)$. You can write $y=x/a-b/a$, So $L\circ M^{-1}(a,b)=L(x=ax+b)=L(y=x/a-b/a) = (1/a,-b/a)$.
With the same method, compute $M\circ L^{-1}(a,b)$.
To show that it is not orientable, compute the determinant of the Jacobian of the transition functions and show that its sign is negative for $a<0$ and positive for $a>0$.