Consider the subspace $Y$ of $\Bbb R^2\times \Bbb RP^1$ defined by $$ Y=\{(x_1,x_2,[y_1:y_2])\in\Bbb R^2\times \Bbb RP^1 : x_1y_2=x_2y_1\} $$
This is clearly a well-defined subspace of $\Bbb R^2\times \Bbb RP^1$. Then consider the space $Z:=Y -(\{0\}\times \Bbb RP^1)$. I am asked to show that $Z$ is diffeomorphic to $\Bbb R^2-\{0\}$. I think I should first define a map from $Z$ to $\Bbb R^2-\{0\}$, but I can't think of an appropriate map. Is there any ideas? Thanks in advance.
As you haven't learned about the tautological bundle Andreas comment does not immediately translate to a map for you. There is a well defined map $f:\mathbb{R}^2\backslash \{0\} \to \mathbb{R}P^1$ defined by $f(y_1,y_2)=[y_1:y_2]$. This map is obviously surjective and can be shown to be smooth. Define $F:\mathbb{R}^2\backslash \{0\} \to Z$ by $$F(x_1,x_2)=\left(x_1,x_2,f(x_1,x_2)\right)$$ As Andreas mentioned, elements $(x_1,x_2,[y_1:y_2]) \in Z$ have the property that $(x_1,x_2)$ is proportional to $(y_1,y_2)$. Use this observation to show that $F$ is surjective.