Quotient space projection

Solution 1:

$\rho=\pi|_V$ means $\rho$ is the restriction of $\pi$ to $V$.

$\pi$ is onto. So any element $y$ of $\mathbb R^{2}/U$ is $\pi (x)$ for some $x$. Now since $U$ and $V$ are two different lines it follows that $U+V$ is two-dimensional, and hence $U+V=\mathbb R^{2}$. Thus, we can write $x=x_1+x_2$ with $x_1 \in U$ and $x_2 \in V$. Now $y=\pi (x)=\pi(x_1)+\pi (x_2)=0+\pi (x_2)$. But $\pi(x_2)=\rho (x_2)$. This proves that $y =\rho (x_2)$ and we have proved that $\rho$ is surjective.

Injectivity: $x \in V, \rho (x)=0$ impies $\pi (x)=0$ which implies $x \in U$. But $x \in V$ also so $x=0$.