Let $V,U$ vector subspaces of $\mathbb{C}^{n+1}$ of dimension $r+1$, $s+1$.
Let $\pi : \mathbb{C}^{n+1}/\{0\} \longrightarrow \mathbb{CP}^n$ the projection map to complex projective space defined by $\pi:(x_0, ... ,x_n)\mapsto(x_0:...:x_n)$.
This is a question that came up while doing homework and I wanted to check some reasoning. It is useful to have that $\pi(V)\cap\pi(U) = \pi(V\cap U)$ for a homework question. Is this true? It seemed to be true based on my currently poor intuition about projective space, so I tried to prove it.
My reasoning for a simple proof was to prove both inclusions. Specifically, if $u\in \pi(U\cap V)$ then there is some $x\in U\cap V$ st $u=\pi(x)$. But then $x$ is in both $U$ and $V$ so $u\in \pi(U)\cap\pi(V)$.
Going in the other direction, if $u\in \pi(U)\cap\pi(V)$ then there is $x\in U$ and $y\in V$ such that $\pi(x)=\pi(y) = u$. But if this is true then $x$ is a scalar multiple of $y$ and vis versa (by the properties of vector subspaces), so $x\in U\cap V$. Thus $u\in\pi(U\cap V)$ and we have proved the sets are equivalent.