WLOG doubt: why can we assume that two disjoint linear subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$ are given by the following equations...

Question

Let $H_1,H_2$ be two linear disjoint subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$.

Let $(x_0:\cdots:x_n:y_0:\cdots:y_n)$ be homogeneous coordinates in $\mathbb{P}^{2n+1}$.

My question is:

Why can we assume that $H_1$ is given by the equations $y_0=\cdots=y_n=0$ and $H_2$ is given by the equations $x_0=\cdots=x_n=0$?

I need to prove this in order to prove a problem of algebraic geometry related to dimension of algebraic/projective varieties. This problem may be very easy, because it seems fair by observing that $H_1$ and $H_2$ are disjoint, bu formally I need some help to prove this step by step.

Thanks!

Answer 1

This is because you are free to fix the coordinate system that suits you most.

Let $V_1$ and $V_2$ the linear subspace in $\Bbb A^{2n+2}$ inverse images of $H_1$ and $H_2$ respectively under the canonical quotient map $$ \Bbb A^{2n+2}\setminus\{0\}\longrightarrow\Bbb P^{2n+1}. $$ Then $\dim(V_1)=\dim(V_2)=n+1$ and choose coordinates with respect to a basis of $\Bbb A^{2n+2}$ that splits as the disjoint union of a basis of $V_1$ and a basis of $V_2$ and you are done.