Let $S$ be a projective space of dimension $n$, and $H$ a hyperplane in $S$ with dimension $n-1$. Consider two projective subspaces, $P_1$ and $P_2$, not contained in $H$. We question if $P_1$ and $P_2$ are identical given their intersections with $S \setminus H$ are the same, i.e., if $P_1 \cap (S \setminus H) = P_2 \cap (S \setminus H)$ implies $P_1 = P_2$.
I made attempts by representing points in $S$ using homogeneous coordinates to explore the algebraic structure of $P_1$ and $P_2$. Given the dimension theorem for vector spaces, which projective spaces generalize, I tried to say something about the dimensions of $P_1$ and $P_2$ based on their intersections with $S \setminus H$. Specifically, I considered whether the dimensionality of their intersections could force $P_1$ and $P_2$ to have the same dimension and thus be potentially equal. My reasoning then is to devise a proof based on how $P_1$ and $P_2$ relate to lines and planes intersecting $H$, but I don't know how to do this.