projective subspaces of a projective space.

Question

Let $E$ be a finite dimensional $\mathbb K$-vector space. Let $P(E)$ the corresponding projective space over $ \mathbb K$. That is, $P(E)$ is the set of all one dimensional vector subspaces of $E$. Is this defintion rich enough to describe subspaces of $P(E)$. I know that each one dimensional subspace of $E$ is called a (projective) point of $P(E)$, but what is a projective line in $P(E)$ and what is a projective plane in $P(E)$ and more generally what is a projective subspace of $P(E)$? If this is not enough should we use the algebraic definition that gives $P(E)$ as the quotient of $E\setminus0$ by the the relation $x\sim y $ if and only if $x=\lambda y$ for some non zero scalar $\lambda \in \mathbb K$? In that case how can one define projective subspaces of $P(E)$?

Answer 1

A projective $k$-dimensional subspace of $P(E)$ is by definition a (possibly empty) subset $K \subseteq P(E)$ of the form $$K=P(S),$$ where $S$ is a $(k+1)$-dimensional subspace of $E$.

Some examples:

1. the empty set $K=\emptyset$ arises by taking $S = 0$ the trivial subspace.
2. A projective point $P$ is a projective $0$-dimensional subspace of the form $S = \langle u \rangle$, with $u\neq 0$.
3. A projective line $\ell$ is a projective $1$-dimensional subspace of the form $S = \langle u_1,u_2 \rangle$, where $u_1$ and $u_2$ are linearly independent.
4. A projective plane $\pi$ is a projective $2$-dimensional subspace of the form $S = \langle u_1,u_2,u_3 \rangle$, where $u_1,u_2,u_3$ are linearly independent.

In general, a $k$-dimensional vector subspace $S$ of $E$ will give you a $(k-1)$-dimensional projective subspace of $P(E)$.

If you want to relate this with the defining equivalence relation $v \sim \lambda v$, take any subspace $S \leq E$, define the corresponding equivalence relation on both spaces $S$ and $E$, and note that the linear inclusion $S \hookrightarrow E$ factors to give an injective map of projective spaces $P(S) \hookrightarrow P(E)$, which is the inclusion of the projective subspace I described above.