Definition : a linear subspace of $\mathbb{P}^n$ is a closed subspace defined by linear homogeneous equations.
Claim: a linear subspace of $\mathbb{P}^n$ of dimension $d$ is isomorphic to $\mathbb{P}^d$.
I have tried building the isomorphism by brute force and failed. But I think there is a more elegant way of proving this. I'll be happy for some help.
Let's work out the case of a hyperplane explicitly. Label the coordinates on $\mathbb{P}^n$ by $x_0,\ldots, x_n$. Then points in a closed subspace are of the form $[x_0:\cdots :x_n]$ solving an equation of the form: $$ a_0x_0+\cdots+a_nx_n=0.$$ We have a map given by $\pi:k^{n+1}\to \mathbb{P}^n_k$ by quotienting. Let's call our hypersurface $H\subseteq \mathbb{P}^n$. Consider $$\pi^{-1}(H)=\{(x_0,\ldots,x_n)\in k^{n+1}: a_0x_0+\cdots +a_nx_n=0\}.$$ This is a linear subspace of $k^{n+1}$, in particular $\pi^{-1}(H)$ is a codimension $1$ linear subspace, isomorphic as vector spaces to $k^n$. So, when we apply the quotient, we get that $H=\pi(\pi^{-1}(H))\cong \mathbb{P}^{n-1}$. You can do the same exact analysis in the general case.