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Let $V$ be a vector space of dim $n+1$ and $\mathbb{P}V=\mathbb{P}^{n}$ be the projective space of $V$. Let $\Lambda \simeq \mathbb{P}^{k} \subset \mathbb{P}^{n}$ be a $k$-dim linear subspace. Why the set of $(k+1)$-planes containing $\Lambda$ is a projective space $\mathbb{P}^{n-k-1}$ and the space of hyperplanes containing $\Lambda$ is the dual projective space $(\mathbb{P}^{n-k-1})^*$? This is a conclusion in the book algebraic geometry (page 6). Thank you very much.

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$\Lambda$ is the set of lines in a subspace $W$ of $V$ of dimension $k+1$.

The set of $(k+1)$ planes in $\mathbb P^n$ containing $\Lambda$ are in bijective correspondence with the subspaces of $V$ of dimension $k+2$ which contain $W$. It follows by one of the isomorphism theorems that this set of subspaces is in bijective correspondence with the set of subspaces of the quotient space $V/W$ of dimension $1$, that is, with the projective space $\mathbb P(V/W)$. This answers your first question.

Your second question can be dealt with in exactly the same way.

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