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My question perhaps is easy but I just cannot answer it myself. Let $V$ be an $n$-dimensional vector space over a field of q elements. Let $\alpha,β,γ$ be subspaces of $V$ of dimensional 2 such that $α∩β=\{0\}$ and $α∩γ$ is $1$-dimensional.

The questions are: How many subspaces $δ$ of $V$ of dimension $2$ are there such that $α∩δ=β∩δ=\{0\}$? How many subspaces $δ$ of $V$ of dimension 2 are there such that $α∩δ=γ∩δ=\{0\}$?

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Let $\alpha\cap \gamma=Span(v)$. Then $\alpha=Span(u,v)$, $\gamma=Span(w,v)$, and $\beta=Span(r,s)$, for some vectors $r,s,u,v,w$. Since $\beta\cap \alpha=\{0\}$, we know that $\{u,v,r,s\}$ is an independent set. We also know that $\{u,v,w\}$ is an independent set, but it's possible that $\{r,s,w\}$ is dependent.

For the first question, you want to avoid $Span(u,v,r,s)$. You can find a basis of size $n-4$ for $Span(u,v,r,s)^\perp$. There are $q^{n-4}$ elements in this subspace; presumably you have a formula for how many subspaces it has of dimension 2.

For the second question, you similarly want to avoid $Span(u,v,w)$.

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  • $\begingroup$ Thank you for your answer but I think you misunderstood my question. $\endgroup$ – Hung Oct 14 '13 at 1:56

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