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Let $V$ be a vector space of dim $n$ over a finite field $F$ with $q$ elements.

(a) Find the no.of dim 1 subspaces of $V$

(b) For each $1\leq k \leq n$, find the no.of dim $k$ subspaces of $V$

My Work:

Let $V=<v_1,v_2,...,v_n>$. Since $v_1,v_2,...,v_n$ linearly independent, it has the following $n$ no. of dim 1 subspaces:

$<v_1>,<v_2>,...,<v_n>$. So, I think answer for part (a) is $n$. Am I correct?

Now, how can I do part (b). Is it $nC_k$? Please give me a hint.

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Fix a basis of $V.$ With respect to that basis, every element of $V$ can be uniquely expressed as an element of $F^n.$ Now a one dimensional subspace of $V$ is of the form $\alpha w_1,$ for some $w_1 \in V \setminus \{0\}.$ Let $w_1= (\alpha_1, \cdots ,\alpha_n)$ (w.r.t. the chosen basis). So at least one of the $\alpha_i$ is non-zero. Suppose $\alpha_1 \neq 0.$ After multiplying by $\alpha^{-1},$ we can assume that $w_1 = (1, \alpha_2, \alpha_3, \cdots , \alpha_n).$ Now note that, for every choice of $(\alpha_2, \cdots , \alpha_n) \in F^{n-1},$ we will get different one dimensional vector subspace of $V.$ So for $w_1$, the number of choice is $q^{n-1}.$ Similarly, if we fix $1$ at each of the $n$ co-ordinates, we will get $q^{n-1}$ different one dimensional vector subspaces of $V.$ Surely, there will be some common subspaces. So we need to count them also.

Let $w_{1,2} \in V$ be of the form $(1, 1, \alpha_3, \cdots , \alpha_n).$ We have $q^{n-2}$ independent choices for such a vector. All of them will give different one dimensional subspaces of $V.$ Also, all these vectors were counted in both $w_1$ and $w_2.$ So we need to subtract them. Similarly for $w_{i,j}, i \neq j.$

Now consider $w_{1,2,3} \in V, w_{1,2,3} = (1,1,1,\alpha_4, \cdots , \alpha_n).$ We have $q^{n-3}$ different choices for this type of vectors and each of them will give different one dimensional subspaces of $V.$ But these were subtracted before. So we need to add them. And so on.

This is a bit of messy. But this is an approach to solve this kind of problems. This is only for one dimensional subspaces. For $k$ dimensional subspaces, though the idea is essentially the same, but more caution is required.

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