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What is the process for finding all subspaces of a vector space over a finite field? Specifically, I want to find all the proper subspaces of the vector space $F^2$ over $\mathbb{Z}_3$. the $0$ vector space is the obvious subspace but I find that I have to guess a subspace then check all the criteria to determine if it is in fact a subspace. Also, is there a way to determine the number of subspaces that exist, apriori, or at least a maximum that can exist given the conditions?

Thanks in advance

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Your example is a vector space of dimension 2, so the only proper subspaces are those of dimensions 0 and 1. You have already accounted for dimension 0. A vector space of dimension 1 consists of a single nonzero vector and all of its scalar multiples. So: pick a nonzero vector, gather all of its scalar multiples, there's a proper subspace. Then pick a vector not in that subspace, and repeat the exercise. Repeat until you have there are no more nonzero vectors left out, and you have all the proper subspaces.

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The number of subspaces of a given dimension $k$ of an $n$-dimensional vector space over a finite field or order $q$ can be determined through a counting argument, in particular, by counting bases (count the number of sets of $k$ linearly independent vectors in an $n$-space, divide by the number of sets of $k$ linearly independent vectors in a $k$-space). The number of such subspaces is given by the Gaussian binomial coefficient $${n \brack k}_{q} = \prod_{i=1}^{k}\frac{q^{n-k+i}-1}{q^{i}-1}$$

(Note that if you wanted to count all of the subspaces, you would need to sum over $0 \leq k \leq n$)

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